A Property of the Gamma Function at its Singularities

TL;DR

This paper investigates specific limits of the Gamma function at its singularities, revealing new identities that deepen understanding of its behavior at nonpositive integers.

Contribution

It derives new limit identities involving the Gamma function at its singularities using Euler and Gauss identities.

Findings

Limits of Gamma function ratios at singularities are explicitly calculated.

New fundamental identities of the Gamma function are established.

Results enhance the mathematical understanding of Gamma function behavior at poles.

Abstract

The singularities of the $\Gamma$ function, a meromorphic function on the complex plane, are known to occur at the nonpositive integers. We show, using Euler and Gauss identities, that for all positive integers $n$ and $k$, $$ \lim_{z\rightarrow 0} \frac{\Gamma(nz)}{\Gamma(z)} = \frac 1 n; \hspace{0.4in} \lim_{z\rightarrow -k} \frac{\Gamma(nz)}{\Gamma(z)} = \f{(-1)^{k}\ \Gamma(k)}{n^2\ \Gamma(nk)}.$$ The above relations add to the list of the known fundamental Gamma function identities.