Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to $\pi^{-1}$

TL;DR

This paper introduces two new series expansions for the logarithm of the gamma function involving Stirling numbers, which have rational coefficients for specific arguments related to pi inverse, and discusses their properties, convergence, and explicit formulas.

Contribution

The paper presents novel series involving Stirling numbers for the gamma function's logarithm with rational coefficients at special arguments, extending known expansions and providing convergence and asymptotic analysis.

Findings

Series converge uniformly at a rate depending on the order of the polygamma function.

Explicit rational coefficient expansions for key gamma and polygamma values at pi inverse.

Sharp bounds and asymptotics for related combinatorial numbers.

Abstract

In this paper, two new series for the logarithm of the $\Gamma$-function are presented and studied. Their polygamma analogs are also obtained and discussed. These series involve the Stirling numbers of the first kind and have the property to contain only rational coefficients for certain arguments related to $\pi^{-1}$. In particular, for any value of the form $\ln\Gamma(\frac{1}{2}n\pm\alpha\pi^{-1})$ and $\Psi_k(\frac{1}{2}n\pm\alpha\pi^{-1})$, where $\Psi_k$ stands for the $k$th polygamma function, $\alpha$ is positive rational greater than $\frac{1}{6}\pi$, $n$ is integer and $k$ is non-negative integer, these series have rational terms only. In the specified zones of convergence, derived series converge uniformly at the same rate as $\sum(n\ln^m\!n)^{-2}$, where $m=1, 2, 3,\ldots$\,, depending on the order of the polygamma function. Explicit expansions into the series with rational coefficients are given for the most attracting values, such as $\ln\Gamma(\pi^{-1})$, $\ln\Gamma(2\pi^{-1})$, $\ln\Gamma(\tfrac{1}{2}+\pi^{-1})$, $\Psi(\pi^{-1})$, $\Psi(\tfrac{1}{2}+\pi^{-1})$ and $\Psi_k(\pi^{-1})$. Besides, in this article, the reader will also find a number of other series involving Stirling numbers, Gregory's coefficients (logarithmic numbers, also known as Bernoulli numbers of the second kind), Cauchy numbers and generalized Bernoulli numbers. Finally, several estimations and full asymptotics for Gregory's coefficients, for Cauchy numbers, for certain generalized Bernoulli numbers and for certain sums with the Stirling numbers are obtained. In particular, these include sharp bounds for Gregory's coefficients and for the Cauchy numbers of the second kind.