Recursion Rules for the Hypergeometric Zeta Functions

TL;DR

This paper explores the properties of the hypergeometric zeta function, including its entire nature, factorization, recurrences, associated polynomials, and probabilistic interpretations related to Beta distributions.

Contribution

It introduces recursion rules for the hypergeometric zeta function and characterizes related polynomials and distributions, expanding understanding of its mathematical structure.

Findings

Hypergeometric zeta function is entire of order 1.

Provides linear and quadratic recurrences for the function.

Characterizes associated polynomials as Appell polynomials.

Abstract

The hypergeometric zeta function is defined in terms of the zeros of the Kummer function M(a, a + b; z). It is established that this function is an entire function of order 1. The classical factorization theorem of Hadamard gives an expression as an infinite product. This provides linear and quadratic recurrences for the hypergeometric zeta function. A family of associated polynomials is characterized as Appell polynomials and the underlying distribution is given explicitly in terms of the zeros of the associated hypergeometric function. These properties are also given a probabilistic interpretation in the framework of Beta distributions.