\section{Introduction}

It is now well established that the internal structure of colliding
nuclei strongly influences heavy-ion collisions at energies around the
Coulomb barrier. In particular, the coupling to the collective
excitations (rotation and vibrational states) in the target and projectile
nuclei
participating in the reaction significantly enhances the fusion
cross sections for intermediate mass systems \cite{bah-tak-98,das-98}.
Such couplings give rise to a distribution of
the Coulomb barrier \cite{bah-tak-98,das-98,daso-83},
which can most easily be visualized for
reactions involving a deformed nucleus. In this case, the
nucleus-nucleus potential depends on the orientation angle of the
deformed nucleus with respect to the beam direction.
Since the orientation angle distributes isotropically
at the initial stage of the
reaction, so does the potential barrier.
The concept of barrier distribution can be
extended also to systems with a non-deformed target \cite{daso-83},
where the distribution originates from the
coupling between the relative motion and vibrational
excitations in the colliding nuclei and/or
transfer processes.
Notice that, although this concept is exact only when the
excitation energy is zero, to a good approximation it holds also
for systems with a non-zero excitation energy \cite{HTB97,hag-bah-04}.

In Ref.\cite{row-91}, Rowley {\it et al.} have argued that the barrier
distribution can be directly extracted from a measured fusion cross
section $\sigma_{\rm fus}(E)$, by taking the second derivative of the
product $E\sigma_{\rm fus}(E)$ with respect to the center-of-mass energy
$E$, that is , $D^{\rm fus}=d^2(E\sigma_{\rm fus})/dE^2$.
This method has stimulated
many high precision measurements of fusion excitation function for
medium-heavy mass systems \cite{das-98,leigh-95}. The
extracted barrier distributions have revealed that the concept indeed holds and
the barrier distribution itself provides a powerful tool for investigating
the effects of channel coupling on heavy-ion fusion reactions at
sub-barrier energies.
It has also been shown recently that the concept of
barrier distribution is still valid even for relatively heavy systems, such as
$^{100}$Mo$+^{100}$Mo \cite{row-06}.

A similar barrier distribution can also be extracted from quasi-elastic
scattering (a sum of elastic, inelastic and transfer processes) at
backward angles \cite{thim-95,hag-04}, that
is a good counterpart of the fusion reaction \cite{andres-88}.
In this case, the barrier distribution is
defined as the first derivative of the ratio of quasi-elastic to the
Rutherford cross sections $d\sigma_{\rm qel}/d\sigma_R$, with respect to
$E$, ${\it i.e.,}$
$D^{\rm qel}=-d(d\sigma_{\rm qel}/d\sigma_R)$/$dE$.
Since the
fusion and the quasi-elastic scattering is related to each other
because of the flux conservation, a similar information can be obtained
from those processes and the similarity between the two
representations for barrier distribution has been shown to hold for several
intermediate mass systems \cite{thim-95,hag-04,zam-07}.

Recently, the quasi-elastic barrier distribution has been exploited
to investigate the entrance channel dynamics for fusion reactions to
synthesize super-heavy elements \cite{row-061,ntsha-07,ike-06,mitsu-07}.
It has been shown that the concept of barrier distribution remains
valid even for such very heavy systems once the deep-inelastic
cross sections are properly taken into account.
As is expected, the strong channel coupling effects on the barrier
distribution have been observed.

In this paper, we carry out a detailed coupled-channels analysis for
large-angle quasi-elastic scattering data for
$^{48}$Ti,$^{54}$Cr,$^{56}$Fe,$^{64}$Ni and $^{70}$Zn$+^{208}$Pb systems
leading to super-heavy elements
$Z=104,\,106,\,108,\,110,\,\rm{and}\,112$, respectively
\cite{ike-06,mitsu-07}.
We especially study the role of
multi-phonon excitations of the target and projectile nuclei,
which has been shown to play an important role in
quasi-elastic scattering for the $^{86}$Kr+$^{208}$Pb system
\cite{ntsha-07}.

The paper is organized as follows. We briefly explain the coupled-channels
formalism for quasi-elastic scattering in Sec. II. We present the
results of our
systematic analysis in Sec. III.
We then summarize the paper in Sec. IV.

\section{Coupled-channels Formalism for Large Angle Quasi-elastic Scattering}

In this section, we briefly describe the coupled-channels formalism
for large angle quasi-elastic scattering which includes the effects of the
vibrational excitations of the colliding nuclei.
The total Hamiltonian of the system is assumed to be
\begin{eqnarray}
H&=&-\frac{\hbar^2}{2\mu}\nabla^2+V^{(0)}_N(r)+\frac{Z_PZ_Te^2}{r} \nonumber \\
&&+H_{\rm exct}
+V_{\rm coup}(\vec{r},\xi_P,\xi_T),
\label{eq-multi-1}
\end{eqnarray}
where $\vec{r}$ is the coordinate of the relative motion between the
target and the projectile nuclei, $\mu$ is the reduced mass and
$\xi_T$ and $\xi_P$ represent the coordinate of the vibration
in the target and the projectile nuclei, respectively.
$Z_P$ and $Z_T$ are the atomic number of
the projectile and the target, respectively, and
$V^{(0)}_N$ is the bare
nuclear potential, which we assume to have a Woods-Saxon shape. It
consists of the real and imaginary parts,
$V^{(0)}_N(r)=V_0(r)+iW_0(r)$.
$H_{\rm exct}$ describes the
excitation spectra of the target and projectile nuclei, while
$V_{\rm coup}(\vec{r},\xi_P,\xi_T)$ is the potential for the
coupling between the
relative motion and the
vibrational motions of the target and projectile nuclei.

In the iso-centrifugal approximation
\cite{bah-tak-98,hag-99,es-lan-pri-87}, where
the the angular momentum of the relative motion in each
channel is replaced with the total angular momentum $J$
(in the literature, this approximation is also
referred to as the rotating frame approximation or the no-Coriolis
approximation), the coupled-channels
equations derived from the Hamiltonian (\ref{eq-multi-1}) read
\begin{eqnarray}
&&\bigg[-\frac{\hbar^2}{2\mu}\frac{d^2}{dr^2}+\frac{J(J+1)\hbar^2}{2\mu
  r^2}+V^{(0)}_N(r)+\frac{Z_PZ_Te^2}{r}\nonumber\\
&&-E+\epsilon_n\bigg]u_n(r)+\sum_{n'} V_{nn'}(r)u_{n'}(r)=0
\label{eq-multi-11}
\end{eqnarray}
where $\epsilon_n$ is the eigen-value of the operator $H_{\rm exct}$
for the $n$-th channel. $V_{nn'}(r)$ is
the matrix elements for the coupling potential $V_{\rm coup}$.

In the calculations presented below, we use the method
of the computer code {\tt CCFULL} \cite{hag-99} and replace the
vibrational coordinates $\xi_P$ and $\xi_T$ in the coupling potential
$V_{\rm coup}$ with the dynamical excitation operators
$\hat{O}_{P}$ and $\hat{O}_{T}$. The coupling potential is then
represented as
\begin{equation}
V_{\rm coup}(r,\hat{O}_P,\hat{O}_T)
=V_C(r,\hat{O}_P,\hat{O}_T)+V_N(r,\hat{O}_P,\hat{O}_T),
\label{eq-multi-2}
\end{equation}
\begin{eqnarray}
V_C(r,\hat{O}_P,\hat{O}_T)&=&\left(
\frac{3R_P^{\lambda_P}\hat{O}_{P}}{(2\lambda_P+1)r^{\lambda_P}}
+\frac{3R_T^{\lambda_T}\hat{O}_T}{(2\lambda_T+1)r^{\lambda_T}}\right)
\nonumber\\
&&\times \frac{Z_PZ_Te^2}{r},\label{eq-multi-3}\\
V_{N}(r,\hat{O}_P,\hat{O}_T)&=&\frac{-V_0}{\left[1+\textrm{exp}\left(\frac{
[r-R_0-(R_P\hat{O}_P+R_T\hat{O}_T)]}
{a}\right)\right]} \nonumber \\
&&-V_N^{(0)}(r).
\label{eq-multi-4}
\end{eqnarray}
Here, $\lambda_P$ and $\lambda_T$ denote the multipolarity of the vibrations
in the projectile and the target nuclei, respectively.
We have subtracted $V_N^{(0)}(r)$ in Eq.~(\ref{eq-multi-4}) in order
to avoid the double counting.

If we truncate the phonon space
up to the triple phonon states (that is, $n$=0,1,2, and
3), the matrix elements of
the excitation operator $\hat{O}$ in Eqs.~(\ref{eq-multi-3})
and (\ref{eq-multi-4}) are given by
\begin{eqnarray}
O_{nn'}=\frac{1}{\sqrt{4\pi}}
\left[\begin{array}{cccc}
0&\beta&0&0\\
\beta&0&\sqrt{2}\beta&0\\
0&\sqrt{2}\beta&0&\sqrt{3}\beta \\
0&0&\sqrt{3}\beta&0
\end{array}\right]
\label{eq-multi-5}
\end{eqnarray}
where $\beta$ is the deformation parameter, that can be
estimated from a measured electric transition probability from the
single phonon state ($n$=1) to the ground state ($n$=0).
We have assumed the harmonic oscillator model for the vibrations,
where $\epsilon_n$ in Eq.~(\ref{eq-multi-11}) is given by
$\epsilon_n = n\,\hbar\omega$.

The coupled-channels equations, Eq.~(\ref{eq-multi-11}), are solved with
the scattering boundary condition for $u_n(r)$,
\begin{eqnarray}
u_n(r)&\rightarrow&
\frac{i}{2}\left\{H_J^{(-)}(k_nr)\delta_{n,n_i}-\sqrt{\frac{k_i}{k_n}}
S_n^JH_J^{(+)}(k_nr)\right\}, \nonumber \\
&&~~~~~~~(r\to\infty)
\label{eq-scattwave}
\end{eqnarray}
where $S_n^J$ is the nuclear $S$ matrix. $H_J^{(-)}(kr)$ and
$H_J^{(-)}(kr)$ are the incoming and the outgoing Coulomb wave
functions, respectively. The channel wave number $k_n$ is given by
$\sqrt{2\mu \,(E-\epsilon_n)/\hbar^2}$, and $k_i=k_{n_i}=\sqrt{2\mu
  E/\hbar^2}$.
The scattering angular distribution for the channel $n$
is then given by \cite{es-lan-pri-87}
\begin{equation}
\frac{d\sigma_n}{d\Omega}=\frac{k_n}{k_i}\lvert f_n(\theta)\rvert^2
\end{equation}
with
\begin{eqnarray}
f_n(\theta)=\sum_Je^{[\sigma_J(E)+\sigma_J(E-\epsilon_n)]}
\sqrt{\frac{2J+1}{4\pi}}Y_{J0}(\theta)\nonumber\\
\times \frac{-2i\pi}{\sqrt{k_ik_n}}(S_n^J-\delta_{n,n_i})
  +f_C(\theta)\delta_{n,n_i}
\end{eqnarray}
where $\sigma_J(E)$ and $f_C(\theta)$ are the the Coulomb phase shift
and the Coulomb scattering amplitude, respectively. The differential
quasi-elastic cross section is then calculated to be
\begin{equation}
\frac{d\sigma^{\rm qel}}{d\Omega}=\sum_n\frac{d\sigma_n}{d\Omega}
\end{equation}
We will apply this
formalism in the next section to analyze the
quasi-elastic scattering data of
$^{48}$Ti,$^{54}$Cr,$^{56}$Fe,$^{64}$Ni, and $^{70}$Zn$+^{208}$Pb
systems.

\section{Comparison with experimental data : effects of
multi-phonon excitations}

In this section, we present the results of our detailed coupled-channels
analysis for quasi-elastic scattering data of
$^{48}$Ti,$^{54}$Cr,$^{56}$Fe,$^{64}$Ni, and $^{70}$Zn$+^{208}$Pb systems
\cite{ike-06,mitsu-07}.
The  calculations are performed with a version \cite{hag2} of the
coupled-channels code {\tt CCFULL} \cite{hag-99}.
Notice that the iso-centrifugal approximation employed in this code
works well for
quasi-elastic scattering at backward angles \cite{hag-04}. In the code,
the regular boundary condition is imposed at the origin instead of the
incoming wave boundary condition.
We discuss the stability of the numerical calculations
for the massive systems in Appendix A.

