Elliptic Selberg integrals

By Giovanni Felder, L. Stevens, Alexander Varchenko Eth Zurich, U. Hill. 2001

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B p (α, β, γ) = ∆p p j=1 t j α−1 (1 − t j) β−1 0≤j<k≤1 (t j − t k) 2γ , where ∆ p = {t ∈ R p | 0 ≤ t p ≤ · · · ≤ t 1 ≤ 1}. The Selberg integral is a generalization of the beta function. It can be calculated explicitly, B p (α, β, γ) = 1 p! p−1 j=0 Γ(1 + γ + jγ)Γ(α + jγ)Γ(β + jγ) Γ(1 + γ)Γ(α + β + (p + j − 1)γ). In this paper, we present an elliptic version of the Selberg integral. Let ϑ 1 (t, τ) be the first Jacobi theta function [WW], ϑ 1 (t, τ) = − j∈Z e πi(j+ 1 2) 2 τ +2πi(j+ 1 2)(t+ 1 2). Introduce special functions σ λ (t, τ) = ϑ 1 (λ − t, τ)ϑ ′ 1 (0, τ) ϑ 1 (λ, τ)ϑ 1 (t, τ) , ρ(t, τ) = ϑ ′ 1 (t, τ) ϑ 1 (t, τ) , E(t, τ) = ϑ 1 (t, τ) ϑ ′ 1 (0, τ). Here ′ denotes the derivative with respect to the first argument. Let κ ≥ 2 be an integer. The theta functions θ κ,m (λ, τ) = j∈Z e 2πiκ(j+ m 2κ) 2 τ +2πiκ(j+ m 2κ)λ , m ∈ Z/2κZ, form a basis of the theta functions of level κ. For a positive integer p, the elliptic Selberg integral I p (λ, τ) is the integral, I p (λ, τ) = J p (λ, τ) + (−1) p+1 J p (−λ, τ),

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b0502f972d0eb530b10f493e51adec03c4183ad5

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