Generalized Stirling Numbers I

Abstract

We consider generalized Stirling numbers of the second kind , , where $a,b,{\alpha }_s,{\beta }_s$ are complex numbers, and $r,p,r_s,p_s$ are non-negative integers given, $s=2,\dots ,L$. (The case $a=1,b=0,r=1,r_s{p}_s=0$, corresponds to the standard Stirling numbers $S\left(p,k\right)$.) The numbers are connected with a generalization of Eulerian numbers and polynomials we studied in previous works. This link allows us to propose (first, and then to prove, specially in the case $r=r_s=1$) several results involving our generalized Stirling numbers, including several families of new recurrences for Stirling numbers of the second kind. In a future work we consider the recurrence and the differential operator associated to the numbers .

Cited by in corpus (1)

• On bi-variate poly-Bernoulli polynomials