Orthogonal Polynomials related to g-fractions with missing terms
Kiran Kumar Behera A. Swaminathan

amsplain





Department of Mathematics

 Indian Institute of Technology Roorkee-247 667,
 Uttarakhand, India
krn.behera@gmail.com



Department of  Mathematics  

Indian Institute of Technology Roorkee-247 667,
Uttarakhand, India

swamifma@iitr.ac.in, mathswami@gmail.com




The purpose of the present paper is to investigate some
structural and qualitative aspects of two different
perturbations of the parameters of g-fractions.

In this context the concept of gap g-fractions is introduced. While tail sequences of a continued fraction play a
significant role in the first perturbation,
Schur fractions are used
in the second perturbation of the g-parameters that are considered.

Illustrations are provided using Gaussian hypergeometric functions.

Using a particular gap g-fraction, some members
of the class of Pick functions are also identified.










Orthogonal Polynomials related to g-fractions with missing terms

    
A. Swaminathan

    January 27, 2017
==================================================================



myheadings  Kiran Kumar Behera and A. Swaminathan 
Orthogonal Polynomials related to g-fractions with missing terms






§ INTRODUCTION



Given an arbitrary real sequence {g_k}_k=0^∞,
a continued fraction expansion of the form


    11[ ; - ](1-g_0)g_1z1[ ; - ](1-g_1)g_2z1[ ; - ](1-g_2)g_3z1[ ; - ]⋯,
       z∈ℂ,


is called a g-fraction if the parameters
g_j∈[0,1], j∈ℕ∪{0}.
It terminates and equals a rational function if
g_j∈{0,1} for some j∈ℕ∪{0}.

If 0<g_j<1, j∈ℕ∪{0},
the g-fraction (<ref>)
still converges uniformly on every compact subsets of the slit
domain ℂ∖[1,∞)
(see <cit.> and
<cit.>),
and in this case,
(<ref>)
will represent an
analytic function, say ℱ(z).

Such g-fractions are found having applications in diverse areas
like number theory
<cit.>,
dynamical systems
<cit.>,
moment problems and analytic
function theory
<cit.>.

In particular, <cit.>,
the Hausdorff moment problem


    ν_j=∫_0^1σ^jdν(σ),
       j≥0,


has a solution if and only if (<ref>)
corresponds to a power series of the form
1+ν_1z+ν_2z^2+⋯, z∈ℂ∖[1,∞).

Further, the g-fractions have also been used to study the geometric properties of ratios of Gaussian hypergeometric functions as well as their q-analogues,
(see the proofs of <cit.>
and <cit.>).

Among several such results,
one of the most fundamental result concerning g-fraction is
<cit.> in which holomorphic functions
having positive real part in ℂ∖[1, ∞) are characterised.

Precisely, Re(√(1+z)  ℱ(z)) is positive
if and only if ℱ(z) has a continued fraction expansion
of the form (<ref>).
Moreover, ℱ(z) has the integral representation


    ℱ(z)=∫_0^1dϕ(t)1-zt,
      
    z∈ℂ∖[1,∞],


where ϕ(t) is a bounded non-decreasing function
having a total increase 1.

Many interesting results are also available in literature if we consider
subsets of ℂ∖[1,∞).
For instance, let 𝕂 be the class of
holomorphic functions having a positive real part on the unit disk
𝔻:={z:|z|<1}. Such functions denoted by 𝒞(z)
are called Carathéodory functions and have the Riesz-Herglotz
representation <cit.>


    𝒞(z)=
    ∫_0^2πe^it+ze^it-zdϕ(t)+qi,


where q=Im  𝒞(0).

Further, if 𝒞(z)∈𝕂 is such that
𝒞(ℝ)⊆ℝ
and normalised by 𝒞(0)=1,
then the following continued fraction expansion can be derived
<cit.>



    1-z1+zC(z)=
    11[ ; - ]g_1ω1[ ; - ](1-g_1)g_2ω1[ ; - ](1-g_2)g_3ω1[ ; - ]⋯,
       z∈𝔻,


where w=-4z/(1-z)^2. Note that here g_0=0.


Closely related to the Carathéodory functions are the Schur functions
f(z) given by


    C(z)=1+zf(z)1-zf(z),
       z∈𝔻.


From (<ref>),
it is clear that f(z) maps the unit disk 𝔻
to the closed unit disk 𝔻̅.
In fact, if


    𝔹={
                 f(𝔻)⫅𝔻̅},


(<ref>)
describes a one-one correspondence between the classes of
holomorphic functions 𝔹 and 𝕂.

The class 𝔹 studied by J. Schur
<cit.>,
is the well-known Schur algorithm.

This algorithm generates a sequence of rational functions
{f_n(z)}_n=0^∞ from a given sequence
{α_n}_n=0^∞ of complex numbers lying in
𝔻̅.

Then, with α_n, n≥0, satisfying some
positivity conditions, f_n(z)→ f(z),
n→∞, where f(z)∈𝔹
is a Schur function
<cit.>.

It is interesting to note that α_n=f_n(0),
n≥0 where f_0(z)≡ f(z). Moreover, using these
parameters {α_n}_n≥0,
the following Schur fraction can be
obtained <cit.>


    α_0+
    (1-|α_0|^2)zα̅_0z[ ; + ]1α_1[ ; + ](1-|α_1|^2)zα̅_1z[ ; + ]1α_2[ ; + ](1-|α_2|^2)zα̅_2z[ ; + ]⋯,


where α_j is related to the g_j
occurring in (<ref>)
by α_j=1-2g_j, j≥1.

Similar to g-fractions the Schur
fraction also terminates if |α_n|=1 for some n∈ℤ_+.
It may be noted that such a case occurs if and only if
f(z) is a finite Blashke product <cit.>.

Let A_n(z) and B_n(z) denote the n^th partial
numerator and denominator of
(<ref>) respectively.

Then, with the initial values A_0(z)=α_0,
B_0(z)=1, A_1(z)=z
and
B_1(z)=α̅_0z,
the following recurrence relations hold
<cit.>


    A_2n(z)   =α_nA_2n-1(z)+A_2n-2(z)
    B_2n(z)   =α_nB_2n-1(z)+B_2n-2(z),
       n≥1,



    A_2n+1(z)   =α̅_nzA_2n(z)+(1-|α_n|^2)zA_2n-1(z)
    B_2n+1(z)   =α̅_nzB_2n(z)+(1-|α_n|^2)zB_2n-1(z),
       n≥1.


Using
(<ref>)
in
(<ref>),
we get


    A_2n+1(z)   =zA_2n-1(z)+α̅_pzA_2n-2(z)
    B_2n+1(z)   =zB_2n-1(z)+α̅_pzB_2n-2(z).


The relations
(<ref>)
and
(<ref>)
are sometimes written in the more
precise matrix form as


    (
      [ A_2p+1 B_2p+1;   A_2p   B_2p;        ])=
    (
      [     z α̅_pz;   α_p     1;       ])
    (
      [ A_2p-1 A_2p-1; A_2p-2 A_2p-2;        ]),
       p≥1.



It is also known that
<cit.>


    A_2n+1(z)   =zB_2n^∗(z)
       ;   B_2n+1(z)=zA_2n^∗(z)
    A_2n(z)   =B_2n+1^∗(z)
       ;   B_2n(z)=A_2n+1^∗(z).


Here, and in what follows,
P_n^∗(z)=z^nP_n(1/z̅)
for any polynomial P_n(z)
with complex coefficients and of degree n.

From (<ref>),
it follows that


    A_2n+1(z)B_2n+1(z)=
    (A_2n^∗(z)B_2n^∗(z))^-1,
      A_2n(z)B_2n(z)=
    (A_2n+1^∗(z)B_2n+1^∗(z))^-1.


