Multiple Gamma Functions and Multiple $q$-Gamma Functions

Abstract

We give an asymptotic expansion (the higher Stirling formula) and an infinite product representation (the Weierstrass canonical product representation) of the Vigneras multiple gamma functions by considering the classical limit of the multiple #-gamma functions.

q-deformed Painlevéτfunction andq-deformed conformal blocks

“…In the paper we use only N = 1, 2 q-Pochhammer symbols. Introduce trigonometric (or q-) Gamma function and trigonometric (or q-) Barnes G function according to [33] Bilinear relations on q-deformed conformal blocks exist not only in case q 1 = q −1 , q 2 = q. In order to write the conjecture we introduce bilinear combination…”

“…where σ = log u 2 log q . It follows form [33,Prop. 3.1] that log G(x; q) ∼ − log(1 − q)x 2 /2, Re(x) → +∞ (this is the first term in the sum in loc.…”

“…(A.6) From (A.2) we have Γ(u + 1; q) = [u] q Γ(u; q), where [u] q = 1 − q u 1 − q (A.7)G(u + 1; q) = Γ(u; q)G(u; q). (A.8) Function [u] q is called q-number.We use part of the Theorem 4.4 from[33] As q → 1 − Γ(u; q) and G(u; q) converge to Γ(u) and G(u). Convergence is uniform on any compact set in the domain C\Z ≤0 .Introduce elliptic Gamma function (see e.g [30]…”

“…We use part of the Theorem 4.4 from [33] Theorem A.1. As q → 1 − Γ(u; q) and G(u; q) converge to Γ(u) and G(u).…”

New asymptotic expansions and Padé approximants related to the triple gamma function

“…It can be observed from the literature that the n plays a vital role in analytic number theory, approximation theory, mathematical physics, and several branches of science and engineering [7]. n satisfies the following recurrence relations [17]:…”