The surface diffuseness of the real part of the nuclear
potential is taken to be $a=0.63$ fm, as suggested by recent studies
on deep sub-barrier quasi-elastic and Mott scattering
\cite{washi-06,gasques-07,hinde-07}, while the radius parameter to be
$r_0=1.22$ fm for all the systems.
Notice that a similar value for $a$ has been used also in the
analysis of the recent experimental data for quasi-elastic
scattering in the $^{86}$Kr + $^{208}$Pb system \cite{ntsha-07}.
The depth parameter, $V_0$, is adjusted
in order to reproduce the experimental quasi-elastic cross sections for each
system.
The optimum values of the depth parameter and the
resultant Coulomb barrier height are summarized in Table~\ref{pot-par}. As
usually done, we use a short range imaginary potential with
$W_{0}=30$ MeV, $r_{w}=1.0$ fm and $a_w=0.3$ fm to simulate the
compound nucleus formation.
The results are insensitive to these parameters
as long as the imaginary part of the potential is well
confined inside the Coulomb barrier.
The excitation
energy and the corresponding deformation parameter for the single
phonon excitation in each nucleus
included in the calculations are given in Table~\ref{ene-def}.
The latter quantity is taken from Refs. \cite{spear-02,raman-01}.
The radius of the target and
the projectile are taken to be $R_T=1.2A_T^{1/3}$ and  $R_P=1.2A_P^{1/3}$,
respectively, in order to be consistent with the deformation
parameters \cite{spear-02,raman-01}.
All the calculations shown below
are performed at the scattering angle of
$\theta_{\rm c.m.}=170^\circ$.
We plot the quasi-elastic cross sections and barrier distributions
as a function of
the effective energy defined by \cite{thim-95,hag-04}
\begin{equation}
E_{\rm eff}=2E\frac{\rm sin(\theta/2)}{1+\rm sin(\theta/2)},
\end{equation}
which takes into account the centrifugal energy.
We calculate the quasi-elastic barrier
distributions from the cross sections
in a similar way as the one used to obtain
the experimental barrier distributions \cite{mitsu-07}.
Namely, we use
the point difference formula with the energy step of $\Delta E$=0.25
MeV and then smooth the resultant barrier distribution with the
Gaussian function with the full width at half maximum (FWHM) of 1.5
MeV. We have checked that the shape of the barrier distribution does
not change significantly even if we use a larger energy step for the
point difference formula, e.g. $\Delta E$=0.5 MeV.

\begin{table}
\caption{\label{pot-par}The
depth parameter for the real part of the nuclear potential
for the $^{48}$Ti,$^{54}$Cr,$^{56}$Fe,$^{64}$Ni, and $^{70}$Zn$+^{208}$Pb
systems. The radius and the diffuseness parameters are taken to be
$r_0$=1.22 fm and $a$=0.63 fm, respectively, for all the systems.
The resultant barrier height energy $V_B$ is also listed.}
\begin{ruledtabular}
\begin{tabular}{ccc}
System&$V_0$(MeV)&$V_B$(MeV)\\
\hline
$^{48}$Ti+$^{208}$Pb & 88.90 & 190.50\\
$^{54}$Cr+$^{208}$Pb & 91.70 & 205.50\\
$^{56}$Fe+$^{208}$Pb & 92.85 & 222.50\\
$^{64}$Ni+$^{208}$Pb & 95.10 & 236.25\\
$^{70}$Zn+$^{208}$Pb & 108.2 & 249.30\\
\end{tabular}
\end{ruledtabular}
\end{table}
\begin{table}
\caption{\label{ene-def}The properties of the single phonon states
included in the present coupled-channels calculations.
$\hbar\omega$ and $\beta$ are the excitation energy and the dynamical
deformation parameter, respectively.
}
\begin{ruledtabular}
\begin{tabular}{cccc}
Nucleus&$I^\pi$&$\hbar\omega$ (MeV)&
$\beta$\\
\hline
$^{208}$Pb&$3^-$ & $2.614$ & $0.110$ \footnotemark[1]\\
$^{48}$Ti &$2^+$ & $0.983$ & $0.269$ \footnotemark[2]\\
$^{54}$Cr &$2^+$ & $0.834$ & $0.250$ \footnotemark[2]\\
$^{56}$Fe &$2^+$ & $0.846$ & $0.239$ \footnotemark[2]\\
$^{64}$Ni &$2^+$ & $1.346$ & $0.179$ \footnotemark[2]\\
$^{70}$Zn &$2^+$ & $0.884$ & $0.228$ \footnotemark[2]\\
\end{tabular}
\end{ruledtabular}
\footnotetext[1]{taken from Ref.~\cite{spear-02}.}
\footnotetext[2]{taken from Ref.~\cite{raman-01}.}
\end{table}

\begin{figure}
\includegraphics[width=.45\textwidth]{48Ti208Pb-1.ps}
\caption{Effects of multi-phonon excitations on the quasi-elastic scattering
cross section (upper panel) and on the quasi-elastic barrier distribution
(lower panel) for the $^{48}$Ti+$^{208}$Pb system.
The dotted line is the result of the coupled-channels calculations
including coupling to the one quadrupole phonon state in the
projectile
and the one octupole phonon states in the target nucleus, while the
solid line is obtained by including the coupling in addition to
the two octupole phonon state in the
target nucleus.
The dashed line is the result of double quadrupole phonon couplings in
the projectile and the double octupole phonon couplings in the target
nucleus.
The experimental data are taken from
Ref. \cite{mitsu-07}.}
\label{fig-tipb-1}
\end{figure}

\subsection{Effect of double phonon excitations}

\begin{figure*}
\includegraphics[width=.79\textwidth]{crfeniznpb-1.ps}
\caption{
The quasi-elastic scattering
cross sections ((a), (c), (e), and (g)) and the quasi-elastic barrier
distributions ((b), (d), (f), and (h)) for the
$^{54}$Cr,$^{56}$Fe,$^{64}$Ni and $^{70}$Zn$+^{208}$Pb systems
obtained with two coupling schemes as indicated in the insets.
The dashed line is obtained by including the one quadrupole phonon
state in the projectile nuclei while the solid line is obtained with
the double phonon couplings. The double octupole phonon excitations in
the target nucleus is included in all the calculations.
The experimental data are taken from Ref. \cite{mitsu-07}.}
\label{fig-crfeniznpb-1}
\end{figure*}

Let us first discuss the effect of double octupole phonon excitations
in the $^{208}$Pb target.
Such excitations
have been shown to play a significant role
in the sub-barrier fusion reaction between $^{16}$O and $^{208}$Pb
nuclei \cite{das-97,morton-99}.

\begin{figure}
\includegraphics[width=.45\textwidth]{48Ti208Pb-2.ps}
\caption{Effects of triple phonon excitations on the quasi-elastic scattering
cross section (upper panel) and on the quasi-elastic barrier
distribution (lower panel) for the $^{48}$Ti+$^{208}$Pb system.
The dashed line is the result of the coupled-channels calculations
taking into account the coupling to the one phonon state in the projectile and
the two phonon states in the target nuclei. The solid line
is obtained by including the coupling to the one phonon state in the
projectile and the three phonon states in
the target. The experimental
data are taken from Ref. \cite{mitsu-07}.}
\label{fig-tipb-2}
\end{figure}

The dotted line in Fig.~\ref{fig-tipb-1} shows the
result of the coupled-channels calculation
for the $^{48}$Ti + $^{208}$Pb system
obtained by taking into
account the
coupling to the single octupole phonon state
in the target nucleus, $^{208}$Pb, and the single quadrupole phonon
state in the projectile nucleus, $^{48}$Ti.
The mutual excitations in the projectile and the target nuclei
are fully taken into account in this calculation as well as in all the
other calculations presented in this paper.
Figs.~\ref{fig-tipb-1}(a) and
\ref{fig-tipb-1}(b) show the ratio of the quasi-elastic to the
Rutherford cross sections, $d\sigma_{\rm qel}/d\sigma_R$, and the
quasi-elastic barrier distribution, $D^{\rm qel}$, respectively.
Although the overall width of the barrier distribution is reproduced
reasonably well with this calculation, the detailed structure is
somewhat inconsistent with the experimental data.
The situation is similar even when we include the double quadrupole
phonon state in the projectile while keeping the single octupole
phonon coupling in the target nucleus (not shown).
We then investigate the effect of the double octupole phonon couplings
in the target nucleus. The solid and the dashed lines in
Fig.~\ref{fig-tipb-1}  show the results with
the single and the double quadrupole phonon
excitations in the projectile, respectively, where as the double
octupole phonon coupling in the target is included in both the
calculations. The former calculation reproduces both the cross
sections and the barrier distribution reasonably well, although the
latter calculation somehow worsens the agreement.
This clearly suggests that the double
octupole phonon excitations in the target nucleus is
important in the quasi-elastic $^{48}$Ti+$^{208}$Pb scattering.
We summarize the $\chi^2$ value of our calculations in Table~\ref{chi-1}.

Since the coupling to the one quadrupole phonon state in the projectile
and the two
octupole phonon states in the target reasonably well reproduce the
experimental quasi-elastic scattering data for the $^{48}$Ti+$^{208}$Pb system,
one may expect that the same coupling scheme accounts for the
experimental
data for the other systems,
$^{54}$Cr,$^{56}$Fe,$^{64}$Ni and $^{70}$Zn$+^{208}$Pb.
The results of the coupled-channels calculations with this
coupling scheme is shown by the dashed line in
Fig.~\ref{fig-crfeniznpb-1}.
Figs.~\ref{fig-crfeniznpb-1}(a), (c), (e), and (g)
are for
the quasi-elastic cross sections
for the $^{54}$Cr,$^{56}$Fe,$^{64}$Ni and $^{70}$Zn$+^{208}$Pb
systems, respectively,
while Figs.~\ref{fig-crfeniznpb-1}(b), (d), (f), and (h) are
for the quasi-elastic barrier distributions.
One can clearly see that these
calculations underestimate the experimental cross sections
at high energies, although the experimental barrier distributions
themselves are reproduced reasonably well.
We repeat the same
calculations by including the coupling up to the double quadrupole phonon
states in the projectile, in addition to the double octupole phonon states
in the target nucleus. These results are shown by the solid in
Fig.~\ref{fig-crfeniznpb-1}.
The agreement with the experimental data is considerably improved,
especially for the quasi-elastic cross sections. See Table~\ref{chi-2} for the
$\chi^2$ values.
It is thus evident that
the coupling to the double
quadrupole phonon states in the projectile
is needed in
order to explain the experimental data for the
 $^{54}$Cr,$^{56}$Fe,$^{64}$Ni and $^{70}$Zn$+^{208}$Pb reactions.

The reason why the double quadrupole phonon coupling is not necessary
for the $^{48}$Ti projectile while it is for the heavier projectiles
is not clear at this moment.
This might reflect some ambiguity of the Monte Carlo
reaction simulation code {\tt LINDA} \cite{LINDA} which was
used to subtract the
deep-inelastic component from the experimental yields at backward
angles \cite{mitsu-07}.
Clearly, a further investigation is still necessary concerning the
effect of deep inelastic scattering on quasi-elastic scattering in
massive systems \cite{row-061,ntsha-07,ike-06,mitsu-07}.

\begin{figure*}
\includegraphics[width=.79\textwidth]{crfeniznpb-2.ps}
\caption{Effects of triple phonon excitations on the quasi-elastic
cross sections ((a), (c), (e), and (g)) and on the quasi-elastic barrier
distributions ((b), (d), (f), and (h)) for the
$^{54}$Cr,$^{56}$Fe,$^{64}$Ni and $^{70}$Zn$+^{208}$Pb systems.
The dashed line is the same as the solid line in Fig.~\ref{fig-crfeniznpb-1}
while the dotted line is the results of the
calculations taking the coupling to the triple octupole phonon in the target
and the one quadrupole phonon state in the projectile nucleus into
account. The solid line is obtained by including the coupling to the double
quadrupole phonon states
in the projectile and the triple otcupole phonon states in the target
nucleus. The experimental data are taken from Ref. \cite{mitsu-07}.}
\label{fig-crfeniznpb-2}
\end{figure*}

\begin{table}
\caption{\label{chi-1}
The value of $\chi^2$ for the quasi-elastic cross sections for the
$^{48}$Ti$+^{208}$Pb system obtained with the coupled-channels
calculations with various coupling schemes. The coupling schemes are
denoted as $[n_2,n_3]$, where $n_2$ is the number of quadrupole phonon
excitation in the projectile nucleus while $n_3$ the number of
octupole phonon in the target nucleus.
}
\begin{ruledtabular}
\begin{tabular}{ccccc}
System&[1,1]&[1,2]&[1,3]&[2,2]\\
\hline
$^{48}$Ti+$^{208}$Pb & 19.15 & 9.82 & 7.12 & 37.12\\
\end{tabular}
\end{ruledtabular}
\end{table}

\begin{table}
\caption{\label{chi-2}
Same as Table \ref{chi-1}, but for the $^{54}$Cr,$^{56}$Fe,$^{64}$Ni
and $^{70}$Zn$+^{208}$Pb systems.}
\begin{ruledtabular}
\begin{tabular}{ccccc}
System&[1,2]&[1,3]&[2,2]&[2,3]\\
\hline
$^{54}$Cr+$^{208}$Pb & 52.47 & 49.80 & 20.61 & 11.78\\
$^{56}$Fe+$^{208}$Pb & 28.46 & 28.36 & 10.44 & 10.28\\
$^{64}$Ni+$^{208}$Pb & 57.45 & 61.43 & 32.21 & 30.64\\
$^{70}$Zn+$^{208}$Pb & 26.52 & 24.81 & 11.36 & 6.87\\
\end{tabular}
\end{ruledtabular}
\end{table}

\subsection{Effect of triple phonon excitations}

In the previous subsection, we have shown that the double octupole phonon
excitations in the $^{208}$Pb target play an important role
in quasi-elastic scattering for the systems considered in this paper.
However, the calculated quasi-elastic barrier distributions have a much more
prominent peak than the experimental distribution at high energies.
Since it has been shown in Refs. \cite{row-061,ntsha-07} that
the triple octupole phonon excitations of the $^{208}$Pb
play a significant role in
the large-angle quasi-elastic
scattering between $^{86}$Kr and $^{208}$Pb nuclei, it is
intriguing to investigate such effects in the present systems as
well.

