Abstract
The subject of this paper is the study of some properties of \(q\)-Stirling numbers of the second kind \(S_q(n,j)\) for \(q\ne 0\) 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 \(S_{\psi,q}(n,j)\) attached to a \(p\)-adic function \(\psi\) that is invariant by \(p^{\ell}\mathbb{Z}_p\). Also, we express the generalized \(q\)-Stirling numbers \(S_{\psi,q}(n,\: j)\) according to generalized Stirling numbers \(S_{\psi}(n,\: j)\).
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Diarra, B., Maïga, H. & Mounkoro, T. Some Identities and Congruences for \(q\)-Stirling Numbers of the Second Kind. P-Adic Num Ultrametr Anal Appl 14, 85–102 (2022). https://doi.org/10.1134/S2070046622020017
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DOI: https://doi.org/10.1134/S2070046622020017