Elliptic hypergeometric functions

TL;DR

This thesis develops a comprehensive theory of elliptic hypergeometric functions, introducing new transcendental functions, integrals, and their connections to orthogonal polynomials and difference equations.

Contribution

It presents a general scheme for constructing special functions via spectral transformations, leading to the discovery of elliptic hypergeometric functions and integrals, including the elliptic beta integral.

Findings

Construction of elliptic hypergeometric functions from integrals

Introduction of the elliptic beta integral as a fundamental building block

Development of a unifying framework for classical and elliptic hypergeometric functions

Abstract

This is author's Habilitation Thesis (Dr. Sci. dissertation) submitted at the beginning of September 2004. It is written in Russian and is posted due to the continuing requests for the manuscript. The content: 1. Introduction, 2. Nonlinear chains with the discrete time and their self-similar solutions, 3. General theory of theta hypergeometric series, 4. Theta hypergeometric integrals, 5. Biorthogonal functions, 6. Elliptic hypergeometric functions with |q|=1, 7. Conclusion, 8. References. It contains an outline of a general heuristic scheme for building univariate special functions through self-similar reductions of spectral transformation chains, which allowed construction of the differential-difference q-Painleve equations, as well as of the most general known set of elliptic biorthogonal functions comprising all classical orthogonal polynomials and biorthogonal rational functions. One of the key results of the thesis consists in the discovery of genuinely transcendental elliptic hypergeometric functions determined by the elliptic hypergeometric integrals. The whole theory of such integrals can be built from the univariate elliptic beta integral -- the most complicated known definite integral with exact evaluation, which generalizes the ordinary binomial theorem and its q-extension, Euler's beta integral, the measure for Askey-Wilson polynomials, and many other previously established results on ordinary and q-hypergeometric functions.