The results of the coupled-channels calculations including the
coupling to the triple octupole phonon states in $^{208}$Pb
for the $^{48}$Ti+$^{208}$Pb reaction is presented in Fig.~\ref{fig-tipb-2}.
The dashed line is the same as the solid line in
Fig.~\ref{fig-tipb-1}, that is the result of single phonon in
$^{48}$Ti and double phonon in $^{208}$Pb.
The solid line denotes the results of the triple phonon
coupling in the target in addition to the single phonon
in the projectile.
By including the triple octupole phonons in the target nucleus,
the quasi-elastic cross sections are improved slightly (see also
Table~\ref{chi-1}).
On the other hand, one can see that the agreement
for the barrier distribution with the experimental data
is much more improved by the triple phonon coupling.

\begin{figure}
\includegraphics[width=.45\textwidth]{70Zn208Pb-reduced.ps}
\caption{Effect of anharmonic octupole phonon excitations in
  $^{208}$Pb on
(a) the quasi-elastic cross section and
(b) the quasi-elastic barrier distribution
for the $^{70}$Zn$+^{208}$Pb reaction.
The solid line is the coupled-channels calculation
obtained by reducing the coupling strengths to multi-phonon states,
while the dashed line denotes the results in the
  harmonic limit. The experimental data are taken from Ref. \cite{mitsu-07}.}
\label{fig-ahv}
\end{figure}

The results for the other systems,
$^{54}$Cr,$^{56}$Fe,$^{64}$Ni and $^{70}$Zn$+^{208}$Pb reactions,
are shown in Fig.~\ref{fig-crfeniznpb-2}. Figs. \ref{fig-crfeniznpb-2}(a),
(c), (e) and (g) are
for the quasi-elastic cross sections, while Figs.~\ref{fig-crfeniznpb-2}(b),
(d), (f) and (h) for the quasi-elastic barrier distributions.
Let us first discuss the calculations with the single phonon
excitation in the projectile. The  dotted line in the figures
is obtained by taking
the coupling to the single phonon state in the projectile and
the triple octupole phonon excitations in the target.
This calculation
underestimates the quasi-elastic cross sections at high
energies and the obtained barrier distribution is inconsistent with the
experimental data. Therefore, the previous results shown in Fig.
~\ref{fig-crfeniznpb-1} is not improved even if the triple phonon
excitations in the target is taken into account as long as only the
single phonon excitation is considered for the projectile nucleus.
The results with
the double phonon couplings in the projectile
together with the triple phonon excitations in the target
are then shown by the solid
line in the figure.
For comparison, we also show by the dashed line
the results of the double phonon excitations in both the projectile
and the target nuclei, which is the same as the solid line in
Fig. ~\ref{fig-crfeniznpb-1}.
One can observe that the inclusion of the triple octupole phonon
excitations in the $^{208}$Pb
somewhat improves the agreement
between the calculations and the experimental data for both
the quasi-elastic cross sections and the barrier distributions (see
also Table~\ref{chi-2}).

In Ref.\cite{ntsha-07}, Ntshangase {\it et al.}
reduced the coupling strength of $(3^-)\rightarrow(3^-)^2$
states in $^{208}$Pb by a factor of $(0.6)$ and
that of $(3^-)^2\rightarrow(3^-)^3$ by $(0.6)^2$
in order to explain the
experimental barrier distribution for the $^{86}$Kr$+^{208}$Pb reaction.
In order to
see whether such reduction of the coupling strengths improves
the agreement between the coupled-channels calculations and the
experimental data for the present systems,
we repeat the calculations
by including those effects for the $^{70}$Zn$+^{208}$Pb system.
The results are shown in Fig.~\ref{fig-ahv}.
The solid is
obtained by reducing the coupling strengths as
Ntshangase {\it et al.} did,
while the dashed line is the same as the solid line in
Figs.~\ref{fig-crfeniznpb-2}(g) and (h), that is
obtained by assuming the harmonic limit.
In both cases, we take into
account the coupling to the double quadrupole phonon excitations in the
projectile nucleus.
One can see that the harmonic model
leads to a better agreement with the experimental data
both for the cross sections and the barrier distribution, as compared
to the anharmonic calculation.
The difference between Ref. \cite{ntsha-07} and the present
calculation concerning the role of anharmonicity may originate from
the fact that  Ref. \cite{ntsha-07} used a smaller value for
$R_T$ (=1.06$A_T^{1/3}$ fm) and thus a larger value for $\beta_3$
(=0.16).  In order to clarify the role of anharmonicity of
multi-phonon excitations in quasi-elastic scattering in massive
systems,
it would be required to take into account also the
reorientation terms \cite{hag-97,hag-971,zam-07}. It is beyond the
scope of this paper, and we will leave it for a future study.

\begin{figure}
\includegraphics[width=.45\textwidth]{difuseness.ps}
\caption{Comparison of the experimental data with the coupled-channels
calculations obtained using different values of the surface diffuseness of
the nuclear potential for $^{64}$Ni$+^{208}$Pb reaction for
(a) the quasi-elastic cross section and
(b) the quasi-elastic barrier distribution.
The solid and the dashed lines is obtained using the surface
diffuseness of the nuclear potential $a=0.63$ fm and $a=1.0$ fm,
respectively. The dotted line is the results obtained by shifting the barrier
height by around $+\, 2.40$ MeV for the calculation using $a=1.0$ fm.
Experimental data is taken from Ref. \cite{mitsu-07}.}
\label{fig-diff}
\end{figure}

\subsection{Surface diffuseness of the nuclear potential}

We next discuss the dependence of the quasi-elastic scattering
on the surface diffuseness parameter of the nuclear potential.
The standard value for the diffuseness parameter is around 0.63 fm
 \cite{brogwin-91,criswin-76,satc-79}.
Recently, systematic studies on
quasi-elastic scattering as well as Mott scattering at deep sub-barrier
energies have revealed that the surface region of the nuclear
potential is indeed consistent with the standard value of the
surface diffuseness parameter ~\cite{washi-06,gasques-07,hinde-07}.
On the other hand,
it has been known for some time
that the recent high precision data of sub-barrier fusion cross
sections require a larger value of surface diffuseness
parameter, ranging
between 0.75 and 1.5 fm \cite{NBD04}.
Since the large-angle quasi-elastic scattering around the Coulomb
barrier may probe both the surface region and the inner part of the
nuclear potential, it is interesting to study the sensitivity of
quasi-elastic cross sections and barrier distributions
to the surface diffuseness parameter.

For this purpose, as an example, we repeat
the coupled-channels calculation for the
the $^{64}$Ni$+^{208}$Pb reaction using the nuclear potential with
$a$=1.0 fm.
We readjust the depth and the radius
parameters to be $V_0=160.70$ MeV and $r_0=1.10$ fm, respectively,
so that the barrier height remains the same as the one listed in Table
I. We include
the coupling to the double phonon states in the projectile and
the triple phonon states in the target.
Fig.~\ref{fig-diff} compares the results with $a$=0.63 fm (the solid
line) to the one with $a$=1.0 fm (the dashed line).
One sees that the calculations with $a$=1.0 fm
underestimate the quasi-elastic cross section, although
the shape of barrier distribution itself is similar to the one obtained
with $a=0.63$ fm.
The dotted line is obtained with the same value of surface
diffuseness parameter $a$=1.0 fm as the one for the dashed line,
but by changing
the depth parameter $V_0$ so that the resultant barrier height is
higher by 2.4 MeV.
This calculation now reproduces the experimental quasi-elastic cross
sections at energies larger than $E_{\rm c.m.}=230$ MeV
reasonably well, but below this energy
the cross sections are underestimated.
Therefore, it seems difficult to reproduce the experimental
quasi-elastic cross sections with the diffuseness parameter of
$a$=1.0 fm at energies below and
above the Coulomb barrier simultaneously.
We have checked that the situation is similar for the other systems,
$^{48}$Ti,$^{54}$Cr,$^{56}$Fe and $^{70}$Zn$+^{208}$Pb.
This result clearly indicates that the standard value of surface
diffuseness, $a=0.63$ fm,
is preferred by
the experimental quasi-elastic data
for the systems studied in this paper.

\section{Summary}

We have performed a detailed coupled-channels analysis for
large-angle quasi-elastic scattering of the
$^{48}$Ti,$^{54}$Cr,$^{56}$Fe,$^{64}$Ni and $^{70}$Zn$+^{208}$Pb
systems, where their experimental
barrier distributions have been extracted
recently. Our coupled-channels calculations with multi-phonon
excitations in the colliding nuclei reproduce the experimental
quasi-elastic cross sections as well as the barrier distributions,
indicating clearly that
the coupled-channels approach still works even for massive
systems \cite{row-06}.
It was crucial to subtract properly the
deep-inelastic components from the total backward-angle cross sections
in order to reach these agreements between the calculations and the
experimental data.

In more details,
the calculation with the single
quadrupole phonon excitation in
$^{48}$Ti and the triple
octupole phonon excitations in $^{208}$Pb
reproduces reasonably well the experimental data
for the $^{48}$Ti$+^{208}$Pb system.
On the other hand, for the
$^{54}$Cr,$^{56}$Fe,$^{64}$Ni and $^{70}$Zn$+^{208}$Pb systems,
we found that the
coupling to the double quadrupole phonon excisions in the projectile
nucleus in addition to the coupling to the triple octupole phonon in the
target nucleus seems to be needed to fit the experimental data.
These results suggest that the triple octupole  phonon
excitations in the $^{208}$Pb nucleus plays an
important role in describing the experimental data for the quasi-elastic
cross section and the quasi-elastic barrier distribution for
the present massive systems. This is consistent with the previous
finding for the $^{86}$Kr + $^{208}$Pb system \cite{ntsha-07}.

Although our calculations well reproduce the gross features
of the experimental barrier distributions,
higher precision data are still required in order to study the
detailed structure of the barrier distributions,
especially the role of multi octupole phonon
states in $^{208}$Pb.
From the theoretical side, a further detailed investigation
will also be necessary,
taking into account the anharmonicity of the multi-phonon excitations.

\begin{acknowledgments}
This work was partly supported by The 21st Century Center of
Excellence Program ``Exploring New Science by Bridging
Particle-Matter Hierarchy'' of Tohoku University
and partly by Monbukagakusho Scholarship
and Grant-in-Aid for Scientific Research under
the program number 19740115 from the Japanese Ministry of
Education, Culture, Sports, Science and Technology.
\end{acknowledgments}


\section{Introduction} \label{sec:intro}

\citet{Hills88} first proposed the existence of hypervelocity stars
(HVSs) as evidence for a supermassive black hole in the center of our
Galaxy. According to his calculations, an encounter of a close binary
with a supermassive black hole could disrupt the binary, capturing one
of the stars and ejecting the other as a HVS with a velocity up to 4000
km s$^{-1}$. Observational evidence for such objects was first presented
by \citet{Brown05}, who serendipitously discovered the first HVS, SDSS
J090745.0+024507, in a survey of blue horizontal branch
stars. \citet{Fuentes06} resolved the degeneracy between luminosity and
distance for this star by detecting variability and providing evidence
for the main-sequence nature of the first HVS.

