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Essays on the theory of elliptic hypergeometric functions

V. P. Spiridonov

TL;DR

This survey consolidates the development of elliptic hypergeometric functions built from the elliptic gamma function, presenting the elliptic beta integral as the umbrella univariate exact integral that encompasses q- and Euler-beta degenerations. It introduces the V-function as an elliptic Gauss analogue tied to the E7 root system and derives the elliptic hypergeometric equation, revealing deep symmetry structures and links to the Sklyanin algebra. The work extends to rich multivariate integral and series theories on root systems C_n and A_n, develops Bailey-type transformation machinery, and constructs biorthogonal systems with both discrete and continuous facets, all while mapping out degenerations, determinant identities, and determinant-based transformations. Together, these results forge a comprehensive framework with broad implications for special functions, exactly solvable models, and discrete integrable systems. The paper also outlines important open problems in convergence, higher-genus extensions, and multivariate classifications, signaling fruitful directions for future research.

Abstract

We give a brief review of the main results of the theory of elliptic hypergeometric functions -- a new class of special functions of mathematical physics. We prove the most general univariate exact integration formula generalizing Euler's beta integral, which is called the elliptic beta integral. An elliptic analogue of the Gauss hypergeometric function is constructed together with the elliptic hypergeometric equation for it. Biorthogonality relations for this function and its particular subcases are described. We list known elliptic beta integrals on root systems and consider symmetry transformations for the corresponding elliptic hypergeometric functions of the higher order.

Essays on the theory of elliptic hypergeometric functions

TL;DR

This survey consolidates the development of elliptic hypergeometric functions built from the elliptic gamma function, presenting the elliptic beta integral as the umbrella univariate exact integral that encompasses q- and Euler-beta degenerations. It introduces the V-function as an elliptic Gauss analogue tied to the E7 root system and derives the elliptic hypergeometric equation, revealing deep symmetry structures and links to the Sklyanin algebra. The work extends to rich multivariate integral and series theories on root systems C_n and A_n, develops Bailey-type transformation machinery, and constructs biorthogonal systems with both discrete and continuous facets, all while mapping out degenerations, determinant identities, and determinant-based transformations. Together, these results forge a comprehensive framework with broad implications for special functions, exactly solvable models, and discrete integrable systems. The paper also outlines important open problems in convergence, higher-genus extensions, and multivariate classifications, signaling fruitful directions for future research.

Abstract

We give a brief review of the main results of the theory of elliptic hypergeometric functions -- a new class of special functions of mathematical physics. We prove the most general univariate exact integration formula generalizing Euler's beta integral, which is called the elliptic beta integral. An elliptic analogue of the Gauss hypergeometric function is constructed together with the elliptic hypergeometric equation for it. Biorthogonality relations for this function and its particular subcases are described. We list known elliptic beta integrals on root systems and consider symmetry transformations for the corresponding elliptic hypergeometric functions of the higher order.

Paper Structure

This paper contains 26 sections, 16 theorems, 279 equations.

Table of Contents

  1. Introduction
  2. Generalized gamma functions
  3. The Barnes multiple gamma function
  4. The elliptic gamma functions
  5. The elliptic beta integral
  6. General elliptic hypergeometric series and integrals
  7. An elliptic analogue of the Meijer function
  8. Well poised and very-well poised integrals
  9. Series
  10. An elliptic analogue of the Gauss hypergeometric function
  11. Definition of the $V$-function and a connection with the root system $E_7$
  12. The elliptic hypergeometric equation
  13. Chains of symmetry transformations for functions
  14. Biorthogonal functions of the hypergeometric type
  15. Discrete biorthogonal functions with the continuous measure
  16. ...and 11 more sections

Key Result

Theorem 1

We consider six complex parameters $t_j,\; j=1,\ldots,6$, and two base variables $p$ and $q$ satisfying the constraints $|p|, |q|, |t_j|<1$ and $\prod_{j=1}^6 t_j=pq$ (the balancing condition). Then the following equality is true where $\mathbb T$ denotes the unit circle with positive orientation and

Theorems & Definitions (25)

  • Theorem 1
  • proof
  • Corollary 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • Theorem 6
  • ...and 15 more