[Paper Review] Short proofs of the elliptic beta integrals

V. P. Spiridonov|arXiv (Cornell University)|Aug 26, 2004

Mathematical functions and polynomials9 references4 citations

TL;DR

This paper presents elementary proofs of univariate and multivariate elliptic beta integrals on $A_n$ and $C_n$ root systems using a method inspired by Wilf and Zeilberger's approach to the Askey-Wilson integral. It establishes new unit circle multiple elliptic beta integrals valid for $|q|=1$ and their $q$-hypergeometric degenerations, extending the domain of elliptic hypergeometric functions beyond $|q|,|p|<1$. The key contribution is a systematic, simplified derivation of these integrals via difference equations and residue calculus.

ABSTRACT

We give elementary proofs of the univariate elliptic beta integral with bases $|q|, |p|&lt;1$ and its multiparameter generalizations to integrals on the $A_n$ and $C_n$ root systems. We prove also some new unit circle multiple elliptic beta integrals, which are well defined for $|q|=1$, and their $p o 0$ degenerations.

Motivation & Objective

• To provide elementary, self-contained proofs of the univariate and multivariate elliptic beta integrals on $A_n$ and $C_n$ root systems, extending beyond previous complex analytic methods.
• To construct and rigorously prove new multiple elliptic beta integrals defined on the unit circle ($|q|=1$), which are not accessible via standard $|q|<1$ techniques.
• To derive $q$-hypergeometric degenerations of these unit circle integrals, establishing connections to known $q$-special functions.
• To unify and simplify existing proofs of elliptic beta integrals using difference equations and residue calculus, inspired by Wilf and Zeilberger's method.
• To extend the domain of elliptic hypergeometric functions to modular parameters ($|q|=1$) via the modified elliptic gamma function $G(u;\boldsymbol{\omega})$.

Proposed method

• Adopts a difference equation approach based on Wilf and Zeilberger's proof technique for the Askey-Wilson integral, applied to kernel functions of elliptic beta integrals.
• Uses the modified elliptic gamma function $G(u;\boldsymbol{\omega})$ defined for $|q|=1$, which satisfies shift equations involving theta functions and is modular-invariant.
• Applies residue calculus to show that integrals are independent of parameters after contour shifts, proving constancy via analytic continuation.
• Derives key difference equations for kernel functions under $\omega_1$ and $\omega_2$ shifts, showing that the integral remains invariant under lattice translations.
• Performs $p\to 0$ limits to obtain $q$-hypergeometric degenerations, recovering known $q$-integrals from the elliptic case.
• Employs exponential parametrization $q = e^{2\pi i \omega_1/\omega_2}$, $p = e^{2\pi i \omega_3/\omega_2}$ to define the modified gamma function and ensure convergence on the unit circle.

Experimental results

Research questions

• RQ1 Can elementary proofs be constructed for the $A_n$ and $C_n$ elliptic beta integrals using difference equations and residue calculus?
• RQ2 What are the conditions under which multiple elliptic beta integrals remain well-defined when $|q|=1$?
• RQ3 How can the $p\to 0$ limit of the unit circle integrals recover known $q$-hypergeometric identities?
• RQ4 What is the role of the modified elliptic gamma function $G(u;\boldsymbol{\omega})$ in extending the domain of elliptic integrals to the unit circle?
• RQ5 Can the invariance of the integral under lattice shifts $g_m \to g_m + \omega_1, \omega_2$ be used to prove constancy and derive exact evaluation?

Key findings

• The univariate elliptic beta integral is proven via elementary difference equations and residue calculus, establishing its validity for $|q|,|p|<1$.
• A new multiparameter $C_n$ elliptic beta integral is derived and proven using the same elementary method, matching known results from Rains.
• A new multiparameter $A_n$ elliptic beta integral is constructed and rigorously proven, with the integral shown to be independent of parameters via contour shifts and residue analysis.
• The paper constructs and proves new unit circle multiple elliptic beta integrals valid for $|q|=1$, using the modified elliptic gamma function $G(u;\boldsymbol{\omega})$.
• The $q$-reduction of the unit circle $A_n$ integral yields a $q$-hypergeometric identity: $\int_{\mathbb{L}^n} \delta(u,g,h;\omega;A_n) \frac{du_1}{\omega_2}\cdots\frac{du_n}{\omega_2} = (-1)^n(n+1)!\frac{(\tilde{q};\tilde{q})_\infty^n}{(q;q)_\infty^n}$.
• The integral is shown to be independent of parameters $g_m$ and $h_m$ through iterative contour shifts and residue calculus, confirming its constancy and exact evaluation.