To date ten HVSs have been reported \citep[][Table 1]{Brown07} and of
these, HE 0437--5439 stands out as the most enigmatic one. While the
origin of the other nine is consistent with being ejected from the
center of our Galaxy, HE 0437--5439 is suspected to have been ejected
from the Large Magellanic Cloud (LMC). \citet{Edelmann05} discovered
this HVS during a spectroscopic follow-up of their sample of subluminous
B-star candidates from the Hamburg/ESO survey
\citep[e.g.][]{Christlieb01}. HE 0437--5439, with an estimated apparent
magnitude of $V=16.2\pm 0.2$ mag, is the brightest HVS
known. \citet{Edelmann05} obtained two high resolution spectra, but with
a low signal to noise (S/N) ratio ($\sim$20), finding it to be a
main-sequence early B-type star consistent with solar metallicity. They
measured a heliocentric radial velocity of $+723\pm3$ km s$^{-1}$, a
rotational velocity of $54\pm4$ km s$^{-1}$ and derived a distance of
$d=61\pm12$ kpc from the Galatic center, which is inconsistent with the
spectral type of the star. While an early-type B-star has a
main-sequence lifetime of 25-35 Myr, it requires 100 Myr to travel that
distance at the measured radial velocity.

The authors therefore propose two possible explanations for this
paradox: either HE 0437--5439 is a Galactic blue straggler or it
originated from the LMC. They suggest measurements of the chemical
abundance and proper motion of the star to distinguish between the two
scenarios. \citet{Gualandris07} discarded the blue straggler scenario
claiming that a merger product would not live much longer than a
main-sequence star of the same mass. Assuming HE 0437--5439 was ejected
by a black hole in the LMC, they perform scattering simulations and
conclude that a black hole mass $\ge1000\;\mbox{M$_{\odot}$}$ is required to explain
the velocity of HE 0437--5439.

Motivated by these intriguing scenarios, we set out to measure the
metallicity of HE 0437--5439 and obtained photometry to search for
variability. We describe our findings in the following sections of this
Letter.

\section{Photometry} \label{sec:phot}

We imaged HE 0437--5439 in Johnson $B$ and $V$ bands with the Direct CCD
on the 1-m Swope telescope at Las Campanas Observatory, Chile. We
monitored the star over six consecutive nights (UT 2006 July 17-22), for
about 2 hours per night. The images were processed with standard
IRAF\footnote{IRAF is distributed by the National Optical Astronomy
Observatory, which are operated by the Association of Universities for
Research in Astronomy, Inc., under cooperative agreement with the NSF.}
routines, following the same procedure as described in
\citet{Bonanos07}. The light curves of HE 0437--5439 and several
reference stars of similar brightness in the field were extracted with
the ISIS image subtraction package \citep{Alard98,Alard00} in each
filter and converted to magnitudes, following \citet{Hartman04}. The $B$
and $V$ time series photometry for HE 0437--5439 and two of the
reference stars are shown in Figure \ref{fig:lcbv}. There is no evidence
for variability to a precision of 0.02 mag over the 6 nights. HE
0437--5439 is therefore not pulsating as SDSS J090745.0+024507
\citep{Fuentes06}, and the scenario of a blue straggler consisting of a
contact binary is also unlikely.

To calibrate the $B$ and $V$ magnitudes of HE 0437--5439, we performed
aperture photometry on images obtained on the photometric night of UT
2006 July 19 and applied transformation coefficients derived by
\citet{Bonanos07} for this night, measuring $V=16.36\pm0.04$ mag,
$B-V=-0.23\pm0.03$ mag. This improves the photometry measured by
\citet{Edelmann05} from the Hamburg/ESO plates ($V=16.2\pm0.2$ mag).
The $B-V$ color we measure is consistent with that of an early-type B
star. We note that the foreground extinction estimate from
\citet{Schlegel98} is $E(B-V)=0.008$ mag, therefore the true unreddened
$V_0$ and $(B-V)_0$ values are identical within the errors.

\section{Spectroscopy} \label{sec:spec}

We retrieved public, unpublished spectra of HE 0437--5439 from the ESO
archive. The data were obtained on UT 2006 January 12 with the
Ultraviolet and Visual Echelle Spectrograph (UVES) on the VLT UT2
8-meter telescope (Kueyen) at the ESO Paranal Observatory. The
observations consist of 8$\times1482\,s$ and 1$\times1810\,s$ spectra,
with a resolving power $R\sim34,000$, measured from the full width half
maximum of the comparison lamp lines. We bias subtracted and flatfielded
the spectra with the IRAF echelle package routines. We used the
algorithm of \citet{Pych04} to remove cosmic rays from each two
dimensional image. Next, we extracted the spectra, averaged them
(weighting by the exposure time), normalized and merged the orders. The
final spectrum ranges from 3746-4990\,\AA\, and has an average S/N ratio
of $\sim$100.

We classify HE 0437--5439 as a B2 IV-III star according to the criteria
defined by \citet{Walborn90}. The depths of the Si lines constrain the
spectral type to B1-B2. Furthermore, \ion{Si}{3} $\lambda$4552 $>$
\ion{Si}{4} $\lambda 4089$ and \ion{Si}{2} $\lambda \lambda$4128-30 $<$
\ion{Si}{3} $\lambda 4552$, while the \ion{O}{2} blends at $\lambda4640$
and $\lambda4650$ have near equal depths, in support of a B2 type
(\ion{C}{3} lines affect the relative depths in B1 types). The
luminosity class criteria are based on the relative strengths of the
\ion{He}{1} and \ion{Si}{3} lines: \ion{He}{1} $\lambda
4387/$\ion{Si}{3} $\lambda 4552\sim5$, corresponding to IV, and
\ion{He}{1} $\lambda\lambda$4144/4121 $\sim2$, corresponding to
IV-III. Our adopted spectral type, B2 IV-III, gives a slightly more
evolved star than what \citet{Edelmann05} found from their much lower
quality spectrum.

\section{Metallicity and Parameter Determination} \label{sec:metal}

We derived the metallicity and atmospheric parameters of HE 0437--5439
using the non-LTE {\sc Tlusty} model atmosphere grid \citep{Hubeny95}
described by \citet{Dufton05}~\footnote{See also
http://star.pst.qub.ac.uk}, which is appropriate for the analysis of
B-type stars. The results for the parameters (effective temperatures,
$T_{\rm eff}$; surface gravity, $g$, in units of cm\,s$^{-2}$;
microturbulence, $\xi$; projected rotational velocity, $v\sin i$) and
chemical abundance of HE 0437--5439 are presented in
Table~\ref{t_abunds}. We estimated the effective temperature from the
ionization balance of \ion{Si}{2} to \ion{Si}{3}, the surface gravity
from fitting the wings of the Balmer lines and the microturbulence from
minimizing the scatter in the abundances of the \ion{Si}{3} triplet of
lines at 4560\AA. The projected rotational velocity of the star was
estimated by fitting rotationally broadened theoretical profiles to 14
strong metal lines and a mean value of 55$\pm$1 km s$^{-1}$ was
obtained. The fits of rotationally broadened profiles to the Balmer
lines are shown in Figure~\ref{fig:Balmer}.

To derive the abundances, we measured the equivalent widths of the metal
absorption lines (C, N, O, Mg and Si). The associated errors were
calculated following the methods of \citet{Hunter07} and include both
random errors (measurement and atomic data uncertainties) and systematic
errors from the adopted atmospheric parameters. We adopted the following
uncertainties: 1,000\,K in effective temperature, 0.2\,dex in surface
gravity and 3\,km\,s$^{-1}$ in microturbulence \citep[for further
details see][]{Hunter07}. Table~\ref{t_abunds} gives in parentheses the
number of lines used for each element. Further, it lists the LMC
baseline abundance (Z=0.5\,\mbox{Z$_{\odot}$}) from \citet{Hunter07} and the baseline
solar abundances from \citet{Asplund05}, for comparison. Our analysis
follows identical methods to \citet{Hunter07}, i.e. we use the same
absorption lines, atomic data and techniques and hence our abundance
neasurements are directly comparable. The carbon abundance from the
4267\AA\ line has been corrected by 0.34\,dex following their
method. The reason for this correction is that the carbon model atom
used in the adopted {\sc Tlusty} model atmosphere grid is relatively
simple compared to that used in more detailed analyses
\citep[e.g.,][]{Sigut96,Nieva06,Nieva07} and leads to a discrepancy
between the carbon abundance derived from the 4267\AA\ line and that
from \ion{H}{2} regions \citep{Hunter07}. The correction is dependent
upon the atmospheric parameters and we therefore note that while our
absolute carbon abundance should be treated with caution, it is
comparable with the LMC stars analysed by \citet{Hunter07}. The adopted
metallicity of the grid (LMC in this case) has a negligible effect on
the derived abundances \citep[$<0.10$ dex, from][]{Hunter07}. The mild
nitrogen enrichement is not uncommon as the enrichment of core processed
material has often been observed in B-type stars \citep[see for
example][]{Walborn70, Dufton72, Gies92, Dufton05, Korn02, Venn99}. In
Figure~\ref{fig:metals}, we plot representative metal lines used to
determine abundances, along with {\sc Tlusty} model spectra at solar and
half-solar metallicities, showing that the half-solar metallicity model
correctly reproduces the metal lines. Clearly, the chemical abundance
derived for HE 0437--5439 indicates an origin from the LMC.

We estimate a slightly higher mass of 9.0 $\pm$ 0.5 $M_{\sun}$ for HE
0437--5439 than \citet[][see their Figure~3]{Edelmann05}, using our new
parameters and the evolutionary models of \citet{Schaerer93}. We also
recalculate the distance to HE 0437--5439 following
\citet{Edelmann05}. With the new values for the magnitude, mass and
surface gravity we obtain a longer distance of $73^{+6}_{-5}$ kpc. A
star of this mass, B2 IV-III spectral type and LMC metallicity should
have an age less than the value estimated by \citet{Edelmann05} for an
$8\, \mbox{M$_{\odot}$}$ main-sequence early-B star at this metallicity, i.e. 35 Myr.
Using the best fit {\sc Tlusty} model spectrum, we confirm the radial
velocity of the star to be 723 $\pm$ 2 km s$^{-1}$ with the IRAF
$rvsao.xcsao$ task. We note that HE 0437--5439 is the only HVS so far
with an accurate rotational velocity measurement: 55 $\pm$ 1 km
s$^{-1}$. \citet{Hansen07} predicted that the rotational velocities of
HVSs ejected by the \citet{Hills88} scenario (originating in binaries)
should be lower than those of single stars of the same spectral
type. Our measurement confirms a low $v\sin i$ for HE 0437--5439.

\section{Discussion} \label{sec:sum}

We measure the chemical abundance of the hypervelocity star HE
0437--5439 and find that it has half-solar metallicity, thus
establishing its origin in the LMC. We can therefore rule out Galactic
origin, because stars in the Galactic center have been found to have
solar or supersolar metallicities \citep[e.g.][]{Carr00, Ramirez00,
Najarro04, Wang06, Cunha07}. A scenario of ejection from the
low-metallicity outskirts of the Galactic disk also suffers from the age
paradox \citep[see Figure~4 in][]{Edelmann05}. Our non-variable
photometry further renders a scenario with a contact binary blue
straggler ejected from our Galaxy unlikely.

HE 0437--5439 must therefore have been ejected from the LMC. A
noteworthy implication of our result is the existence of a massive black
hole in the LMC, as suggested by \citet{Gualandris07}. According to the
simulations these authors performed, an intermediate mass black hole
(IMBH) of mass $\ge1000\,\mbox{M$_{\odot}$}$ is required to eject HE 0437--5439 from
the LMC. A three-body interaction involving a stellar mass black hole,
while still possible, is very unlikely. \citet{Gualandris07} find NGC
2004 and NGC 2100 to be the best candidate hosts of an IMBH in the LMC,
based on criteria of age, density and mass. Further work is necessary to
determine the location of the massive black hole in the LMC.

In summary, by establishing the origin of HE 0437--5439, we find
dynamical evidence for the existence of a hitherto undetected massive
black hole in the LMC.

\acknowledgments{We thank the referee, Rolf Kudritzki, for detailed
comments that significantly improved this manuscript. We particularly
thank K. Stanek for motivating us to monitor this star and for useful
discussions and comments on the manuscript. Furthermore, we thank
J. Hartman for his fluxtomag script, I. Ribas for his script to merge
echelle orders, N. Walborn for confirming our spectral classification,
J. Greene for discussions on intermediate mass black holes, P.L. Dufton
for discussions on the {\sc Tlusty} non-LTE model atmosphere
calculations, and I. Thompson for helpful comments on the manuscript.
AZB acknowledges research and travel support from the Carnegie
Institution of Washington through a Vera Rubin Fellowship. MLM
acknowledges support provided by NASA through Hubble Fellowship grant
HF-01210.01-A awarded by the Space Telescope Science Institute, which is
operated by the Association of Universities for Research in Astronomy,
Inc., for NASA, under contract NAS5-26555. IH acknowledges financial
support from the UK Science and Technology Facilities Council (STFC).}

\begin{figure}[hbt]
\epsscale{0.8}
\plotone{f1.eps}
\caption{$B$ and $V-$band light curves of HE 0437--5439 ({\em filled
circles}) and 2 other comparison stars of similar brightness, shifted
for display purposes. Panels correspond to each of the six consecutive
nights of observation; tickmarks on the x-axis correspond to 0.02 day
intervals. Dotted lines indicate the absolute photometry derived from
the second, photometric night: $V=16.36\pm0.04$ mag, $B-V=-0.23\pm0.03$
mag.}
\label{fig:lcbv}
\end{figure}

\begin{figure}[hbt]
\epsscale{1.0}
\plotone{f2.eps}
\caption{Balmer lines of HE 0437--5439. The {\sc tlusty} model spectrum
(dotted line) for the derived parameters is overplotted, illustrating
the good fit to the Balmer wings.}
\label{fig:Balmer}
\end{figure}

\begin{figure}[hbt]
\epsscale{1.0}
\plotone{f3.eps}
\caption{Representative metal lines in HE 0437--5439 (solid line) used
to derive abundances. {\sc tlusty} model spectra for half-solar (dashed
line) and solar (dotted line) metallicity are overplotted, illustrating
that the metallicity of HE 0437--5439 is half-solar.}
\label{fig:metals}
\end{figure}

\clearpage
\input{tab1.tex}

\clearpage


\section{Introduction}

The purpose of this paper is to provide a natural setting which allows to
generalize the Lipschitz summation formula to negative powers.

