A gamma function in two variables

TL;DR

This paper introduces a two-variable gamma function extending the classical gamma function, establishing its properties, functional equations, and asymptotic behavior.

Contribution

It defines a new two-variable gamma function and extends many classical properties and formulas to this broader context.

Findings

The new gamma function $ ext{Ga}(x,z)$ converges to the classical gamma function as $x o 1$.

Functional equations and multiplication formulas are established for $ ext{Ga}(x,z)$.

Analogues of Stirling's formula with asymptotic estimates are derived.

Abstract

We introduce a gamma function $\Ga(x,z)$ in two complex variables which extends the classical gamma function $\Ga(z)$ in the sense that $\lim_{x\to 1}\Ga(x,z)=\Ga(z)$. We will show that many properties which $\Ga(z)$ enjoys extend in a natural way to the function $\Ga(x,z)$. Among other things we shall provide functional equations, a multiplication formula, and analogues of the Stirling formula with asymptotic estimates as consequences.