Further, the even approximants of the Schur fraction
(<ref>)
coincide with the the n^th approximant of the Schur algorithm,
so that
A_2n(z)/B_2n(z) converges to the
Schur function f(z) as n→∞.

From the point of view of their applications, it is obvious
that the parameters g_n of the g-fraction
(<ref>) contain hidden
information about the properties of the dynamical systems or the
special functions they represent.

One way to explore this hidden
information is through perturbation;
that is, through a study of the consequences when some disturbance
is introduced in the parameter sequence {g_n}.

The main objective of the present manuscript is to study the
structural and qualitative aspects of two
perturbations.

The first is when a finite number of parameters g_j's are missing
in which case we call the corresponding g-fraction as a gap-g-fraction.

The second case is replacing {g_n}_n=0^∞ by
a new sequence {g_n^(β_k)}_n=0^∞ in which the
j^th term g_j is replaced by g_j^(β_k).

The first case is illustrated
using Gaussian hypergeometric functions,
where we use the fact that
many g-fractions converge to ratios of
Gaussian hypergeometric functions in slit complex domains.

The second case is studied by applying the technique of coefficient
stripping <cit.> to the sequence of Schur parameters {α_j}.
This follows from the fact that the Schur fraction and the g-fraction are
completely determined by the related Schur parameters α_k's
and the g-parameters respectively, and that a perturbation in α_j
produces a unique change in the g_j and vice-versa.


The manuscript is organised as follows.

Section <ref>
provides structural relations for the three different cases
provided by the gap g-fraction. A particular ratio
of Gaussian hypergeometric functions is used to illustrate the results.

The modified g-fractions given in
Section <ref>
has a shift in g_k to g_k+1 and so on for any fixed k.

Instead, the effect of changing g_k to any another value
g_k^(β_k) is discussed in
Section <ref>.

Illustrations of the results obtained in
Section <ref>
leading to characterization of a class of ratio of hypergeometric functions
such as Pick functions is outlined in
Section <ref>.






§ GAP G-FRACTIONS AND STRUCTURAL RELATIONS



As the name suggests, gap-g-fractions correspond to the
g-sequence {g_k}_k=0^∞
with missing parameters.
We study three cases in this section and in each the concept of tail sequences
of a continued fraction plays an important role. For more information on the tails
of a continued fraction, we refer to
<cit.>.

For z∈ℂ∖[1,∞), let ℱ(z) be the continued fraction
(<ref>) and


    ℱ(k;z)=
          11[ ; - ](1-g_0)g_1z1[ ; - ]
    
    ⋯(1-g_k-2)g_k-1z1[ ; - ](1-g_k-1)g_k+1z1[ ; - ](1-g_k+1)g_k+2z1[ ; - ]⋯.


Note that
(<ref>)
is obtained from
(<ref>)
by removing g_k for some arbitrary k
which cannot be obtained by letting g_k=0.
Let,


    ℋ_k+1(z)=
    g_k+1z1[ ; - ](1-g_k+1)g_k+2z1[ ; - ](1-g_k+2)g_k+3z1[ ; - ]⋯,


so that -(1-g_k)ℋ_k+1(z) is the (k+1)^th
tail of ℱ(z).
We note that
<cit.>,
the existence of ℱ(z) guarantees the
existence of ℋ_k+1(z).

Further, if


    h(k;z)=(1-g_k-1)ℋ_k+1(z),
       k≥1,


then, from (<ref>)
and (<ref>)
we obtain the rational function


    𝒳_k(h(k;z);z)𝒴_k(h(k;z);z)=
    11[ ; - ](1-g_0)g_1z1[ ; - ]⋯(1-g_k-2)g_k-1z1-h(k;z).


It is known
<cit.>,
that the k^th approximant of
(<ref>)
is given by the rational function


    𝒳_k(0;z)𝒴_k(0;z)=
    11[ ; - ](1-g_0)g_1z1[ ; - ]⋯(1-g_k-2)g_k-1z1=
    𝒮_k(0,z),
      ,


and that


    𝒳_k(h(k;z);z)𝒴_k(h(k;z);z)=
    𝒳_k(0;z)-h(k;z)𝒳_k-1(0;z)𝒴_k(0;z)-h(k;z)𝒴_k-1(0;z).


Then,


    𝒳_k(h(k;z);z)𝒴_k(h(k;z);z)-
    𝒳_k(0;z)𝒴_k(0;z)   =
    h(k;z)[𝒳_k(0;z)𝒴_k-1(0;z)-𝒳_k-1(0;z)𝒴_k(0;z)]𝒴_k(0;z)[𝒴_k(0;z)-h(k;z)𝒴_k-1(0;z)]
       =
    h(k;z)z^k-1∏_j=1^k-1(1-g_j-1)g_j𝒴_k(0;z)[𝒴_k(0;z)-h(k;z)𝒴_k-1(0;z)],


where the last equality follows from <cit.>.
Denoting d_j=(1-g_j-1)g_j, j≥1, we have from
(<ref>)


    𝒳_k(h(k;z);z)𝒴_k(h(k;z);z)=
    𝒳_k(0;z)𝒴_k(0;z)-
                     ∏_j=1^k-1d_jz^k-1h(k;z)𝒴_k-1(0;z)𝒴_k(0;z)h(k;z)-
                           [𝒴_k(0;z)]^2.




In the sequel, by ℱ(z) we will mean the
unperturbed g-fraction as given in
(<ref>)
with g_k∈[0,1], k∈ℤ_+.

Further, as the notation suggests, the rational function
𝒮_k(0;z) is independent of the parameter g_k
and is known whenever ℱ(z) is given.

The information of the missing parameter g_k
at the k^th position
is stored in h(k;z) and hence the notation
ℱ(k;z).


It may also be noted that the polynomials
𝒴_k(0;z)
can be easily computed from the Wallis recurrence
<cit.>


    𝒴_j(0;z)=𝒴_j-1(0;z)-
                         (1-g_j-2)g_j-1z𝒴_j-2(0;z),
       j≥2,


with the initial values 𝒴_0(0;z)=𝒴_1(0;z)=1.
Thus, we state our first result.



Suppose ℱ(z) is given.
Let ℱ(k;z) denote the
perturbed g-fraction in which the parameter g_k is missing. Then, with
d_j=(1-g_j-1)g_j, j≥1


    ℱ(k;z)=𝒮_k(0;z)-
                     ∏_j=1^k-1d_jz^k-1h(k;z)𝒴_k-1(0;z)𝒴_k(0;z)h(k;z)-[𝒴_k(0;z)]^2,


where 𝒴_k(0;z), 𝒮_k(0;z) and
-(1-g_k-1)^-1(1-g_k)h(k;z) are respectively,
the k^th partial denominator,
the k^th approximant and
the (k+1)^th tail of ℱ(z).



It may be observed that the right side of
(<ref>)
is of the form


    a(z)h(k;z)+b(z)c(z)h(k;z)+d(z),


with a(z), b(z), c(z), d(z) being well defined polynomials.
Rational functions of such form are said to be
rational transformation of h(k;z) and occur frequently in
perturbation theory of orthogonal polynomials.
For example, see <cit.>.


A similar result for the perturbed g-fraction in which a finite number
of consecutive parameters are missing can be obtained by an analogous
argument that is stated directly as



Let ℱ(z) be given. Let
ℱ(k,k+1,⋯,k+l-1;z) denote the perturbed g-fraction in which
the l consecutive parameters g_k,g_k+1,⋯,g_k+l-1 are missing.
Then,


    ℱ(k,k+1,⋯,k+l-1;z)=
          𝒮_k(0;z)-∏_j=1^k-1d_jz^k-1h(k,k+1,⋯,k+l-1;z)𝒴_k-1(0;z)𝒴_k(0;z)h(k,k+1,⋯,k+l-1;z)-
                           [𝒴_k(0;z)]^2,


where -(1-g_k-1)^-1(1-g_k+l-1)h(k,k+1,⋯,k+l-1;z)
is the (k+l)^th tail of ℱ(z).



