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

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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Correspondence to Bertin Diarra, Hamadoun Maïga or Tongobé Mounkoro.

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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

Keywords

  • \(p\)-adic measures
  • moments sequence
  • Laplace transform
  • Stirling numbers of the second kind
  • \(p^\ell\mathbb{Z}_p\)-invariant function
  • congruences
  • identities