On some expansions for the Euler Gamma function and the Riemann Zeta function

TL;DR

This paper introduces new series expansions for the Euler Gamma and Riemann Zeta functions using falling factorials, employing advanced combinatorial tools, and analyzes their convergence and numerical properties.

Contribution

It presents novel expansions for these functions based on falling factorials, utilizing combinatorial polynomials and special numbers, with convergence analysis and numerical illustrations.

Findings

New series expansions for Gamma and Zeta functions

Analysis of convergence rates of the series

Numerical examples demonstrating the expansions' effectiveness

Abstract

In the present paper we introduce some expansions, based on the falling factorials, for the Euler Gamma function and the Riemann Zeta function. In the proofs we use the Fa\'a di Bruno formula, Bell polynomials, potential polynomials, Mittag-Leffler polynomials, derivative polynomials and special numbers (Eulerian numbers and Stirling numbers of both kinds). We investigate the rate of convergence of the series and give some numerical examples.