The next result is about the perturbation in which only two parameters
g_k and g_l are missing, where l need not be k±1.




Let ℱ(z) be given.
Let ℱ(k,l;z) denote the perturbed g-fraction
in which two parameters g_k and g_l are missing, where
we assume l=k+m+1, m≥1.
Then


    ℱ(k,l;z)=𝒮_k(0;z)-
                     ∏_j=1^k-1d_jz^k-1h(k,l;z)𝒴_k-1(0;z)𝒴_k(0;z)h(k,l;z)-[𝒴_k(0;z)]^2


where -(1-g_k-1)^-1(1-g_k)h(k,l;z) is the perturbed (k+1)^th tail of
ℱ(z) in which g_l is missing and is given by


    -(1-g_k-1)^-1(1-g_k)h(k,k+m+1;z)=
          𝒮_m^(k+1)(0,z)-
    ∏_j=k+1^k+md_jz^m h(k+m+1;z)[𝒴^(k+1)_m(0;z)]^2-
    𝒴^(k+1)_m-1(0;z)𝒴^(k+1)_m(0;z)h(k+m+1;z)


where 𝒴^(k+1)_m(0;z) and 𝒮_m^(k+1)(0,z)
are respectively, the
m^th partial denominator and
m^th approximant of the
(k+1)^th tail of ℱ(z).
Here, -(1-g_l-1)^-1(1-g_l)h(l;z) is the (l+1)^th tail
of ℱ(z).





Let


    ℋ_k+1(l;z)=
    g_k+1z1[ ; - ](1-g_k+1)g_k+2z1[ ; - ]⋯(1-g_l-1)g_l+1z1[ ; - ](1-g_l+1)g_l+2z1[ ; - ]⋯


so that -(1-g_k)ℋ_k+1(l;z) is the perturbed
(k+1)^th tail of ℱ(z) in which g_l is missing.
Then we can write


    ℱ(k,l;z)=
    𝒳_k(h(k,l;z);z)𝒴_k(h(k,l;z);z)=
    11[ ; - ](1-g_0)g_1z1[ ; - ]⋯(1-g_k-2)g_k-1z1-h(k,l;z),


where h(k,l;z)=(1-g_k-1)ℋ_k+1(l;z).
Now, proceeding as in
Theorem <ref>,
we obtain


    ℱ(k,l;z)=𝒮_k(0;z)-
                     ∏_j=1^k-1d_jz^k-1h(k,l;z)𝒴_k-1(0;z)𝒴_k(0;z)h(k,l;z)-[𝒴_k(0;z)]^2


Hence all that remains is to find the expression for h(k,l;z)
or ℋ_k+1(l;z).

Now, let

    ℋ_l+1(z)=
    g_l+1z1[ ; - ](1-g_l+1)g_l+2z1[ ; - ](1-g_l+2)g_l+3z1[ ; - ]⋯,


and h(l;z)=(1-g_l-1)ℋ_l+1(z).
From
(<ref>)
and
<cit.>,
we have


    -(1-g_k)ℋ_k+1(l;z)
       =
    -(1-g_k)g_k+1z1[ ; - ](1-g_k+1)g_k+2z1[ ; - ]⋯(1-g_l-2)g_l-1z1-h(l;z)
       =
    𝒳^(k+1)_l-k-1(h(l;z);z)𝒴^(k+1)_l-k-1(h(l;z);z).


It is clear that, the rational function
[𝒳^(k+1)_l-k-1(0;z)/𝒴^(k+1)_l-k-1(0;z)]
is the approximant of the (k+1)^th tail
-(1-g_k)ℋ_k+1(z)
of ℱ(z).
Then, using <cit.>
we obtain


    𝒳^(k+1)_l-k-1(h(l;z);z)𝒴^(k+1)_l-k-1(h(l;z);z)-
    𝒳^(k+1)_l-k-1(0;z)𝒴^(k+1)_l-k-1(0;z)=
    -h(l;z)∏_k^l-1[d_jz]𝒴^(k+1)_l-k-1(0;z)[𝒴^(k+1)_l-k-1(0;z)-h(l;z)𝒴^(k+1)_l-k-2(0;z)]


Finally, using the fact that l=k+m+1, we obtain


    -(1-g_k)ℋ_k+1(k+m+1;z)=
          𝒮_m^(k+1)(0,z)-
    ∏_j=k+1^k+md_jz^m h(k+m+1;z)[𝒴^(k+1)_k+m+1(0;z)]^2-𝒴^(k+1)_k+m(0;z)𝒴^(k+1)_k+m+1(0;z)h(k+m+1;z),


where


    𝒮_m^(k+1)(0,z)=
    -(1-g_k)g_k+1z1[ ; - ](1-g_k+1)g_k+2z1[ ; - ]⋯(1-g_k+m-1)g_k+mz1


is the m^th approximant of the (k+1)^th tail of ℱ(z).



As mentioned earlier, from
(<ref>),
(<ref>) and
(<ref>),
it is clear that tail sequences play a
significant role in deriving the
structural relations for the gap g-fractions.
We now illustrate the role of tail
sequences


using particular g-fraction expansions.



 §.§ Tail sequences using hypergeometric functions


The Gaussian hypergeometric function, with the
complex parameters a, b and c is defined by the power series

    F(a,b;c;ω)=
    ∑_n=0^∞(a)_n(b)_n(c)_n(1)_nω^n,
       |ω|<1

where c≠0,-1,-2,⋯ and
(a)_n=a(a+1)⋯(a+n-1) is the Pochhammer symbol.

Two hypergeometric functions
F(a_1,b_1;c_1;ω) and F(a_2,b_2;c_2,ω)
are said to be contiguous if the difference between the corresponding parameters
is at most unity. A linear combination of two contiguous hypergeometric
functions is again a hypergeometric function. Such relations are called contiguous relations
and have been used to explore many hidden properties of
the hypergeometric functions; for example, many special functions
can be represented by ratios of Gaussian hypergeometric functions.
For more details, we refer to <cit.>.


Consider the Gauss continued fraction <cit.>
(with b↦ b-1 and c↦ c-1)


    F(a,b;c;ω)F(a,b-1;c-1;ω)=
    11[ ; - ](1-g_0)g_1ω1[ ; - ](1-g_1)g_2ω1[ ; - ](1-g_2)g_3ω1[ ; - ]⋯


where


    g_2p=c-a+p-1c+2p-1,
      
    g_2p+1=c-b+pc+2p,
       p≥0.


Let k_p=1-g_p, p≥0. We aim to find the ratio of
hypergeometric functions given by the continued fraction


    11[ ; - ]k_1ω1[ ; - ](1-k_1)k_2ω1[ ; - ](1-k_2)k_3ω1[ ; - ]⋯


For this, first note that from
(<ref>),
we can write


    ℛ(ω)=
    1-1k_0[1-F(a,b-1;c-1;ω)F(a,b;c;ω)]=
    1-(1-k_1)ω1-k_1(1-k_2)ω1-k_2(1-k_3)ω1-⋯


Now, replacing b↦ b-1 and c↦ c-1 in
the contiguous relation
<cit.>
we obtain


    F(a,b;c;ω)-F(a,b-1;c-1;ω)=
    a(c-b)(c-1)cω F(a+1,b;c+1;ω)


Hence, with k_0=(c-a-1)/(c-1), we have


    ℛ(ω)
       =1-c-1a[F(a,b;c;ω)-F(a,b-1;c-1;ω)F(a,b;c;ω)]
       =1-c-bcF(a+1,b;c+1;ω)F(a,b;c;ω)
       =(1-ω)F(a+1,b;c;ω)F(a,b;c;ω),


where the last equality follows from the relation


    F(a,b;c;ω)=(1-ω)F(a+1,b;c;ω)+c-bcω F(a+1,b;c+1;ω),


which is easily proved by comparing the coefficients of
ω^k on both sides.
Finally, using the well known result
<cit.>, we obtain


    F(a+1,b;c;ω)F(a,b;c;ω)=
    11[ ; - ]b/cω1[ ; - ](c-b)(a+1)/c(c+1)ω1[ ; - ](c-a)(b+1)/(c+1)(c+2)ω1[ ; - ]⋯


Note that the continued fraction
(<ref>)
has also been derived by different means in
<cit.> and studied
in the context of geometric properties of
hypergeometric functions.

