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 \(S_{\psi}(n,\: j)\).

中文翻译：

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$$q$$ 的一些恒等式和同余-第二类斯特林数

摘要

本文的主题是研究 \(q\) - 第二类斯特林数 \(S_q(n,j)\) 的一些性质，对于\ ( q \ne 0\)复数或\(p \) -adic 复数。正如我们所知，在\(p\) -adic 设置中，拉普拉斯变换在一些等差数列的研究中起着重要作用。我们提醒一下\(p\)进数测度的拉普拉斯变换的定义及其与该测度时刻的联系。当\(q\)为非零\(p\ ) 时，借助特定度量，我们为\(q\) -Stirling 数\(S_q(n,j)\)建立了一些恒等式和同余-adic 复数和对于广义\(q\) -第二类斯特林数\(S_{\psi,q}(n,j)\)附加到\(p\) -adic 函数\(\ psi\)是不变的\(p^{\ell}\mathbb{Z}_p\)。此外，我们根据广义斯特林数\ ( S_{\psi}(n,\: j )\)。