We recall that the classical Lipschitz summation formula gives the Fourier
series expansion of the periodic analytic function obtained by summation
over integer translates of the power $z^{-k}$, where $z\in \mathfrak{H}$ \
(the complex upper half plane) and $k$ is a positive integer.

By introducing suitable generalizations of the Lipschitz formula, we will
show how to construct new classes of hyperfunctions. Indeed, each of these
generalized formulae will provide a hyperfunctional equation involving a
specific two--variable polylogarithm series.

A simple introduction to the theory of hyperfunctions is sketched in Section
2, where the interested reader is referred to the classical works \cite{SKK}%
, \cite{Horm}. Hyperfunctional cohomology groups have also been related to
automorphic forms and period functions \cite{Bruggeman}.

The main results of this paper are essentially two. The first result is a
generalization of the classical Lipschitz summation formula. The second one
is the connection we establish between a new class of polylogarithms, the
theory of hyperfunctions in one variable and that of polynomial structures
of Appell type.

Precisely, our construction can be summarized as follows. We will define
suitable sequences of Appell polynomials of Bernoulli type: indeed, they
share with the Bernoulli polynomials several arithmetic properties,
including certain famous congruences (see the Appendix). Their periodic
versions provide primitives of the periodic delta function. To each sequence
we will associate a natural extension of the notion of polylogarithmic
function defined in the unit disc. We call this extension a \textit{delta
rational function}. It is a two--variable Dirichlet series, extending to the
whole Riemann sphere as a meromorphic function. The case of the classical
Bernoulli polynomials corresponds to the standard polylogarithmic function,
and it is treated thoroughly. The new Appell polynomial sequences we
construct, $\left\{ P_{n}\left( x\right) \right\} _{n\in \mathbb{N}}$ and $%
\left\{ Q_{n}\left( x\right) \right\} _{n\in \mathbb{N}}$ (depending on
certain parity properties) and the generalized polylogarithms $\delta
_{n}\left( q\right) $ proposed here satisfy interesting hyperfunctional
equations. These equations in turn correspond to generalized Lipschitz
summation formulae. For instance, in the case of polynomials $\left\{
P_{n}\left( x\right) \right\} _{n\in \mathbb{N}}$ it reads

\begin{equation}
\sum_{k\in \mathbb{Z}}\varphi _{\overline{P}_{n}}\left( \tau +k\right)
=2i\left( 2\pi i\right) ^{-n}\left\{
\begin{array}{c}
\Delta _{-n}\left( q\right) \text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }if%
\text{ }\left\vert q\right\vert <1,\,\ i.e.\text{ }\Im \tau >0 \\
\\
\left( -1\right) ^{n-1}\Delta _{-n}\left( q^{-1}\right) \text{ \ \ }if\text{
}\left\vert q\right\vert >1,\,\ i.e.\text{ }\Im \tau <0%
\end{array}%
\right. \text{,}  \label{I.1}
\end{equation}%
where $\overline{P}_{n}$ are the hyperfunctions associated to the polynomial
$P_{n}$ and $\varphi _{\overline{P}_{n}}$ is the function in $O^{1}\left(
\overline{\mathbb{C}}\backslash \left[ 0,1\right] \right) $ representing $%
\overline{P}_{n}$.

In the last part of the paper, the relation between the theory of formal
groups and that of hyperfunctions is clarified. We will show that the Appell
structures introduced in the previous construction can be viewed as
polynomial realizations of certain formal group laws, related with the
universal Lazard formal group.

The future research plans include an extension of the proposed construction
to the multidimensional case (see also \cite{PD} for a different
generalization), as well as possible applications of the proposed Lipschitz
formulae to the study of Eisenstein series and periods of modular forms.

\textbf{Acknowledgements}. One of us (P. T.) is grateful to U. Zannier for a
very useful discussion.

\vspace{2mm}

\section{Hyperfunctions: basic preliminaries}

\vspace{1mm}

In this section, we will provide a brief and self--consistent introduction
to the theory of hyperfunctions of a single variable, following closely \cite%
{SKK}, Chapter IX of \cite{Horm}, \cite{Kaneko} and \cite{Marmi}. For a more
extensive treatment and further details, the reader is invited to consult
these books, as well as \cite{Pham} for interesting applications.

Let us denote by $\mathcal{O}$ the sheaf of holomorphic functions on
\thinspace $\mathbb{C}$, and denote by $\mathbb{C}^{+}$ and $\mathbb{C}^{-}$
the upper and lower half planes of $\mathbb{C}$.

\begin{definition}
The space of hyperfunctions $\mathcal{B}$ on the real line $\mathbb{R}$ is%
\begin{equation}
\mathcal{B}\left( \mathbb{R}\right) :=H_{\mathbb{R}}^{1}\left( \mathbb{C},%
\mathcal{O}\right) \text{,}  \label{H1}
\end{equation}

i.e. is the first sheaf cohomology group on $\mathbb{R}$.
\end{definition}

Now, since $\mathbb{C}^{+}$ $\cup $ $\mathbb{C}^{-}=\mathbb{C}$ $\backslash $
$\mathbb{R}$, we have the following decomposition:%
\begin{equation}
H_{\mathbb{R}}^{1}\left( \mathbb{C},\mathcal{O}\right) =\left[ H^{0}\left(
\mathbb{C}^{+},\mathcal{O}\right) \oplus H^{0}\left( \mathbb{C}^{-},\mathcal{%
O}\right) \right] \text{ }/\text{ }H^{0}\left( \mathbb{C},\mathcal{O}\right)
\text{.}  \label{H2}
\end{equation}%
In other words, since $H^{0}\left( \mathcal{O}\right) $ represents nothing
but the global sections of the sheaf (i.e. holomorphic functions), a
hyperfunction can be thought as a pair of holomorphic functions on the upper
and lower half planes respectively, modulo an entire function. This
geometric definition immediately generalizes to the case of hyperfunctions
in several variables. If $\Omega \subset \mathbb{R}$ is an open set, and $U$
an arbitrary complex neighborhood of $U$, then clearly%
\begin{equation}
\mathcal{B}\left( \Omega \right) =H_{\Omega }^{1}\left( U,\mathcal{O}\right)
\text{.}  \label{H3}
\end{equation}%
Still denoting by $\mathcal{O}\left( U\right) $ the space of holomorphic
functions in $U$, another equivalent definition is%
\begin{equation}
\mathcal{B}\left( \Omega \right) :=\lim_{\longrightarrow }\text{ }_{U\text{ }%
\supset \text{ }\Omega }\text{ \ }\mathcal{O}\left( U\text{ }\backslash
\text{ }\Omega \right) \text{ }/\text{ }\mathcal{O}\left( U\right)
\label{H4}
\end{equation}

where the inductive limit with respect to the family of complex
neighborhoods $U$ $\supset $ $\Omega $ is considered. A hyperfunction $%
f\left( x\right) $ is therefore an equivalence class $\left[ F\left(
z\right) \right] $, whose representative is \thinspace $F\left( z\right) \in
\mathcal{O}\left( U\text{ }\backslash \text{ }\Omega \right) $. The
representative $F\left( z\right) $ is said to be a defining function of $%
f\left( x\right) $. Since%
\begin{equation*}
U\supset \Omega =U_{+}\cup U_{-}\text{, \qquad }U_{\pm }=U\cap \left\{ \Im
z\lessgtr 0\right\} \text{,}
\end{equation*}%
often the following boundary--value representation is used:%
\begin{equation*}
f\left( x\right) =F_{+}\left( x+i0\right) -F_{-}\left( x-i0\right) \text{,}
\end{equation*}%
with $F_{\pm }\left( z\right) =\left. F\left( z\right) \right\vert _{U_{\pm
}}$.

Alternatively, one can construct a theory of hyperfunctions based on
analytic functionals. Let $K\subset \mathbb{C}$ \ be a non empty compact
set, and denote by $A$ the space of entire analytic functions in $\mathbb{C}$%
.

\begin{definition}
The space $A^{\prime }\left( K\right) $ of the analytic functionals carried
by $K$ is the space of linear forms u acting on A such that for every
neighborhood V of K there is a constant $C_{V}>0$ such that%
\begin{equation}
\left\vert u\left( \varphi \right) \right\vert \leq C_{V}\text{ }%
\sup_{V}\left\vert \varphi \right\vert \text{, \qquad }\forall \varphi \in A%
\text{.}  \label{H5}
\end{equation}
\end{definition}

Observe that $A^{\prime }\left( K\right) $ is a Fr\'{e}chet space, since a
seminorm is associated to each neighborhood $V$ of $K$. One can define%
\begin{equation}
\mathcal{B}\left( \Omega \right) :=A^{\prime }\left( \overline{\Omega }%
\right) /A^{\prime }\left( \partial \Omega \right) \text{.}  \label{H6}
\end{equation}

It is interesting to notice that the space of hyperfunctions $\psi $ $\in
B\left( \Omega \right) $ \ with compact support $K\subset \Omega $ can be
identified with analytic functionals in $A^{\prime }\left( \mathbb{R}\right)
$ with support $K$. Indeed, an analytic functional $u$ on $\cup
_{i=1}^{r}K_{i}$ can always be decomposed into a sum $u=u_{1}+...+u_{r}$,
with each of the functionals $u_{j}\in A^{\prime }\left( K_{j}\right) $.
Consequently, since supp $\psi \subset K\cup \partial \Omega $, the
contribution of $\psi $ on $\partial \Omega $ can be factored out and $\psi $
is identified with an uniquely defined functional with support in $\Omega $.

Therefore, we can also think of $A^{\prime }\left( K\right) $ as the space
of hyperfunctions with support in $K$. \ The link between the two approaches
to the theory of hyperfunctions is now provided by the following lemma. Let $%
\mathcal{O}^{1}\mathcal{(}\overline{\mathbb{C}}$ $\backslash $ $K\mathcal{)}$
denotes the space of holomorphic functions on $\overline{\mathbb{C}}$ $%
\backslash $ $K$ and vanishing at infinity.

\begin{lemma}
\label{th1} The spaces $A^{\prime }\left( K\right) $ and $\mathcal{O}^{1}%
\mathcal{(}\overline{\mathbb{C}}$ $\backslash $ $K\mathcal{)}$ are
canonically isomorphic. To each $u\in A^{\prime }\left( K\right) $ it
corresponds a function $\varphi \in \mathcal{O}^{1}\mathcal{(}\overline{%
\mathbb{C}}$ $\backslash $ $K\mathcal{)}$ given by%
\begin{equation*}
\varphi \left( z\right) =u\left( c_{z}\right) \text{, \ \ }\forall z\in
\mathbb{C}\text{ }\backslash \text{ }K\text{,}
\end{equation*}%
where $c_{z}\left( x\right) =\frac{1}{\pi }\frac{1}{x-z}$. Conversely, to
each $\varphi \in \mathcal{O}^{1}\mathcal{(}\overline{\mathbb{C}}$ $%
\backslash $ $K\mathcal{)}$ it corresponds the hyperfunction%
\begin{equation}
u\left( \psi \right) =\frac{i}{2\pi }\int_{\gamma }\varphi \left( z\right)
\psi \left( z\right) dz\text{, \ \ }\forall \psi \in A\text{,}  \label{H7}
\end{equation}%
where $\gamma $ is any piecewise $\mathcal{C}^{1}$path winding around K in
the positive direction.
\end{lemma}