For further analysis, we establish the following
formal continued fraction expansion:


    F(a+1,b;c+1;ω)F(a,b;c;ω)=
    
    
    
    
    11[ ; - ]b(c-a)/c(c+1)ω1[ ; - ](a+1)(c-b+1)/(c+1)(c+2)ω1[ ; - ](b+1)(c-a+1)/(c+2)(c+3)ω1[ ; - ]⋯


Suppose for now, the left hand side of
(<ref>)
is denoted by 𝒢_1^(a,b,c)(ω). Again using, Gauss
continued fraction <cit.>
with a↦ a+1 and c↦ c+1,
we arrive at


    [𝒢_1^(a,b,c)]^-1(ω)=1-b(c-a)c(c+1)ωF(a+1,b+1;c+2;ω)F(a+1,b;c+1;ω).


In the relation
(<ref>),
interchanging a and b and then replacing a↦ a+1 and
c↦ c+1, we get


    1-b(c-a)c(c+1)ωF(a+1,b+1;c+2;ω)F(a+1,b;c+1;ω)=
    F(a,b;c;ω)F(a+1,b;c+1;ω),


which implies


    𝒢_1^(a,b,c)(ω)=
    F(a+1,b;c+1;ω)F(a,b;c;ω).



As mentioned, the right hand side is only a
formal expansion for the left hand side in
(<ref>).
However, note the fact that the sequence {𝒫_j(ω)}_j=0^∞,

    𝒫_2j(ω)   =F(a+j,b+j;c+2j;ω),
    𝒫_2j+1(ω)   =F(a+j+1,b+j;c+2j+1;ω),
       j≥0


satisfies the difference equation


    𝒫_j(ω)=𝒫_j+1(ω)-
    d_j+1ω𝒫_j+2(ω),
       j≥0,


where


    d_n=
    {[ (b+j)(c-a+j)(c+2j)(c+2j+1),              n=2j+1≥1, j≥0;; (a+j)(c-b+j)(c+2j-1)(c+2j),                n=2j≥2, j≥1. ].


Thus, using
<cit.>,
we can conclude that the right side of
(<ref>)
indeed corresponds as well as converges to the
left side of
(<ref>),
which we state as




  The following correspondence and convergence properties hold.

  
  * With a, b, c≠0,-1,-2,⋯ complex constants,
        
    F(a+1,b;c+1;ω)F(a,b;c;ω)∼11[ ; - ](b)(c-a)/c(c+1)ω1[ ; - ](a+1)(c-b+1)/(c+1)(c+2)ω1[ ; - ](b+1)(c-a+1)/(c+2)(c+3)ω1[ ; - ]⋯

  
  * The continued fraction on the right side of
        (<ref>)
        converges to the meromorphic function f(ω) in the cut-plane 𝔇
        where f(ω)=F(a+1,b;c+1;ω)/F(a,b;c;ω) and
        𝔇={ω∈ℂ:|(1-ω)|<π}. The convergence is uniform
        on every compact subset of {ω∈𝔇:f(ω)≠∞}.




We would like to mention here that the polynomial sequence
{𝒬_n(ω)}
corresponding to {𝒫_n(ω)}
and that arises during the discussion of the convergence of
Gauss continued fractions
<cit.>
is given by



    𝒬_2j(ω)   =F(a+j,b+j;c+2j;ω),    j≥0 
    𝒬_2j+1(ω)   =F(a+j,b+j+1;c+2j+1;ω),    j≥0.


Also note that the continued fraction used in the right side of
(<ref>)
is


    11[ ; - ]k_1(1-k_2)ω1[ ; - ]k_2(1-k_3)ω1[ ; - ]k_3(1-k_4)ω1[ ; - ]⋯.


Here, we recall that k_n=1-g_n, n≥0, where {g_n} are the parameters appearing in the Gauss continued fraction
(<ref>).

The following result gives a kind of generalization of
Proposition <ref>.
The correspondence and convergence properties of the
continued fractions involved can be discussed similar to the one for
𝒢_1^(a,b,c)(ω).





Let,

    𝒢_n^(a,b,c)(ω)=
    11[ ; - ]k_n(1-k_n+1)ω1[ ; - ]k_n+1(1-k_n+2)ω1[ ; - ]k_n+2(1-k_n+3)ω1[ ; - ]⋯


Then,


    𝒢_2j^(a,b,c)(ω)   =
    F(a+j,b+j;c+2j;ω)F(a+j,b+j-1;c+2j-1;ω)
       j≥1,
    𝒢_2j+1^(a,b,c)(ω)   =
    F(a+j+1,b+j;c+2j+1;ω)F(a+j,b+j;c+2j;ω)
       j≥0






The case j=0 has already been established in
Proposition <ref>.

Comparing the continued fractions for
𝒢_2j+1^(a,b,c)(ω)
and 𝒢_2j-1^(a,b,c)(ω),
j≥1,
it can be seen that
𝒢_2j+1^(a,b,c)(ω)
can be obtained for
𝒢_2j-1^(a,b,c)(ω) j≥1
by shifting
a↦ a+1, b↦ b+1 and c↦ c+2.

For n=2j, j≥1,
we note that the continued fraction in right side of
(<ref>)
is nothing but the Gauss continued fraction
<cit.>
with the shifts a↦ a+j, b↦ b+j-1
and c↦ c+2j-1
in the parameters.



Instead of starting with k_n(1-k_n+1), as the first partial numerator
term in the continued fraction
(<ref>),
a modification by inserting a new term changes the hypergeometric ratio given in Proposition <ref>, thus leading to interesting consequences. We state this result as follows.




Let


    ℱ_n^(a,b,c)(ω)=
    11[ ; - ]k_nω1[ ; - ](1-k_n)k_n+1ω1[ ; - ](1-k_n+1)k_n+2ω1[ ; - ]⋯ .


Then,


    ℱ_2j+1^(a,b,c)(ω)
       =F(a+j+1,b+j;c+2j;ω)F(a+j,b+j;c+2j;ω),
       j≥0,
    ℱ_2j+2^(a,b,c)(ω)
       =F(a+j+1,b+j+1;c+2j+1;ω)F(a+j+1,b+j;c+2j+1;ω),
       j≥0.







Denoting,


    ℰ_n+1^(a,b,c)(ω)=
    1-1k_n(1-1𝒢_n^(a,b,c)(ω))=
    1-(1-k_n+1)ω1-k_n+1(1-k_n+2)ω1-⋯,
       n≥1,


we find from <cit.>,


    ℱ_n+1^(a,b,c)(ω)=
    ℰ_n+1^(a,b,c)(ω)1-z=
    11-k_n+1ω1-(1-k_n+1)k_n+2ω1-⋯,
       n≥1.


