On generalized Stirling numbers and special functions

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.