TL;DR
This paper introduces q-deformed Stirling numbers of both kinds, explores their relations with classical special numbers and functions, and establishes new recurrence relations and interpolation functions.
Contribution
It presents the first definitions of q-deformed Stirling numbers, explores their relations with q-Bernoulli and q-Bell numbers, and derives new recurrence and interpolation formulas.
Findings
Relations between q-deformed Stirling numbers and Riemann zeta function established.
New recurrence relations for q-deformed Stirling numbers derived.
A novel interpolating function for q-deformed Stirling numbers at negative integers introduced.
Abstract
The purpose of this article is to introduce q-deformed Stirling numbers of the first and second kinds. Relations between these numbers, Riemann zeta function and q-Bernoulli numbers of higher order are given. Some relations related to the classical Stirling numbers and Bernoulli numbers of higher order are found. By using derivative operator to the generating function of the q-deformed Stirling numbers of the second kinds, a new function is defined which interpolates the q-deformed Stirling numbers of the second kinds at negative integers. The recurrence relations of the Stirling numbers of the first and second kind are given. In addition, relation between q-deformed Stirling numbers and q-Bell numbers is obtained.
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Taxonomy
TopicsAdvanced Mathematical Identities · Advanced Combinatorial Mathematics · Mathematical functions and polynomials