Hence, we need to derive the functions
ℰ_n+1^(a,b,c)(ω). For n=2j, j≥1,
using (<ref>)
and k_2j=(a+j)/(c+2j-1), we find that


    1k_2j[1-1𝒢_2j^(a,b,c)(ω)]=
    c-b+jc+2jωF(a+j+1,b+j;c+2j+1;ω)F(a+j,b+j;c+2j;ω)


Shifting a↦ a+j, b↦ b+j and c↦ c+2j in
(<ref>),
we find that


    ℰ_2j+1^(a,b,c)(ω)=(1-z)F(a+j+1,b+j;c+2j;ω)F(a+j,b+j;c+2j;ω)


so that


    ℱ_2j+1^(a,b,c)(ω)=
    F(a+j+1,b+j;c+2j;ω)F(a+j;b+j;c+2j;ω)=
    11-k_2j+1ω1-(1-k_2j+1)k_2j+2ω1-⋯,
       j≥1.


Repeating the above steps, we find that for n=2j+1, j≥0 and
k_2j+1=(b+j)/(c+2j), j≥0,


    ℰ_2j+2^(a,b,c)(ω)=(1-z)F(a+j+1,b+j+1;c+2j+1;ω)F(a+j+1,b+j;c+2j+1;ω),
       j≥0


so that


    ℱ_2j+2^(a,b,c)(ω)=
    F(a+j+1,b+j+1;c+2j+1;ω)F(a+j+1;b+j;c+2j+1;ω)=
    11-k_2j+2ω1-(1-k_2j+2)k_2j+3ω1-⋯,
       j≥0.


















For particular values of ℱ_n^(a,b,c)(ω), further properties of the ratio of hypergeometric function can be discussed. One particular case and few ratios of hypergeometric functions are given in
Section <ref>
with some properties. Before proving such specific case, we consider another type of perturbation in g-fraction in the next section.






§ PERTURBED SCHUR PARAMETERS



As mentioned in Section <ref>,
the case of a single parameter
g_k being replaced by g_k^(β_k) can be studied using the
Schur parameters. It is obvious that this is equivalent to studying the
perturbed sequence {α_j^(β_k)}_j=0^∞,
where


    α_j^(β_k)=
    {[  α_j, j≠ k;;  β_k,  j=k. ].


Hence, we start with a given Schur function and study the perturbed
Carathéodory function and its corresponding g-fraction.
The following theorem gives the structural relation between the Schur
function and the perturbed one. The proof follows the transfer matrix
approach, which has also been used earlier
in literature (see for example
<cit.>).




Let A_k(z) and B_k(z) be the n^th partial numerators
and denominators of the Schur fraction associated with the sequence
{α_k}_n=0^∞. If A_k(z;k)
and B_k(z;k) are the n^th partial numerators
and denominators of the Schur fraction associated with the sequence
{α_j^(β_k)}_j=0^∞ as defined in
(<ref>),
then the following structural relations hold for
p≥ 2k, k≥1.


    z^k-1∏_j=0^k(1-|α_j|^2)
    (
      [ A_2p+1(z;k)   A_2p(z;k); B_2p+1(z;k)   B_2p(z;k);             ])=
    𝔗(z;k)
    (
      [ A_2p+1(z)   A_2p(z); B_2p+1(z)   B_2p(z);           ]),


where the entries of the transfer matrix
𝔗(z;k) are given by


    (
      [ 𝔗_(1,1) 𝔗_(1,2); 𝔗_(2,1) 𝔗_(2,2);         ])
    
    =   (
       [ p_k(z,k)A_2k-1(z)+q_k^∗(z,k)A_2k-2(z) q_k(z,k)A_2k-1(z)+p_k^∗(z,k)A_2k-2(z); p_k(z,k)B_2k-1(z)+q_k^∗(z,k)B_2k-2(z) q_k(z,k)B_2k-1(z)+p_k^∗(z,k)B_2k-2(z);                                       ]),


with


    p_k(z,k)   =(α_k-β_k)B_2k-1(z)+(1-βα̅_k)B_2k-2(z)
    
    q_k(z;k)   =(β_k-α_k)A_2k-1(z)-(1-α̅_kβ_k)A_2k-2(z).






Let


    Ω_p(z;α)=
    (
      [ A_2p+1(z) B_2p+1(z);   A_2p(z)   B_2p(z);           ])
        Ω_p(z;α;k)=
    (
      [ A_2p+1(z;k) B_2p+1(z;k);   A_2p(z;k)   B_2p(z;k);             ]).


Then the matrix relation
(<ref>)
can be written as


    Ω_p(z;α)   =T_p(α_p)·Ω_p-1(z;α)
       =T_p(α_p)· T_p-1(α_p-1)·⋯·
                       T_1(α_1)·Ω_0(z;α),
       p≥1,


with


    T_p(α_p)=
    (
      [     z α̅_pz;   α_p     1;       ])
        Ω_0(z;α)=
    
    
    
    
    
    
    (
      [     z α̅_0z;   α_0     1;       ])=
    T_0(α_0).


From (<ref>), it is clear that


    Ω_p(z;α;k)=
    Ω_0(z;α) T_k(β_k)∏_j≠ kj=1^pT_j(α_j).


Defining the associated polynomials of order k+1 as


    Ω_p-(k+1)^(k+1)(z;α)=
    T_p(α_p)T_p-1(α_p-1)⋯ T_k+1(α_k+1)Ω_0(z;α),


we have


    T_p(α)T_p-1(α_p-1)⋯ T_k+1(α_k+1)=
    Ω_p-(k+1)^(k+1)(z;α)[Ω_0(z;α)]^-1,


where [Ω_0(z;α)]^-1 denotes the matrix inverse of
Ω_0(z;α).
Now, using
(<ref>)
and
(<ref>)
in
(<ref>),
we get


    Ω_p(z;k;α)=Ω_p-(k+1)^(k+1)(z;α)·[Ω_0(z;α)]^-1· T_k(β_k)·Ω_k-1(z;α).


Again from (<ref>),


    Ω_p(z;α)   =T_p(α_p)⋯ T_k+1(α_k+1)·T_k(α_k)⋯ T_1(α_1)Ω_0(z;α)
       =Ω_p-(k+1)^(k+1)(z;α)Ω_0^-1(z;α)·Ω_k(z;α),


which means


    Ω_p-(k+1)^(k+1)(z;α)=
    Ω_p(z;α)[Ω_k(z;α)]^-1Ω_0(z;α).


Using
(<ref>)
in
(<ref>), we get


    Ω_p(z;k;α)=Ω_p(z,α)[Ω_k(z,α)]^-1Ω_0(z;α)·
                         [Ω_0(z;α)]^-1· T_k(β_k)·Ω_k-1(z,α),


which implies


    [Ω_p(z;k;α)]^T=[T_k(β_k)Ω_k-1(z,α)]^T·
                             [Ω_k(z,α)]^-T·
                             [Ω_p(z,α)]^T.


where [Ω_p(z,α)]^T denotes the matrix transpose of
Ω_p(z,α).

After a brief calculation, and using the relations
(<ref>),
it can be proved that the product
[T_k(β_k)Ω_k-1(z,α)]^T·Ω_k^-T(z,α)
precisely gives the transfer matrix
𝔗(z;k)
leading to
(<ref>).



As an important consequence of Theorem
(<ref>),
we have,


    z^k-1∏_j=0^k(1-|α_j|^2)
    (
      [ A_2p(z;k); B_2p(z;k);           ])=
    𝔗_k(z;k)
    (
      [ A_2p(z); B_2p(z);         ]),


which implies,


    A_2p(z;k)B_2p(z;k)=
    𝔗_(1,2)+𝔗_(1,1)(A_2p(z)/B_2p(z))𝔗_(2,2)+𝔗_(2,1)(A_2p(z)/B_2p(z)).


This gives the perturbed Schur function as,


    f^(β_k)(z;k)=
    𝔗_(1,2)+𝔗_(1,1)f(z)𝔗_(2,2)+𝔗_(2,1)f(z).



