Interpolating the Derivatives of the Gamma Function

TL;DR

This paper introduces a function that interpolates the derivatives of the Gamma function and analyzes its asymptotic behavior as the interpolation parameter grows large.

Contribution

It defines a new entire function interpolating Gamma derivatives and derives its asymptotic properties for large interpolation parameters.

Findings

Derived asymptotic formulas for the interpolating function as   o +.

Established the entire nature of the interpolating function.

Connected the interpolation to classical Gamma function derivatives.

Abstract

We consider a function $G(\lambda, z)$, entire in $\lambda$, which interpolates the derivatives of the Gamma function in the sense that $G(m, z) = \Gamma^{(m)}(z)$ for any integer $m \geq 0$ and we calculate the asymptotics of $G(\lambda, z)$ as $\lambda \to +\infty$.