Calculation of some determinants using the s-shifted factorial

TL;DR

This paper evaluates determinants involving gamma functions using the s-shifted factorial, a generalization of factorial functions, to compute determinants relevant in random matrix theory and combinatorics.

Contribution

It introduces properties and composition laws of the s-shifted factorial and applies them to evaluate generalized Vandermonde determinants with gamma function elements.

Findings

Derived formulas for determinants with gamma function elements

Extended the s-shifted factorial to complex and negative integers

Provided methods for computing determinants in random matrix contexts

Abstract

Several determinants with gamma functions as elements are evaluated. This kind of determinants are encountered in the computation of the probability density of the determinant of random matrices. The s-shifted factorial is defined as a generalization for non-negative integers of the power function, the rising factorial (or Pochammer's symbol) and the falling factorial. It is a special case of polynomial sequence of the binomial type studied in combinatorics theory. In terms of the gamma function, an extension is defined for negative integers and even complex values. Properties, mainly composition laws and binomial formulae, are given. They are used to evaluate families of generalized Vandermonde determinants with s-shifted factorials as elements, instead of power functions.