We next consider a non-constant Schur function of the  form f(z)=cz+d where
|c|+|d|≤1 and have a perturbation α_1↦β_1. Then


    p_1(z,1)   =(α_1-β_1)α̅_0z+(1-α̅_1β_1);
      
    p_1^∗(z,1)=(1-α_1β̅_1)z+(α̅_1-β̅_1)α_0;
    
    q_1(z,1)   =(β_1-α_1)z-(1-α̅_1β_1)α_0;
      
    q_1^∗(z,1)=(β̅_1-α̅_1)-(1-α_1β̅_1)α̅_0z.


The matrix entries are


    τ_(1,1)   =(α_1-β_1)z^2+[(1-β_1α̅_1)-(1-α_1β̅_1)|α_0|^2]z+
      α_0(β̅_1-α̅_1),
    
    τ_(1,2)   =(β_1-α_1)z^2+[(1-α_1β̅_1)α_0-(1-α̅_1β_1)α_0]z+
      (α̅_1-β̅_1)α_0^2,
    
    τ_(2,1)   =(α_1-β_1)(α̅_0)^2z^2+[(1-β_1α̅_1)α̅_0-(1-α_1β̅_1)α̅_0]z+
      (β̅_1-α̅_1),
    
    τ_(2,2)   =(β_1-α_1)α̅_0z^2+[(1-α_1β̅_1)-(1-α̅_1β_1)|α_0|^2]z+
       (α̅_1-β̅_1)α_0.


The transformed Schur function is a rational function given by


    f^(β_1)(z,1)=
    Az^3+Bz^2+Cz+DÂz^3+B̂z^2+Ĉz+D̂,


where


    A   =(α_1-β_1)α̅_0c,
      
    B=(β_1-α_1)(1-α̅_0d)+
        c(1-β_1α̅_1)-
        c|α_0|^2(1-α_1β̅_1),
    
    C   =(1-α_1β̅_1)(α_0-d|α_0|^2)+
       (1-α̅_1β_1)(d-α_0)+
       cα_0(β̅_1-α̅_1),
    
    D   =(β̅_1-α̅_1)(d-α_0)α_0,


and


    Â   =(α_1-β_1)(α̅_0)^2c,
      B̂=(β_1-α_1)(1-α̅_0d)α̅_0+
             c(1-β_1α̅_1)α̅_0-
             c(1-α_1β̅_1)α̅_0,
    Ĉ   =(1-α_1β̅_1)(1-dα̅_0)+
             (1-β_1α̅_1)(dα̅_0-|α_0|^2)+
             c(β̅_1-α̅_1),
    D̂   =(β̅_1-α̅_1)(d-α_0).


This leads to the following easy consequence of Theorem
<ref>.



Let f(z)=cz+α_0, where |c|≤ 1-|α_0|
denote the class of Schur functions.
Then with the perturbation α_1↦β_1,
the resulting Schur function is the rational function given by


    f^(β_1)(z;1)=Az^2+Bz+CÂz^2+B̂z+Ĉ,
       A≠0,  Â≠0.





We now consider an example illustrating the above discussion.





Consider the sequence of Schur parameters
{α_n}_n=0^∞
given by
α_0=1/2 and α_n=2/(2n+1), n≥1.
Then, as in <cit.>,
the Schur function is f(z)=(1+z)/2

with


    A_2m(z)     =12+2z^m+2-2(m+1)z^2+2mz(2m+1)(z-1)^2,
    B_2m(z)     =1+z^m+2+z^m+1-(2m+1)z^2+(2m-1)z(2m+1)(z-1)^2,
    A_2m+1(z)   =z+z^2-(2m+3)z^m+2+(2m+1)z^m+3(2m+1)(z-1)^2,
    B_2m+1(z)   =z^m+12+2z-(m+1)z^m+1+mz^m+2(2m+1)(z-1)^2.


We study the perturbation α_1↦β_1=1/2.
For the transfer matrix 𝔗(z;k),
the following polynomials are required.


    p_1(z)   =z12+23;
      
    p_1^∗(z)=23z+112;
    
    q_1(z)   =-z6-13;
      
    q_1^∗(z)=-z3-16.


The entries of 𝔗(z;k) are


    𝔗_(1,1)   =z^212+z2-112;
      𝔗_(1,2)=-z^26+112;
    𝔗_(2,1)   =z^224-16;
      𝔗_(2,2)=-z^212+z2+112.


Hence, the transformed Schur function using
(<ref>)
is


    f^(1/2)(z;1)=2z^2-3z+5z^2-3z+20.


Observe that f(z) and f^(1/2)(z;1) are analytic in 𝔻 with f(0)=f^(1/2)(0;1) and


    ω(z)=f^-1(f^(1/2)(z;1))=
    3z(z-3)z^2-3z+20,


where ω(z) is analytic in 𝔻 with |ω(z)|<1.
Further, by Schwarz lemma |ω(z)|<|z| for 0<|z|<1 unless ω(z)
is a pure rotation. In such a case the range of f^(1/2)(z;1) is contained in the range of f(z). The function f^(1/2)(z;1) is said to be subordinate to f(z) and written as f^(1/2)(z;1)≺ f(z)
for z∈𝔻
<cit.>.

We plot the ranges of both the Schur functions below.












In Figure <ref>, the outermost circle
is the unit circle while the middle one is the image of
|z|=0.9 under f(z) which is again a
circle with center at 1/2. The innermost figure is the image of
|z|=0.9 under f^(1/2)(z;1).









 §.§ The change in Carathéodory function


Let the Carathéodory function associated with the
perturbed Schur function f^(β_k)(z;k) be
denoted by 𝒞^(β_k)(z;k).
Then, using
(<ref>),
we can write


    𝒞^(β_k)(z;k)   =1+zf^(β_k)(z;k)1-zf^(β_k)(z;k)
       =(𝔗_2,2+z𝔗_1,2)+(𝔗_2,1+z𝔗_1,1)f(z)(𝔗_2,2-z𝔗_1,2)+(𝔗_2,1-z𝔗_1,1)f(z).


Further, using the relation
(<ref>),
we have


    𝒞^(β_k)(z;k)=
    𝒴^-(z)+𝒴^+(z)𝒞(z)𝒲^-(z)+𝒲^+(z)𝒞(z),


where


    𝒴^±(z)   =
    z(𝔗_(2,2)+z𝔗_(1,2))
    ± (𝔗_(2,1)+z𝔗_(1,1))
    𝒲^±(z)   =
    z(𝔗_(2,2)-z𝔗_(1,2))
    ± (𝔗_(2,1)-z𝔗_(1,1)).


As an illustration, for the Schur function f(z)=(1+z)/2,
it is easy to verify that


    𝒞(z)=2+z+z^22-z-z^2    𝒞^(1/2)(z;1)=2z^3-5z^2+7z+20-2z^3+7z^2-13z+20.



We plot these Carathéodory functions below.












In Figures  <ref> and <ref>,
the ranges of both the original and perturbed Carathéodory functions
are plotted for |z|=0.9.
Interestingly, the range of 𝒞(z) is unbounded
(Figure <ref>) which is clear as z=1 is a pole of
𝒞(z).

However 𝒞^1/2(z;1) has simple poles at 5/2 and
(1± i√(15))/2
and hence
with the use of perturbation we are able to make the
range bounded (Figure <ref>).





As shown in
<cit.>,
the sequence
{γ_j}_j=0^∞
satisfying the recurrence relation


    γ_p+1=γ_p-α̅_p1-α_pγ_p,
       p≥0.


where γ_0=1 and α_j's are the Schur parameters
plays an important role in the g-fraction expansion for a special
class of Carathéodory functions.

Let {γ_j^(β_k)} correspond to the perturbed
Carathéodory function 𝒞^(β_k)(z;k).

