Some Identities and Congruences for q-Stirling Numbers of the Second Kind

The subject of this paper is the study of some properties of q-Stirling numbers of the second kind S-q(n, j) for q not equal 0 a complex or a p-adic complex number. In the p-adic setting, as we known, the Laplace transform plays an important role in the study of some arithmetic sequences. We remind the definition of the Laplace transform of a p-adic measure and its link with the moment of this measure. With the aid of a specific measure we establish some identities and congruences for the q-Stirling numbers S-q(n, j) when q is a non zero p-adic complex number and for the generalized q-Stirling numbers of the second kind S-psi,S-q (n, j) attached to a p-adic function psi that is invariant by p(l)Z(p). Also, we express the generalized q-Stirling numbers S-psi,S-q(n, j) according to generalized Stirling numbers S-psi(n, j).