On Certain Positive Semidefinite Matrices of Special Functions

TL;DR

This paper explores how special functions like Gamma, Beta, and hypergeometric functions relate to positive definite matrices derived from their Fourier or Laplace transforms of positive measures, illustrating their mathematical properties.

Contribution

It provides explicit examples of positive definite matrices associated with various well-known special functions, highlighting their connection to positive measures.

Findings

Demonstrates positive definiteness of matrices from special functions

Connects special functions to Fourier and Laplace transforms of positive measures

Provides explicit matrix examples for Gamma, Beta, hypergeometric, and other functions

Abstract

Special functions are often defined as a Fourier or Laplace transform of a positive measure, and the positivity of the measure manifests as positive definiteness of certain matrices. The purpose of this expository note is to give a sample of such positive definite matrices to demonstrate this connection for some well-known special functions such as Gamma, Beta, hypergeometric, theta, elliptic, zeta and basic hypergeometric functions.