Since only α_j is perturbed,
it is clear that γ_j remains unchanged for j=0,1,⋯,k.
The first change,
γ_k+1 to γ_k+1^(β_k),
occurs when α_k is replaced by β_k.
Consequently, γ_k+j, j≥2,
change to
γ_k+j^(β_k), j≥2, respectively.

We now show that γ_j
can be expressed as a bilinear transformation
of γ_j^(β_k) for j≥ k+1.





Let {γ_j}_j=0^∞ be the sequence corresponding to
{α_j}_j=0^∞ and {γ_j^(β_k)}_j=0^∞
that to {α_j^(β_k)}_j=0^∞. Then,


    γ_k+j^(β_k)=a̅_k+jγ_k+j-b_k+j-b̅_k+jγ_k+j+a_k+j,
       j≥1,


where



  * a_k+1=1-α̅_kβ_k1-|β_k|^2    
       b_k+1=β̅_̅k̅-α̅_k1-|β_k|^2, (j=1).



  * For j≥2,
      
    (
                        [ a_k+j; b_k+j;       ])=
                      11-|α_k+j-1|^2(
                        [        1  α_k+j-1; α̅_k+j-1        1;          ])
                      (
                        [ a_k+j-1-α̅_k+j-1b̅_k+j-1; b_k+j-1-α̅_k+j-1a̅_k+j-1;                          ]).







Consider first the expression


    a_k+1γ_k+1^(β_k)+b_k+1b̅_k+1γ_k+1^(β_k)+a̅_k+1.

Substituting
γ_k+1^(β_k)=(γ_k-β̅_k)/(1-β_kγ_k)
and the given values of a_k+1 and b_k+1,
it simplifies to


    (1-α̅_kβ_k)(γ_k-β̅_k)+(β̅_k-α̅_k)(1-α_kγ_k)(β_k-α_k)(γ_k-β̅_k)+(1-α_kγ̅_k)(1-β_kγ_k)
    =γ_k(1-|β_k|^2)-α̅_k(1-|β_k|^2)(1-|β_k|^2)-α_kγ_k(1-|β|^2)
    =γ_k+1.


Since |a_k+1|^2-|b_k+1|^2=(1-|α_k|^2)/(1-|β_k|^2)≠0,
(<ref>)
is proved for j=1.

Next, let


    a_k+2γ_k+2^(β_k)+b_k+2b̅_k+2γ_k+2^(β_k)+a̅_k+2
    =(a_k+2-α_k+1b_k+2)γ_k+1^(β_k)+(b_k+2-α̅_k+1a_k+2)(b̅_k+2-α_k+1a̅_k+2)γ_k+1^(β_k)+(a̅_k+2-α̅_k+1b̅_k+2)
    =N(γ_k)D(γ_k).


Substituting first the given values of
a_k+2 and b_k+2,
the numerator becomes


    N(γ_k)=(1-|α_k+1|^2)[(a_k+1-α̅_k+1b̅_k+1)γ_k+1^(β_k)+
                                     (b_k+1-α̅_k+1a̅_k+1)],


and then using,
γ_k+1^(β_k)=(a̅_k+1-b_k+1)/
(-b̅_k+1γ_k+1+a_k+1),


    N(γ_k)=(1-|α_k+1|^2)(|α_k+1|^2-
    |b_k+1|^2)(γ_k+1-α̅_k+1).


With similar calculations, we obtain


    D(γ_k)=(1-|α_k+1|^2)(|α_k+1|^2-
    |b_k+1|^2)(1-α_k+1γ_k+1).


This means


    N(γ_k)D(γ_k)=
    γ_k+1-α̅_k+11-α_k+1γ_k+1=
    γ_k+2,


where
|a_k+2|^2-|b_k+2|^2=|a_k+1|^2-|b_k+1|^2≠0,
thus proving
(<ref>)
for j=2.

The remaining part of the proof
is follows by a simple induction on j.





With the condition |α_p|<1, it is clear that
(<ref>)
gives analytic self-maps of the unit disk.
Similar to the changes in mapping properties obtained as
a consequence of perturbation,
it is expected that
(<ref>)
may lead to interesting results in fractal geometry and complex dynamics.




Since γ_j and γ_j^(β_k) are related by a bilinear
transformation, it is clear that the expressions for a_k+j and
b_k+j, j≥1, are not unique.



It is known that if 𝒞(z) is real for real z, then
the α_p's are real and
γ_p=1, p=0,1,⋯.
Further, it is clear from
(<ref>)
that γ_j^(β_k)=1 whenever γ_j=1.
In this case, the following g-fraction is obtained
for k≥0.


    1-z1+z𝒞^(β_k)(z)=
    11[ ; - ]g_1ω1[ ; - ](1-g_1)g_2ω1[ ; - ]⋯(1-g_k)g_k+1^(β_k)ω1[ ; - ](1-g_k+1^(β_k))g_k+2ω1[ ; - ]⋯


where g_j=(1-α_j-1)/2, j=1,⋯,k,k+2⋯,
g_k+1^(β_k)=(1-β_k)/2
and ω=-4z/(1-z)^2.






§ A CLASS OF PICK FUNCTIONS AND SCHUR FUNCTIONS



Let the Hausdorff sequence {ν_j}_j≥0
with ν_0=1 be given so that there
exists a bounded non-decreasing measure ν on [0,1]
satisfying


    ν_j=∫_0^1σ^jdν(σ),
       j≥0.


By <cit.>, the existence of
dν(σ) is equivalent to the power series


    F(ω)=∑_j≥0ν_jω^j=
    ∫_0^111-σωdν(σ)


having a continued fraction
expansion of the form


    ∫_0^111-σωdν(σ)=
    ν_01[ ; - ](1-g_0)g_1ω1[ ; - ](1-g_1)g_2ω1[ ; - ](1-g_2)g_3ω1[ ; - ]⋯


where ν_0≥0 and 0≤ g_p≤ 1, p≥0.
Such functions F(ω) are analytic in the slit domain
ℂ∖[1,∞)
and belong to the class of Pick functions.
We note that the Pick functions are analytic in the
upper half plane and have a positive imaginary part
<cit.>.


In the next result, we characterize some members of the class of Pick
functions using the gap g-fraction ℱ_2^(a,b,c)(ω).
The proof is similar to that of
<cit.> and
follows from <cit.>, given earlier
as <cit.>.




If a, b, c∈ℝ with -1<a≤ c and 0≤ b≤ c,
then the functions


    ω   ↦F(a+1,b+1;c+1;ω)F(a+1,b;c+1;ω)   ;   ω↦ω F(a+1,b+1;c+1;ω)F(a+1,b;c+1;ω)
    ω   ↦F(a+2,b+1;c+2;ω)F(a+1,b;c+1;ω)   ;   ω↦zF(a+2,b+1;c+2;ω)F(a+1,b;c+1;ω)
    ω   ↦F(a+2,b+1;c+2;ω)F(a+1,b+1;c+1;ω)   ;   ω↦ω F(a+2,b+1;c+2;ω)F(a+1,b+1;c+1;ω)


are analytic in ℂ∖[1,∞) and each
function map both the open unit disk 𝔻 and the
half plane {ω∈ℂ: Re ω<1} univalently
onto domains that are convex in the direction of the
imaginary axis.













We would like to note here that by a domain convex in the direction
of imaginary axis, we mean that every line parallel to the imaginary
axis has either connected or empty intersection with the
corresponding domain
<cit.>,
(see also
<cit.>.
).




With the given restrictions on a, b and c, ℱ_2^(a,b,c)(ω)
has a g-fraction expansion and hence by <cit.>, there
exists a non-decreasing function ν_0:[0,1]↦ [0,1] with a total increase of 1 and


    F(a+1,b+1;c+1;ω)F(a+1,b;c+1;ω)=
    ∫_0^111-σωdν_0(σ),
      ω∈ℂ∖[1,∞),


which implies


    ω F(a+1,b+1;c+1;ω)F(a+1,b;c+1;ω)=
    ∫_0^1ω1-σωdν_0(σ),
      ω∈ℂ∖[1,∞).


