Series and Product Representations of Gamma, Pseudogamma and Inverse Gamma Functions

TL;DR

This paper develops new series and product representations for the gamma and related functions, introduces a novel pseudogamma function, and proposes a conjecture for the inverse gamma function, advancing analytical tools in special functions.

Contribution

It presents new series and product formulas for gamma and related functions, introduces the $ ext{Lambda}$ pseudogamma function, and conjectures a series for the inverse gamma function.

Findings

New series and product representations for gamma functions

A novel pseudogamma function interpolating factorials and reciprocals

Conjectured series for the inverse gamma function

Abstract

We derive product and series representations of the gamma function using Newton interpolation series. Using these identities, a new formula for the coefficients in the Taylor series of the reciprocal gamma function is found. We also find two new series representations for the Euler-Mascheroni constant, containing only rational terms. After that, we introduce a new pseudogamma function which we call the $\Lambda$ function. This function interpolates the factorial at the positive integers, the reciprocal factorial at the negative integers, and is convergent for the entire real axis. Finally, we conjecture a novel series representation for the principal branch of the inverse gamma function $\text{inv}\Gamma_0(z)$.