On generalized Stirling numbers and special functions
Kamel Mezlini, Tahar Moumni, Najib Ouled Azaiez
TL;DR
The paper introduces a new generalized family of Stirling numbers of the second kind, $S_n^p$, defined by $S_n^p = \frac{(-1)^n}{n!}\sum_{k=0}^{n}\frac{(-1)^k\binom{n}{k}}{(k+1)^p}$, and develops their horizontal generating function, integral representation, and recurrence. It establishes deep connections to the Riemann and Hurwitz zeta functions, polylogarithms, harmonic sums, and multiple sums, enabling exponential-rate rational approximations to zeta values and providing several hypergeometric and Bell-polynomial representations. The study further extends to a $q$-deformation, defining generalized $q$-Stirling numbers $S_q(n,p)$ and a generalized $q$-zeta function, with generating functions, $q$-hypergeometric expressions, and a $q$-zeta framework that parallels the classical theory. Overall, the work contributes new algebraic tools for number theory and combinatorics, linking special functions, zeta values, and $q$-calculus in a unified setting and offering practical rational approximants for zeta evaluations.
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.