Now, if we define


    ν_1(σ)=1k_2∫_0^σρ dν_0(ρ),


where k_2=(a+1)/(c+1)>0, it can be easily seen that
ν_1:[0,1]↦ [0,1] is again a non-decreasing
map with ν_1(1)-ν_1(0)=1.
Further, using the contiguous relation


    F(a+1,b;c;ω)-F(a,b;c;ω)=bcω F(a+1,b+1;c+1;ω),


we obtain


    ω F(a+2,b+1;c+2;ω)F(a+1,b;c+1;ω)=
    ∫_0^1ω1-σωdν_1(σ),
      ω∈ℂ∖[1,∞),


and hence


    F(a+1,b+1;c+1;ω)F(a+1,b;c+1;ω)=
    1+k_2
    ∫_0^1ω1-σωdν_1(σ),
      ω∈ℂ∖[1,∞).


Further, noting that the coefficient of ω in
F(a+2,b+1;c+2;ω)/F(a+1,b;c+1;ω) is
[(b+1)(c-a)]/[(c+1)(c+2)]=k_3+(1-k_3)k_2,
we define


    ν_2(σ)=1k_3+k_2(1-k_3)∫_0^σρ dν_1(ρ),


and find that


    F(a+2,b+1;c+2;ω)F(a+1,b;c+1;ω)=
    1+[k_3+k_2(1-k_3)]
    ∫_0^1ω1-σωdν_2(σ).


Finally from Gauss continued fraction
(<ref>),
we conclude that
F(a+2,b+1;c+2;ω)/F(a+1,b+1;c+1;ω) has a g-fraction
expansion and so there exists a map ν_3:[0,1]↦[0,1]
which is non-decreasing, ν_3(1)-ν_3(0)=1 and


    ω F(a+2,b+1;c+2;ω)F(a+1;b+1;c+1;ω)=
    ∫_0^1ω1-σωdν_3(σ),
      ω∈ℂ∖ [1,∞).


Defining for a<c


    ν_4(σ)=1(1-k_2)k_3∫_0^σρ dν_3(ρ),


so that (1-k_2)k_3>0, and using the fact that the coefficient
of ω in F(a+2,b+1;c+2;ω)/F(a+1,b+1;c+1;ω) is (1-k_2)k_3,
we obtain


    F(a+2,b+1;c+2;ω)F(a+1,b+1;c+1;ω)=
    1+[(1-k_2)k_3]
    ∫_0^1ω1-σωdν_4(σ).


Thus, with ν_j, j=0,1,2,3,4, satisfying
the conditions of
<cit.>,
<cit.>,
the proof of the theorem is completed.














Ratios of Gaussian hypergeometric functions having mapping properties described in Theorem <ref>
are also found in <cit.>
but for the ranges -1≤ a≤ c and 0<b≤ c. Hence for the common range -1<a≤ c and 0<b≤ c, two different ratios of hypergeometric functions belonging to the class of Pick functions can be obtained leading to the expectation of finding more such ratios for every possible range.





It may be noted that the ratio of Gaussian hypergeometric functions in (<ref>)
denoted here as ℱ(z)
has the mapping properties given in Theorem
<ref>,
which is proved in <cit.>.
We now consider its g-fraction expansion with the parameter k_2 missing.

Using the contiguous relation
(<ref>)
and the notations used in Theorems
<ref>
and
<ref>,
it is clear that
ℱ_3^(a,b,c)(ω)=F(a+2,b+1;c+2;ω)/F(a+1,b+1;c+2;ω)
and


    ℋ_3(ω)=
    1-1ℱ_3^(a,b,c)(w)=
    b+1c+2ωF(a+2,b+2;c+3;ω)F(a+2,b+1;c+2;ω).


Then


    h(2;ω)=(1-k_1)ℋ_3(ω)=
    (c-b)(b+1)(c)(c+2)ωF(a+2,b+2;c+3;ω)F(a+2,b+1;c+2;ω)


Then, from Theorem <ref>,


    ℱ(2;ω)
       =11-(1-k_0)k_1ω-
    (1-k_0)k_1ω h(2;ω)[1-(1-k_0)k_1z]h(2;ω)-[1-(1-k_0)k_1z]^2
       =cc-bz-bczh(2;z)c(c-bz)h(2;z)-(c-bz)^2


which implies


    ℱ(2;ω)=
    cc-bω-
    b(b+1)(c-b)c+2ω^2F(a+2,b+2;c+3;ω)F(a+2,b+1;c+2;ω)(c-b)(b+1)(c-bω)c+2ωF(a+2,b+2;c+3;ω)F(a+2,b+1;c+2;ω)-(c-bω)^2


that is ℱ(2;ω) is given as a rational transformation of a new ratio of hypergeometric functions. It may also be noted that
for -1≤ a≤ c and 0<b≤ c,
<cit.>,
both ℱ(ω) and
ℱ(2;ω)
will map both the unit disk 𝔻 and the half plane
{ω∈ℂ:Re ω<1} univalently onto domains that are convex in the direction of the imaginary axis.

As an illustration, we plot both these functions in figures
(<ref>) and (<ref>).













 §.§ A class of Schur functions


From Theorem <ref>
we obtain


    k_2j+1ω1-(1-k_2j+1)k_2j+2ω1-(1-k_2j+2)k_2j+3ω1-⋯   =1-F(a+j,b+j;c+2j;ω)F(a+j+1,b+j;c+2j;ω)
       =b+jc+2jω F(a+j+1,b+j+1;c+2j+1;ω)F(a+j+1,b+j,c+2j;ω)


where the last equality follows from the contiguous relation
(<ref>)
Hence using <cit.>
we get


    1-z21-f_2j(z)1+zf_2j(z)=
    b+jc+2jF(a+j+1,b+j+1;c+2j+1;ω)F(a+j+1,b+j,c+2j;ω),
       j≥1,


where f_n(z) is the Schur function and
ω and z are related as ω=-4z/(1-z)^2.

Similarly, interchanging a and b in
(<ref>)
we obtain


    1-z21-f_2j+1(z)1+zf_2j+1(z)=
    a+j+1c+2j+1F(a+j+2,b+j+1;c+2j+2;ω)F(a+j+1,b+j+1,c+2j+1;ω),
       j≥0,


where ω=-4z/(1-z)^2.

Moreover, using the relation α_j-1=1-2k_j, j≥1,
the related sequence of Schur parameters is given by


    α_j=
    {[     c-2bc+j,   j=2n, n≥0;;   c-2a-1c+j, j=2n+1, n≥1. ].


We note the following particular case. For a=b-1/2 and
c=b, the resulting Schur parameters are
α_j^(b)=-b/(b+j), j≥0.
Such parameters have been considered in
<cit.>
(when b∈ℝ) in the
context of orthogonal polynomials on the unit circle.
These polynomials are known in modern literature as
Szegö polynomials and we suggest the interested readers to
refer <cit.> for further information.

Finally, as an illustration we note that while the Schur function
associated with the parameters
{α_j^(b)}_j≥0 is
f(z)=-1,
that associated with the parameters
{α^(b)_j}_j≥1
is given by


    1-z21-f^(b)(z)1+zf^(b)(z)=
    b+1/2b+1F(b+3/2,b+1;b+2;ω)F(b+1/2,-;-;ω)


where ω=-4z/(1-z)^2.



We remark that in this section, specific illustration of the results given in Section <ref> are discussed leading to characterizing a class of ratio of hypergeometric functions. Similar characterization of functions involving the function ω=-4z/(1-z)^2, given in Section <ref>
may provide some important consequences of such perturbation of g-fractions.







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