[Paper Review] A note on the p-adic log-gamma functions

Taekyun Kim|ArXiv.org|Oct 26, 2007

Advanced Mathematical Identities12 references3 citations

TL;DR

This paper establishes a p-adic q-analogue of the log-gamma function using q-p-adic integrals and derives a Stirling-type series expansion where q-Euler numbers appear as coefficients. The key contribution is Theorem A, which expresses the p-adic q-log-gamma function in terms of [x]_q, logarithmic terms, and q-Euler numbers E_{n+1,q}, generalizing classical log-gamma asymptotics to the p-adic q-setting.

ABSTRACT

In this paper we prove that q-Euler numbers are occured in the coefficients of some stirling type seies for p-adic analytic q-log gamma function

Motivation & Objective

• To develop a p-adic q-analogue of the classical log-gamma function using p-adic q-integrals.
• To derive a Stirling-type asymptotic expansion for the p-adic q-log-gamma function.
• To investigate the role of q-Euler numbers in the coefficients of this expansion.
• To generalize classical log-gamma asymptotic formulas to the p-adic q-setting using q-integer and q-logarithmic functions.

Proposed method

• Define the p-adic q-log-gamma function G_{p,q}(x) as a q-p-adic integral over ℤ_p involving [x+z]_q and log[x+z]_q.
• Use the fermionic q-p-adic measure μ_{-q} to express the integral I_{-q}(f) as a limit of finite sums involving (-q)^x.
• Derive a functional equation for G_{p,q}(x) using the property qI_{-q}(f_1) + I_{-q}(f) = [2]_q f(0), where f_1(x) = f(x+1).
• Expand [x+z]_q(log[x+z]_q - 1) using the identity [x+z]_q = [x]_q + q^x[z]_q and the series (1+x)log(1+x) = x + ∑_{n=1}^∞ (-1)^{n+1}/(n(n+1)) x^{n+1}.
• Apply the integral formula for q-Euler polynomials E_{m,q}(x) and evaluate E_{n,q} = E_{n,q}(0) to extract coefficients.
• Derive Theorem A by combining series expansion with q-integral identities, resulting in a closed-form expression involving E_{n+1,q} and [x]_q terms.

Experimental results

Research questions

• RQ1 How can the classical Stirling-type expansion of the log-gamma function be generalized to the p-adic q-setting?
• RQ2 What role do q-Euler numbers play in the coefficients of p-adic q-log-gamma function expansions?
• RQ3 How do q-p-adic integrals and the μ_{-q} measure contribute to constructing a p-adic q-analogue of the log-gamma function?
• RQ4 What functional equation governs the p-adic q-log-gamma function G_{p,q}(x)?
• RQ5 Can the q-log-gamma function be expressed as a series involving [x]_q, log[x]_q, and q-Euler numbers?

Key findings

• The p-adic q-log-gamma function G_{p,q}(x) satisfies the functional equation qG_{p,q}(x+1) + G_{p,q}(x) = [2]_q ([x]_q (log[x]_q - 1)).
• Theorem A provides an explicit series expansion: G_{p,q}(x) = ([x]_q - q^x / [2]_{q^2}) log[x]_q - [x]_q + ∑_{n=1}^∞ [(-q^x)^{n+1} / (n(n+1))] [E_{n+1,q} / [x]_q^n].
• The coefficients in the Stirling-type series for G_{p,q}(x) are explicitly shown to be q-Euler numbers E_{n+1,q}.
• The expression for G_{p,q}(x) is valid for x ∈ ℂ_p \ ℤ_p and relies on the convergence of the q-p-adic integral under |1-q|_p < p^{-1/(p-1)}.
• The construction generalizes classical log-gamma asymptotics to the p-adic q-setting, with the q-Euler numbers E_{n,q} replacing classical Bernoulli numbers in the expansion.
• The paper establishes a p-adic q-analogue of the log-gamma function that reduces to the classical case as q → 1.