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Theta hypergeometric integrals

V. P. Spiridonov

TL;DR

This work develops a comprehensive framework for theta hypergeometric integrals built from Jacobi theta functions, bridging to theta hypergeometric series via residue calculus and yielding theta-extensions of the Meijer function in one variable. It constructs and analyzes multivariable elliptic beta integrals for the $C_n$ and $A_n$ root systems, including a new $C_n$ formula and a conjectured $A_n$ integral, and demonstrates rich structures such as very-well-poised/total ellipticity and determinant-based proofs, giving rise to new multivariable elliptic hypergeometric sums and biorthogonality relations. It further relates these integrals to generalized eigenvalue problems, introducing biorthogonal families tied to terminating $_{12}V_{11}$ series and Frenkel–Turaev type sums, with implications for integrable systems and modularity. The results position theta beta integrals as a unifying framework connecting elliptic gamma, double sine, and Jacobi theta theories, and open avenues for further multivariable and geometric generalizations.

Abstract

We define a general class of (multiple) integrals of hypergeometric type associated with the Jacobi theta functions. These integrals are related to theta hypergeometric series through the residue calculus. In the one variable case, we get theta function extensions of the Meijer function. A number of multiple generalizations of the elliptic beta integral [S2] associated with the root systems $A_n$ and $C_n$ is described. Some of the $C_n$-examples were proposed earlier by van Diejen and the author, but other integrals are new. An example of the biorthogonality relations associated with the elliptic beta integrals is considered in detail.

Theta hypergeometric integrals

TL;DR

This work develops a comprehensive framework for theta hypergeometric integrals built from Jacobi theta functions, bridging to theta hypergeometric series via residue calculus and yielding theta-extensions of the Meijer function in one variable. It constructs and analyzes multivariable elliptic beta integrals for the and root systems, including a new formula and a conjectured integral, and demonstrates rich structures such as very-well-poised/total ellipticity and determinant-based proofs, giving rise to new multivariable elliptic hypergeometric sums and biorthogonality relations. It further relates these integrals to generalized eigenvalue problems, introducing biorthogonal families tied to terminating series and Frenkel–Turaev type sums, with implications for integrable systems and modularity. The results position theta beta integrals as a unifying framework connecting elliptic gamma, double sine, and Jacobi theta theories, and open avenues for further multivariable and geometric generalizations.

Abstract

We define a general class of (multiple) integrals of hypergeometric type associated with the Jacobi theta functions. These integrals are related to theta hypergeometric series through the residue calculus. In the one variable case, we get theta function extensions of the Meijer function. A number of multiple generalizations of the elliptic beta integral [S2] associated with the root systems and is described. Some of the -examples were proposed earlier by van Diejen and the author, but other integrals are new. An example of the biorthogonality relations associated with the elliptic beta integrals is considered in detail.

Paper Structure

This paper contains 10 sections, 11 theorems, 170 equations.

Table of Contents

  1. Introduction
  2. A general definition of theta hypergeometric integrals
  3. A theta analog of the Meijer function
  4. Known $C_n$ elliptic beta integrals
  5. A new $C_n$ integration formula
  6. An elliptic beta integral for the $A_n$ root system
  7. Some other $A_n$ integrals
  8. Relations to the generalized eigenvalue problems
  9. Proof of a biorthogonality relation
  10. Integral representations for $_{12}E_{11}$ series

Key Result

Theorem 1

Let five complex parameters $t_m, m=0, \dots,4,$ satisfy the inequalities $|t_m|<1,\; |pq|<|A|,$ where $A\equiv\prod_{r=0}^4t_r.$ Define the elliptic beta integral as the following contour integral: where $\mathbb{T}$ is the positively oriented unit circle, and Then

Theorems & Definitions (23)

  • Theorem 1
  • Definition
  • Theorem 2
  • proof
  • Definition
  • Definition
  • Theorem 3
  • proof
  • Conjecture
  • Theorem 4
  • ...and 13 more