Title: Iterated Elliptic and Hypergeometric Integrals for Feynman Diagrams,
Abstract: We calculate 3-loop master integrals for heavy quark correlators and the
3-loop QCD corrections to the $\rho$-parameter. They obey non-factorizing
differential equations of second order with more than three singularities,
which cannot be factorized in Mellin-$N$ space either. The solution of the
homogeneous equations is possible in terms of convergent close integer power
series as $_2F_1$ Gau\ss{} hypergeometric functions at rational argument. In
some cases, integrals of this type can be mapped to complete elliptic integrals
at rational argument. This class of functions appears to be the next one
arising in the calculation of more complicated Feynman integrals following the
harmonic polylogarithms, generalized polylogarithms, cyclotomic harmonic
polylogarithms, square-root valued iterated integrals, and combinations
thereof, which appear in simpler cases. The inhomogeneous solution of the
corresponding differential equations can be given in terms of iterative
integrals, where the new innermost letter itself is not an iterative integral.
A new class of iterative integrals is introduced containing letters in which
(multiple) definite integrals appear as factors. For the elliptic case, we also
derive the solution in terms of integrals over modular functions and also
modular forms, using $q$-product and series representations implied by Jacobi's
$\vartheta_i$ functions and Dedekind's $\eta$-function. The corresponding
representations can be traced back to polynomials out of Lambert--Eisenstein
series, having representations also as elliptic polylogarithms, a $q$-factorial
$1/\eta^k(\tau)$, logarithms and polylogarithms of $q$ and their $q$-integrals.
Due to the specific form of the physical variable $x(q)$ for different
processes, different representations do usually appear. Numerical results are
also presented.