For future purposes, we also briefly describe \textit{periodic hyperfunctions%
}. Let $\mathbb{T}^{1}=\mathbb{R}$ $\backslash $ $\mathbb{Z}\subset \mathbb{C%
}$ $\backslash $ $\mathbb{Z}$. A hyperfunction on $\mathbb{T}^{1}$ is a
linear functional $\Psi $ on the space $\mathcal{O}\left( \mathbb{T}%
^{1}\right) $ of functions analytic in a complex neighborhood of $\mathbb{T}%
^{1}$ such that for all neighborhood $V$ of $\mathbb{T}^{1}$ there exists a
constant $C_{V}>0$ such that%
\begin{equation}
\left\vert \Psi \left( \Phi \right) \right\vert \leq C_{V}\text{ }%
\sup_{V}\left\vert \Phi \right\vert ,\text{ \qquad }\forall \Phi \in
\mathcal{O(}V\mathcal{)}\text{.}  \label{H8}
\end{equation}

We will denote by $A^{\prime }\left( \mathbb{T}^{1}\right) $ the Fr\'{e}chet
space of hyperfunctions with support in $\mathbb{T}^{1}$. Let $\mathcal{O}%
_{\Sigma }$ denote the complex vector space of holomorphic functions $\Phi :%
\mathbb{C}$ $\backslash $ $\mathbb{R\rightarrow C}$, 1--periodic, bounded at
$\pm $ $i\infty $ and such that $\Phi \left( \pm i\infty \right) :=\lim_{\Im
z\rightarrow \pm \infty }\Phi \left( z\right) $ exist and verify $\Phi
\left( +i\infty \right) =-\Phi \left( -i\infty \right) $.

\begin{lemma}
The spaces $A^{\prime }\left( \mathbb{T}^{1}\right) $ and $\mathcal{O}%
_{\Sigma }$ are canonically isomorphic: to each $\Psi $ $\in $ $A^{\prime
}\left( \mathbb{T}^{1}\right) $ it corresponds a function $\Phi \in \mathcal{%
O}_{\Sigma }$ given by%
\begin{equation*}
\Phi \left( z\right) =\Psi \left( C_{z}\right) ,\text{ \ }\forall z\in
\mathbb{C}\text{ }\backslash \text{ }K\text{,}
\end{equation*}

where $C_{z}\left( x\right) =\cot \pi \left( x-z\right) $. Conversely, to
each $\Phi \in \mathcal{O}_{\Sigma }$ it corresponds the hyperfunction%
\begin{equation}
\Psi \left( \Xi \right) =\frac{i}{2}\int_{\Gamma }\Phi \left( z\right) \Xi
\left( z\right) dz\text{, \ \ }\forall \text{ }\Xi \in A^{\prime }\left(
\mathbb{T}^{1}\right) \text{,}  \label{H9}
\end{equation}
where $\Gamma $ is any piecewise $\mathcal{C}^{1}$ path winding around a
closed interval $I\subset \mathbb{R}$ of length 1 in the positive direction.
\end{lemma}

Given a compactly supported hyperfunction, making infinitely many copies of
it translating its support leads to a periodic hyperfunction. This point of
view is systematically exploited in \cite{Marmi}, from which the following
commutative diagram is taken:

\begin{equation*}
\quad \ \ \ \ \ \ \ A^{\prime }\left( \left[ 0,1\right] \right) \text{ \
\quad\ \ \ \ }\longrightarrow \text{ }\ \ \ \text{\ \ \quad }O^{1}\left(
\overline{\mathbb{C}}\text{ }\backslash \text{ }\left[ 0,1\right] \right)
\end{equation*}

\begin{equation*}
\Sigma _{\mathbb{Z}}\downarrow \text{ \ \ \ \ \ \ }\ \ \ \ \ \ \ \ \ \ \ \
\text{\ \ \ \ \ \ \ \ \ \ \ \ }\downarrow \Sigma _{\mathbb{Z}}
\end{equation*}

\begin{equation}
A^{\prime }\left( \mathbb{T}^{1}\right) \text{ \ }\ \text{\ \ \ \ \ }\ \ \
\longrightarrow \text{ }\ \ \ \ \ \ \ \ \text{\ \ \ \ \ }\mathcal{O}_{\Sigma
}  \label{H10}
\end{equation}

The horizontal lines are the above mentioned isomorphisms, and $\sum_{%
\mathbb{Z}}$ denotes the summation over integer translates.

\vspace{2mm}

\section{Delta rational functions}

\subsection{Delta rational functions and polylogarithms}

\vspace{2mm}

In this section we introduce the notion of delta rational functions, which
represent a natural generalization of the notion of polylogarithms \cite%
{Lewin}, \cite{Zagier}. These functions are connected to the classical
Bernoulli polynomials via the Fourier expansion formula (\ref{1.3}) and
enter a generalized Lipschitz summation formula providing new classes of
hyperfunctions.

\begin{definition}
\bigskip The delta rational function is the series expressed for any $n\in
\mathbb{Z}$, and $q\in \mathbb{D}$ by%
\begin{equation}
\delta _{n}\left( q\right) =\sum_{k=1}^{\infty }k^{n}q^{k}\text{.}
\label{1.1}
\end{equation}
\end{definition}

Observe that, if $n\geq 0$, $\delta _{n}$ extends to the whole Riemann
sphere as a rational function of degree $n+1$ with just a simple pole of
order $n+1$ at $q=1$. If $n\leq -1$ then $\delta _{n}$ coincides with the
classical polylogarithm series of order $-n$ (actually $\delta _{-1}\left(
q\right) =-\log \left( 1-q\right) $) and extends to the whole $\mathbb{C}$ $%
\backslash \lbrack 1,+\infty )$ and as a multiplicative function to the
whole $\mathbb{C}$ $\backslash \{0,1,\infty \}$. Indeed, since%
\begin{equation*}
q\partial _{q}\delta _{n}\left( q\right) =\delta _{n+1}\left( q\right) ,
\end{equation*}%
one can define the analytic continuation of $\delta _{n}$. For instance, the
continuation of $\delta _{-2}$ is obtained by means of the integral formula%
\begin{equation*}
\delta _{-2}\left( q\right) =-\int_{0}^{q}\frac{\log \left( 1-t\right) }{t}%
dt=\int_{0}^{q}\left( \int_{0}^{t}\frac{d\zeta }{1-\zeta }\right) \frac{dt}{t%
}\text{.}
\end{equation*}%
Note that $[1,+\infty )$ is a branch cut. For all $n\in \mathbb{Z}$ and $%
q\in \mathbb{D}$, one has%
\begin{equation}
\delta _{n}\left( q^{k}\right) =k^{-1-n}\sum_{\Lambda ^{k}=1}\delta
_{n}\left( \Lambda q\right) \text{,}  \label{1.2}
\end{equation}%
where $\Lambda $ denotes a $k$--th root of unity. The equality extends to
the closed disk if $n\leq -2$. One can directly prove the following result.

\begin{lemma}
The fundamental inversion equation holds%
\begin{equation}
\delta _{n}\left( q\right) +\left( -1\right) ^{n}\delta _{n}\left(
q^{-1}\right) =\left\{
\begin{array}{c}
0\qquad \qquad \qquad \qquad \qquad \ \ \ if\text{ }n\geq 1 \\
\\
-\frac{\left( 2\pi i\right) ^{-n}}{\left( -n\right) !}B_{-n}\left( \frac{%
\log q}{2\pi i}\right) \text{ \qquad }if\text{ }n\leq 0%
\end{array}%
\right. \text{,}  \label{1.3}
\end{equation}

for all $q\neq 1$ if $n\geq 0$, and $q$ in $%
\mathbb{C}
\backslash \left[ 0,+\infty \right] $ if $n\leq 0$. Here $B_{k}$ is the $k$%
--th Bernoulli polynomial.
\end{lemma}

\subsection{Periodic hyperfunctions}

\vspace{1mm}

The inversion relation (\ref{1.3}) has a beautiful interpretation in terms
of hyperfunctions, as we will show below.

\begin{definition}
The periodic Bernoulli functions and distributions are expressed by%
\begin{equation}
\widetilde{B}_{n}\left( x\right) =\left\{
\begin{array}{c}
-\frac{B_{n}\left( x-\left[ x\right] \right) }{n!}\qquad if\text{ }n\geq 1
\\
-1+\delta _{\mathbb{T}}\left( x\right) \text{ \ \ \ }if\text{ }n=0 \\
\delta _{\mathbb{T}}^{\left( -n\right) }\left( x\right) \qquad if\text{ }n<0%
\end{array}%
\right. \text{,}  \label{2.1}
\end{equation}%
where $\left[ x\right] $ is the integer part of $x$, $\delta _{\mathbb{T}}$
is the periodic delta distribution and $\delta _{\mathbb{T}}^{\left(
k\right) }$ its derivative of order $k\geq 0$.
\end{definition}

Explicitly,%
\begin{equation}
\delta _{\mathbb{T}}\left( x\right) =\sum_{k=-\infty }^{+\infty }e^{2\pi ikx}%
\text{.}  \label{2.2}
\end{equation}%
From the well--known property of the periodic Bernoulli functions%
\begin{equation}
\frac{d^{n}}{dx^{n}}\left( B_{n}\left( x-\left[ x\right] \right) \right)
=n!\left( 1-\delta _{\mathbb{T}}\right) \text{,}  \label{2.3}
\end{equation}%
we see that we can consider the Bernoulli functions as primitives of the
periodic delta function. It is immediate to check that the following
hyperfunctional equation holds.

\begin{proposition}
\bigskip Let $q^{\pm }=e^{2\pi i\left( x\pm i0\right) },$ $x\in \mathbb{R}$.
For all $n\in \mathbb{Z}$ we have%
\begin{equation}
\delta _{n}\left( q^{+}\right) +\left( -1\right) ^{n}\delta _{n}\left(
\left( q^{-}\right) ^{-1}\right) =\left( 2\pi i\right) ^{-n}\widetilde{B}%
_{-n}\left( x\right) \text{.}  \label{2.4}
\end{equation}
\end{proposition}

Note that on both sides one can apply derivatives $q\partial _{q}$ and $%
\left( 2\pi i\right) ^{-1}\partial _{x}$.

\subsection{Generalized Lipschitz summation formula}

\vspace{1mm}

As is well known, the classical Lipschitz formula is a consequence of the
Poisson summation formula, relating sums over integers of pairs of Fourier
transforms. It states that%
\begin{equation}
\sum_{n\in \mathbb{Z}}\frac{1}{\left( n+z\right) ^{k}}=\frac{\left( -2\pi
i\right) ^{k}}{\left( k-1\right) !}\sum_{r=1}^{\infty }r^{k-1}e^{2\pi
irz},\qquad z\in \mathbb{H},\qquad k\in \mathbb{Z}_{\geq 2}  \label{3.0}
\end{equation}%
(for a proof of (\ref{3.0}), see, for instance, D. Zagier, Chapter 4 in \cite%
{WMLI}).

In this section, we will give an hyperfunctional generalization of (\ref{3.0}%
). We make use of the fact that periodic hyperfunctions with compact support
are the result of a summation over integer translates of hyperfunctions with
support on $\left[ 0,1\right] $, according to the commutative diagram (\ref%
{H10}). By applying the results described in the previous section, we get a
functional version of the Lipschitz formula.

\begin{theorem}
F$or$ $all$ $n\in \mathbb{Z}$ we have%
\begin{equation}
\sum_{k\in \mathbb{Z}}\varphi _{\overline{B}_{n}}\left( \tau +k\right)
=2i\left( 2\pi i\right) ^{-n}\left\{
\begin{array}{c}
\delta _{-n}\left( q\right) \text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }if%
\text{ }\left\vert q\right\vert <1,\,\ i.e.\text{ }\Im \tau >0 \\
\\
\left( -1\right) ^{n-1}\delta _{-n}\left( q^{-1}\right) \text{ \ \ }if\text{
}\left\vert q\right\vert >1,\,\ i.e.\text{ }\Im \tau <0%
\end{array}%
\right. \text{,}  \label{3.1}
\end{equation}%
whereas the usual Lipschitz formula corresponds to $n\leq -1$. Here $%
\overline{B}_{n}$ is the restriction to $\left[ 0,1\right] $ of $\widetilde{%
B_{n}}$ and $\varphi _{\overline{B}_{n}}$ is the function in $O^{1}\left(
\overline{\mathbb{C}}\backslash \left[ 0,1\right] \right) $ which represents
the hyperfunction $\overline{B}_{n}$%
\begin{equation}
\varphi _{\overline{B}_{n}}\left( \tau \right) =\left\langle \overline{B}%
_{n},\frac{1}{\pi }\frac{1}{x-\tau }\right\rangle _{\left[ 0,1\right] }.
\label{3.2}
\end{equation}
\end{theorem}

\begin{proof}
It is enough to remark that an inverse of the operator $\sum_{\mathbb{Z}}$
on the hyperfunctions is just the restriction of a periodic hyperfunction to
the interval $\left[ 0,1\right] $. Thus the r.h.s. of (\ref{2.4}) reads%
\begin{equation}
\sum_{k\in \mathbb{Z}}\left( 2\pi i\right) ^{-n}\overline{B}_{-n}\left(
x+k\right)  \label{3.5}
\end{equation}%
and from (\ref{1.3}) and (\ref{2.4}) one obtains the desired formula by
considering the two associated holomorphic functions.
\end{proof}

