On generalized Stirling numbers and special functions

TL;DR

This paper introduces a new generalization of Stirling numbers, explores their properties, and applies them to approximate zeta values, establishing connections with special functions and extending to q-analogues.

Contribution

It presents a novel generalization of Stirling numbers, analyzes their properties, and links them to various special functions and q-analogues, advancing combinatorics and number theory.

Findings

New generalized Stirling numbers with explicit properties

Effective approximation of Riemann zeta values using these numbers

Connections established between generalized Stirling numbers, special functions, and q-analogues

Abstract

We introduce a new generalization of Stirling numbers of the second kind and analyze their properties, including generating functions, integral representations, and recurrence relations. These numbers are used to approximate Riemann zeta values by rationals with exponentially decreasing error. We establish connections with Hurwitz zeta functions, polylogarithms, harmonic sums, and multiple sums. Finally, we extend our study to q-Stirling numbers, linking them to q-hypergeometric functions and a q-zeta function, revealing new insights in combinatorics and number theory.