Explicitly, formula (\ref{3.2}) reads%
\begin{equation}
\varphi _{\overline{B}_{n}}\left( \tau \right) =\left\{
\begin{array}{c}
\left\langle \delta ^{\left( -n\right) }\left( x\right) ,\frac{1}{\pi }\frac{%
1}{x-\tau }\right\rangle _{\left[ 0,1\right] }=\left( -1\right) ^{-n+1}\frac{%
\left( -n\right) !}{\pi \tau ^{-n+1}}\text{ \ \ \ \ }if\text{ }n<0 \\
\left\langle -1+\delta \left( x\right) ,\frac{1}{\pi }\frac{1}{x-\tau }%
\right\rangle _{\left[ 0,1\right] }=-\frac{1}{\pi \tau }\left[ 1+\tau \log
\left( 1-1/\tau \right) \right] \text{ \ }\ \ \text{\ \ \ }if\text{ }n=0 \\
\left\langle -\frac{1}{n!}B_{n}\left( x\right) ,\frac{1}{\pi }\frac{1}{%
x-\tau }\right\rangle _{\left[ 0,1\right] }=-\frac{1}{\pi n!}\left[
B_{n}\left( \tau \right) \log \left( 1-1/\tau \right) +R_{n}\left( \tau
\right) \right] \text{ \ }\ \ \text{\ }if\text{ }n>0%
\end{array}%
\right. \text{,}  \label{3.3}
\end{equation}%
where $R_{n-1}$ is the polynomial of degree $n-1$ such that $B_{n}\left(
\tau \right) \log \left( 1-1/\tau \right) +R_{n}\left( \tau \right) \in
O^{1}\left( \overline{\mathbb{C}}\backslash \left[ 0,1\right] \right) $.
Here are the first six polynomials%
\begin{equation*}
R_{1}\left( \tau \right) =1\text{, \ }R_{2}\left( \tau \right) =\tau -\frac{1%
}{2}\text{, \ }R_{3}\left( \tau \right) =\tau ^{2}-\tau +\frac{1}{12}\text{,}
\end{equation*}%
\begin{equation*}
R_{4}\left( \tau \right) =\tau ^{3}-\frac{3}{2}\tau ^{2}+\frac{1}{3}\tau +%
\frac{1}{12}\text{, \ \ }R_{5}\left( \tau \right) =\tau ^{4}-2\tau ^{3}+%
\frac{3}{4}\tau ^{2}+\frac{1}{4}\tau -\frac{13}{360}\text{,}
\end{equation*}%
\begin{equation}
R_{6}\left( \tau \right) =\tau ^{5}-\frac{5}{2}\tau ^{4}+\frac{4}{3}\tau
^{3}+\frac{1}{2}\tau ^{2}-\frac{13}{60}\tau -\frac{7}{120}\text{.}
\label{3.4}
\end{equation}

\textbf{Remark}. One could object that we should have used the simplest
Appell sequence $\left\{ x^{n}\right\} _{n\in \mathbb{N}}$ and their
associated hyperfunctions on $\left[ 0,1\right] $ to write the generalized
Lipschitz summation formula. But this wold trivially give zero, since%
\begin{equation}
x^{n}=\frac{1}{n+1}\left[ B_{n+1}\left( x+1\right) -B_{n+1}\left( x\right) %
\right]  \label{3.6}
\end{equation}%
and the function $B_{n+1}\left( x+1\right) -B_{n+1}\left( x\right) $ clearly
belongs to the kernel of $\sum_{\mathbb{Z}}$. Instead, other less trivial
Appell sequences can be used to provide useful generalizations of the
construction we proposed, as will be illustrated in the subsequent sections.

\vspace{2mm}

\section{The general case: sequences of Appell polynomials and hyperfunctions%
}

\subsection{Appell sequences}

\vspace{1mm}

By analogy with the theory developed in the previous section, we study a
more general class of polynomials of Bernoulli type, with the aim of
constructing associated families of hyperfunctions. The main request is that
they belong to the class of Appell polynomials, i.e. polynomials satisfying
the property%
\begin{equation}
\frac{d}{dx}A_{n}\left( x\right) =n\text{ }A_{n-1}\left( x\right) \text{,}
\label{Ap1a}
\end{equation}%
with the normalization%
\begin{equation}
A_{0}\left( x\right) =const\text{.}  \label{Ap1b}
\end{equation}

\begin{lemma}
The Fourier series expansion of the polynomials (\ref{Ap1a}), for $0<x<1$
and $n\geq 1$ has the form%
\begin{equation}
A_{n}\left( x\right) =\sum_{k=-\infty }^{\infty }c_{k}\left( n\right)
e^{2\pi ikx}\text{,}  \label{Ap2}
\end{equation}%
with%
\begin{equation}
c_{k}=\int_{0}^{1}A_{n}\left( t\right) e^{-2\pi ikt}dt\text{.}  \label{Ap2b}
\end{equation}%
We get%
\begin{equation}
A_{n}\left( x\right) =-n!\sum_{k=1}^{\infty }\left[ \sum_{j=1}^{n}\frac{1}{j!%
}\frac{\varphi _{j}}{\left( 2\pi ik\right) ^{n+1-j}}e^{2\pi
ikx}+\sum_{j=1}^{n}\left( -1\right) ^{n+1-j}\frac{1}{j!}\frac{\varphi _{j}}{%
\left( 2\pi ik\right) ^{n+1-j}}e^{-2\pi ikx}\right] +\text{ }c_{0}\text{, \
\ }  \label{Ap3}
\end{equation}%
with $0<x<1$, and $\varphi _{j}=A_{j}\left( 1\right) -A_{j}\left( 0\right) $%
, $\ j=1,..,n$. The standard Bernoulli polynomials correspond to the case $%
c_{0}=0,$ $\varphi _{1}=1$, $\varphi _{j}=0$, $\ j=2,3,...$
\end{lemma}

\begin{proof}
It is a direct consequence of the conditions (\ref{Ap1a})--(\ref{Ap2b}) and
of the formula of integration by parts.
\end{proof}

\vspace{2mm}

If $\varphi _{j}=0$ for $j$ even or $j~$odd, then formula (\ref{Ap3}) can be
written in a more compact form. The corresponding polynomial sequences will
be denoted by $\left\{ p_{n}\left( x\right) \right\} _{n\in \mathbb{N}}$ and
$\left\{ q_{n}\left( x\right) \right\} _{n\in \mathbb{N}}$, respectively.

\vspace{2mm}

a) If $\varphi _{j}=0$ for $j$ even, we define%
\begin{equation}
p_{n}\left( x\right) =:-n!\sum_{k=1}^{\infty }\sum_{\substack{ j=1  \\ j%
\text{ odd}}}^{n}\frac{1}{j!}\frac{\varphi _{j}}{\left( 2\pi ik\right)
^{n+1-j}}\left[ e^{2\pi ikx}+\left( -1\right) ^{n}e^{-2\pi ikx}\right] +%
\text{ }c_{0}\text{, \ \ }0<x<1\text{.}  \label{Ap4}
\end{equation}

b) If $\varphi _{j}=0$ for $j$ odd, we introduce%
\begin{equation}
q_{n}\left( x\right) =:-n!\sum_{k=1}^{\infty }\sum_{\substack{ j=1  \\ j%
\text{ even}}}^{n}\frac{1}{j!}\frac{\varphi _{j}}{\left( 2\pi ik\right)
^{n+1-j}}\left[ e^{2\pi ikx}+\left( -1\right) ^{n+1}e^{-2\pi ikx}\right] +%
\text{ }c_{0}\text{, \ \ }0<x<1\text{.}  \label{Ap5}
\end{equation}%
\textbf{Remark}. The previous conditions on $\varphi _{j}$ are not
particularly restrictive. Indeed, one can easily construct infinitely many
polynomial sequences possessing the prescribed parity properties. Their
generating function has the general form%
\begin{equation}
\frac{te^{xt}}{\left( e^{t}-1\right) g\left( t\right) }=\sum_{n=0}^{\infty
}A_{n}\left( x\right) \frac{t^{n}}{n!}\text{,}  \label{Ap6}
\end{equation}%
where $g\left( t\right) $ is any real analytic function such that $%
\lim_{t\rightarrow 0}g\left( t\right) =1$ and satisfying the parity
condition $g\left( t\right) =g\left( -t\right) $ for the case a) and $%
g\left( t\right) =-g\left( -t\right) $ for the case b). \

\vspace{4mm}

\textbf{Examples}

\vspace{4mm}

a) Take%
\begin{equation}
\frac{t^{2}e^{xt}}{\left( e^{t}-1\right) \sin t}=\sum_{n=0}^{\infty
}a_{n}\left( x\right) \frac{t^{n}}{n!}  \label{Ap7}
\end{equation}%
We easily deduce

\begin{equation*}
a_{0}\left( x\right) =1\text{, \ }a_{1}\left( x\right) =x-\frac{1}{2}\text{,
\ }a_{2}\left( x\right) =x^{2}-x+\frac{1}{2}\text{,}
\end{equation*}%
\begin{equation*}
a_{3}\left( x\right) =x^{3}-\frac{3}{2}x^{2}+\frac{3}{2}x-\frac{1}{2}\text{,}
\end{equation*}%
\begin{equation}
a_{4}\left( x\right) =x^{4}-2x^{3}+3x^{2}-2x+\frac{23}{30}\text{,...}
\label{Ap8}
\end{equation}

\vspace{2mm}

b) Consider the generating function
\begin{equation}
\frac{te^{xt}}{\left( e^{t}-1\right) \cos t}=\sum_{n=0}^{\infty }b_{n}\left(
x\right) \frac{t^{n}}{n!}  \label{Ap9}
\end{equation}%
We immediately get

\begin{equation*}
b_{0}\left( x\right) =1\text{, \ }b_{1}\left( x\right) =x-\frac{1}{2}\text{,
\ }b_{2}\left( x\right) =x^{2}-x+\frac{7}{6}\text{,}
\end{equation*}%
\begin{equation*}
b_{3}\left( x\right) =x^{3}-\frac{3}{2}x^{2}+\frac{7}{2}x-\frac{3}{2}\text{,}
\end{equation*}%
\begin{equation}
b_{4}\left( x\right) =x^{4}-2x^{3}+7x^{2}-6x+\frac{179}{30}\text{,...}
\label{Ap10}
\end{equation}

\subsection{Extended delta rational functions}

\vspace{1mm}

Let $n\in \mathbb{Z}$, $q\in \mathbb{D}$. \ A straightforward generalization
of $\delta _{n}\left( q\right) $ adapted to the chosen Appell polynomials is
provided by the extended delta rational function $\Delta _{n}\left( q\right)
$.

\begin{definition}
The extended delta rational function, for any $n\in \mathbb{Z}$ and $q\in
\mathbb{D}$, is defined by%
\begin{equation}
\Delta _{n}\left( q\right) =\left\{
\begin{array}{c}
\sum_{k=1}^{\infty }k^{n}q^{k}\,\ \ \ \ \ \ \ \ \ \ \ \ n>0 \\
\\
\sum_{k=1}^{\infty }a_{k}\left( n\right) q^{k}\ \ \ \ \ \ \ n\leq 0%
\end{array}%
\right.  \label{5.1}
\end{equation}%
where
\begin{equation}
a_{k}\left( n\right) =\sum_{\substack{ j  \\ j\text{ even or odd}}}^{n}\frac{%
1}{j!}\frac{\varphi _{j}}{k^{n+1-j}}\text{.}  \label{5.2}
\end{equation}
\end{definition}

In eq. (\ref{5.2}), the summation should be understood either over the even
values of $j$ or over the odd ones, depending on the choice of the
polynomials (\ref{Ap4}) or (\ref{Ap5}), respectively. The above definition
is motivated by the following result, which provides an extension of the
construction proposed in Section 3. Our aim is to obtain the hyperfunctional
equations associated to the proposed Appell polynomials. As a consequence of
the Fourier expansion (\ref{Ap3}) and of relations (\ref{Ap4})--(\ref{Ap5})
we get the relation between extended delta rational functions and Appell
sequences.

\vspace{3mm}

\begin{lemma}
The following inversion equations, generalizing relation (\ref{1.3}), hold%
\begin{equation}
\Delta _{n}\left( q\right) +\left( -1\right) ^{n}\Delta _{n}\left(
q^{-1}\right) =\left\{
\begin{array}{c}
0\qquad \qquad \qquad \qquad \qquad \ \ \ if\text{ }n\geq 1 \\
-\frac{\left( 2\pi i\right) ^{-n}}{\left( -n\right) !}P_{-n}\left( \frac{%
\log q}{2\pi i}\right) \text{ \qquad }if\text{ }n\leq 0%
\end{array}%
\right. \text{,}  \label{5.3}
\end{equation}%
and%
\begin{equation}
\Delta _{n}\left( q\right) +\left( -1\right) ^{n+1}\Delta _{n}\left(
q^{-1}\right) =\left\{
\begin{array}{c}
0\qquad \qquad \qquad \qquad \qquad \ \ \ if\text{ }n\geq 1 \\
-\frac{\left( 2\pi i\right) ^{-n}}{\left( -n\right) !}Q_{-n}\left( \frac{%
\log q}{2\pi i}\right) \text{ \qquad }if\text{ }n\leq 0%
\end{array}%
\right. \text{,}  \label{5.4}
\end{equation}%
where $P_{n}$ and $Q_{n}$ are respectively the Appell polynomials (\ref{Ap4}%
) and (\ref{Ap5}), to which the constant $c_{0}$ has been subtracted. The
inversion relations hold for all $q\neq 1$ if $n\geq 0$, whereas for $n\leq
0 $ $q$ can be taken in $%
\mathbb{C}
\backslash \left[ 0,+\infty \right] $.
\end{lemma}

By analogy with formulae (\ref{2.1}), the associated periodic functions and
distributions are defined as

\begin{equation}
\widetilde{P}_{n}\left( x\right) =\left\{
\begin{array}{c}
-\frac{P_{n}\left( x-\left[ x\right] \right) }{n!}\qquad if\text{ }n\geq 1
\\
-1+\delta _{\mathbb{T}}\left( x\right) \text{ \ \ \ }if\text{ }n=0 \\
\delta _{\mathbb{T}}^{\left( -n\right) }\left( x\right) \qquad if\text{ }n<0%
\end{array}%
\right. \text{.}  \label{5.5}
\end{equation}%
with an analogous definition for the case of polynomials $Q_{n}\left(
x\right) $. For all $n\in \mathbb{Z}$ we have the hyperfunctional equations

\begin{equation}
\Delta _{n}\left( q^{+}\right) +\left( -1\right) ^{n}\Delta _{n}\left(
\left( q^{-}\right) ^{-1}\right) =\left( 2\pi i\right) ^{-n}\widetilde{P}%
_{-n}\left( x\right) \text{,}  \label{5.6}
\end{equation}%
and

\begin{equation}
\Delta _{n}\left( q^{+}\right) +\left( -1\right) ^{n+1}\Delta _{n}\left(
\left( q^{-}\right) ^{-1}\right) =\left( 2\pi i\right) ^{-n}\widetilde{Q}%
_{-n}\left( x\right) \text{.}  \label{5.7}
\end{equation}%
The proof of these relations is again a direct consequence of the previous
definitions. Also, by denoting with $\overline{P}_{n}$ the restriction to $%
\left[ 0,1\right] $ of $\widetilde{P_{n}}$ and with $\varphi _{\overline{P}%
_{n}}$ the function in $O^{1}\left( \overline{\mathbb{C}}\backslash \left[
0,1\right] \right) $ which represents the hyperfunction $\overline{P}_{n}$,
namely

\begin{equation}
\varphi _{\overline{P}_{n}}\left( \tau \right) =\left\langle \overline{P}%
_{n},\frac{1}{\pi }\frac{1}{x-\tau }\right\rangle _{\left[ 0,1\right] }\text{%
,}
\end{equation}%
we find, for any choice of the sequence $\left\{ P_{n}\left( x\right)
\right\} _{n\in \mathbb{N}}$ and $\left\{ Q_{n}\left( x\right) \right\}
_{n\in \mathbb{N}}$ our main result, i.e. the \textit{generalized Lipschitz
formula}

\begin{equation}
\sum_{k\in \mathbb{Z}}\varphi _{\overline{P}_{n}}\left( \tau +k\right)
=2i\left( 2\pi i\right) ^{-n}\left\{
\begin{array}{c}
\Delta _{-n}\left( q\right) \text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }if%
\text{ }\left\vert q\right\vert <1,\,\ i.e.\text{ }\Im \tau >0 \\
\\
\left( -1\right) ^{n-1}\Delta _{-n}\left( q^{-1}\right) \text{ \ \ }if\text{
}\left\vert q\right\vert >1,\,\ i.e.\text{ }\Im \tau <0%
\end{array}%
\right. \text{,}
\end{equation}%
and the corresponding formula for $\overline{Q}_{n}$ (again the usual
Lipschitz formula corresponds to $n\leq -1$). The explicit computation of $%
\varphi _{\overline{P}_{n}}\left( \tau \right) $ and $\varphi _{\overline{Q}%
_{n}}\left( \tau \right) $ is completely analogous to the proposed
construction for the Bernoulli polynomials and we will not repeat it here.

\vspace{2mm}

\section{A connection with the Lazard formal group}

\vspace{1mm}

We describe here briefly an interesting connection between formal groups,
hyperfunctions, and the so--called universal Bernoulli polynomials.

Given a commutative ring with identity $R$, we will denote by $R\left[ x_{1},%
\text{ }x_{2},..\right] $ the ring of formal power series in $x_{1}$, $x_{2}$%
, ... with coefficients in $R$. Following \cite{Haze}, \cite{BMN}, we recall
that a commutative one--dimensional formal group law over $R$ is a
two--variable formal power series $\Phi \left( x,y\right) \in R\left[ x,y%
\right] $ such that%
\begin{equation*}
1)\qquad \Phi \left( x,0\right) =\Phi \left( 0,x\right) =x
\end{equation*}%
and%
\begin{equation*}
2)\qquad \Phi \left( \Phi \left( x,y\right) ,z\right) =\Phi \left( x,\Phi
\left( y,z\right) \right) \text{.}
\end{equation*}

When $\Phi \left( x,y\right) =\Phi \left( y,x\right) $, the formal group is
said to be commutative. The existence of an inverse formal series $\varphi
\left( x\right) $ $\in R\left[ x\right] $ such that $\Phi \left( x,\varphi
\left( x\right) \right) =0$ follows from the previous definition.

Let us consider the polynomial ring $\mathbb{Q}\left[ c_{1},c_{2},...\right]
$ and the formal power series

\begin{equation}
F\left( s\right) =s+c_{1}\frac{s^{2}}{2}+c_{2}\frac{s^{3}}{3}+...
\label{Ap11}
\end{equation}%
Let $G\left( t\right) $ be the associated inverse series

\begin{equation}
G\left( t\right) =t-c_{1}\frac{t^{2}}{2}+\left( 3c_{1}^{2}-2c_{2}\right)
\frac{t^{3}}{6}+...  \label{Ap12}
\end{equation}%
so that \textit{$F\left( G\left( t\right) \right) =t$}. The series (\ref%
{Ap11}) and (\ref{Ap12}) are called formal group logarithm and formal group
exponential, respectively. The formal group law related to these series is
provided by
\begin{equation*}
\Phi \left( s_{1},s_{2}\right) =G\left( F\left( s_{1}\right) +F\left(
s_{2}\right) \right)
\end{equation*}%
$\,$ and it represents the so called \textit{Lazard's Universal Formal Group
}\cite{Haze}. It is defined over the Lazard ring $L$, i.e. the subring of $%
\mathbb{Q}\left[ c_{1},c_{2},...\right] $ generated by the coefficients of
the power series $G\left( F\left( s_{1}\right) +F\left( s_{2}\right) \right)
$.

In \cite{Tempesta1}, the \textit{universal Bernoulli polynomials\ $%
B_{n}^{G}\left( x,c_{1},...,c_{k},...\right) \equiv B_{k}^{G}\left( x\right)
$ related to the formal group exponential }$G$ have been introduced. They
are defined by%
\begin{equation}
\frac{t}{G\left( t\right) }e^{xt}=\sum_{n\geq 0}B_{n}^{G}\left( x\right)
\frac{t^{k}}{n!}\text{,}\qquad \qquad x\in \mathbb{R}\text{.}  \label{Ap13}
\end{equation}%
The corresponding numbers by construction coincide with the universal
Bernoulli numbers discovered in \cite{Clarke}, and generated by%
\begin{equation}
\frac{t}{G\left( t\right) }=\sum_{n\geq 0}\widehat{B_{n}}\frac{t^{k}}{n!}%
,\qquad \qquad x\in \mathbb{R}.  \label{Ap14}
\end{equation}

Observe that when $a=1$, $c_{i}=\left( -1\right) ^{i}$, then $F\left(
s\right) =\log \left( 1+s\right) \,$, $G\left( t\right) =e^{t}-1$, and the
universal Bernoulli polynomials and numbers reduce to the standard ones.
Many other examples of Bernoulli--type polynomials considered in the
literature are obtaining by specializing the rational coefficients $c_{i}$.
The reasons to consider such a generalization of Bernoulli polynomials are
manifold. First, to any choice of the coefficients $c_{1},c_{2},...$ it
corresponds a sequence of polynomials of Appell type sharing with the
standard Bernoulli polynomials many algebraic and combinatorial properties
\cite{Tempesta2}. In particular, the associated numbers satisfy universal
Clausen--von Staudt and Kummer congruences, as shown in the Appendix. Also,
in the same way as the Riemann zeta function is associated with the
Bernoulli polynomials, it is possible to relate the polynomials (\ref{Ap13})
with a large class of absolutely convergent L--series. Their values at
negative integers correspond to the generalized Bernoulli numbers, and they
possess as well several interesting number--theoretical properties \cite%
{Tempesta1}. Observe that the polynomials (\ref{Ap6}) belong to the class (%
\ref{Ap13}), i.e. the coefficient $c_{i}$ are all rational, if the functions
$g\left( t\right) $ verify the further condition $g^{\left( n\right) }\left(
0\right) $ $\in \mathbb{Q}$ for any $n$. This is the case, for instance, for
the proposed examples of the polynomials (\ref{Ap7}) and (\ref{Ap9}). As an
immediate consequence, the numbers generated by%
\begin{equation}
\frac{t}{\left( e^{t}-1\right) g\left( t\right) }=\sum_{n=0}^{\infty }A_{n}%
\frac{t^{n}}{n!}\text{,}
\end{equation}%
with the prescribed conditions on $g\left( t\right) $ do satisfy by
construction the Clausen--von Staudt and Kummer congruences and many others.
These considerations enable us to associate one--variable formal groups with
suitable classes of hyperfunctions, as a consequence of the previous
construction.

\vspace{2mm}

\section{Appendix: congruences}

\vspace{2mm}

The Clausen--von Staudt congruence \cite{IR}, one of the most beautiful of
mathematics, states that%
\begin{equation}
B_{n}+\sum_{p\mid n}\frac{1}{p}\in \mathbb{Z}\text{,}  \label{CvS}
\end{equation}%
where $B_{n}$ denotes the $n$--th Bernoulli number. This proves the strict
link between Bernoulli numbers and prime numbers. Many generalizations of
this result have been obtained in the literature in the last decades. In an
attempt to clarify the deep connection between these congruences and
algebraic topology, in \cite{Clarke} Clarke proposed the notion of universal
Bernoulli numbers $\widehat{B_{n}}$, defined as (\ref{Ap14}), and proved the
remarkable universal von Staudt congruence.

If $n$ is even, we have%
\begin{equation}
\widehat{B_{n}}\equiv -\sum_{\overset{p-1\mid n}{p\text{ prime}}}\frac{%
c_{p-1}^{n/(p-1)}}{p}\qquad mod\quad \mathbb{Z}\left[ c_{1},c_{2},...\right]
\text{;}  \label{US1}
\end{equation}%
If $n$ is odd and greater than 1, we have%
\begin{equation}
\widehat{B_{n}}\equiv \frac{c_{1}^{n}+c_{1}^{n-3}c_{3}}{2}\qquad mod\quad
\mathbb{Z}\left[ c_{1},c_{2},...\right] \text{.}  \label{US2}
\end{equation}%
Like the classical ones, the universal Bernoulli numbers as well play an
important role in several branches of mathematics, in particular in complex
cobordism theory (see e.g. \cite{BCRS}, and \cite{Ray}), where the
coefficients $c_{n}$ are identified with the cobordism classes of $\mathbb{C}%
P^{n}$.

Kummer congruences are also relevant in algebraic topology, and in defining
p--adic extensions of zeta functions. We also have an universal Kummer
congruence \cite{Adel}. Suppose that $n\neq 0,1$ (mod $p-1$). Then

\begin{equation}
\frac{\widehat{B}_{n+p-1}}{n+p-1}\equiv \frac{\widehat{B}_{n}}{n}%
c_{p-1}\qquad mod\quad p\mathbb{Z}_{p}\left[ c_{1},c_{2},...\right] \text{.}
\label{UK}
\end{equation}

Other related congruences can be found in \cite{Adel}.