Congruences for degenerate number sequences.
Following Carlitz, the degenerate Stirling numbers \(S(n,k| \lambda)\) of the second kind are implicitly defined via the relations \[ (x| \lambda)_n=\sum_{k=0}^n S(n,k| \lambda)(x| 1)_k, \] where \((x| \lambda)_n=x(x-\lambda)\dots(x-(n-1)\lambda)\). In a similar vein, the degenerate Eulerian polynomials \(A_n(\lambda,x)\) are defined as \[ \sum_{k=0}^{\infty} (k| \lambda)_nx^k={{A_n(\lambda,x)}\over {(1-x)^{n+1}}} \] for \(x\neq 1\). The usual Stirling numbers and Eulerian polynomials are obtained when \(\lambda=0\). The main results of this paper are the two Theorems from Section 1. The first one asserts, among other things, that if \(\lambda\in {\mathbb Z}_p^*\), \(\mu=1/\lambda\), \(h\in {\mathbb Z}_pT-1\) and if one defines \(\alpha_n(\lambda)\) implicitly by \[ h((1+\lambda t)^{\mu})=\sum_{n=0}^{\infty} \alpha_n(\lambda){{t^n}\over {n!}}, \] then \(\alpha_n(\lambda)\in {\mathbb Z}_p\). Here, \({\mathbb Z}_p\) denotes the ring of \(p\)-adic integers. The second Theorem asserts that with the above notations, if \(\lambda\in p{\mathbb Z}_p\) and \(m\equiv n\pmod{\varphi(q)p^a}\), where \(a\geq 0\), then \(\alpha_m(\lambda)\equiv \alpha_n(\lambda)\pmod {p^A {\mathbb Z}_p}\), where \(A=\min \{m,n,a+1\}\), whereas if \(\lambda\in {\mathbb Z}_p^{\times}\), then for all \(m\geq 0\) one has \(\alpha_m(\lambda) \equiv 0\pmod {m!{\mathbb Z}_p}\). Here, \(q=p\) if \(p\) is odd and \(q=4\) otherwise. In Section 3, the above Theorems are applied to a very general class of Stirling numbers \(S(n,k;\alpha,\beta,r)\) defined by the relations \[ (x| \alpha)_n=\sum_{k=0}^n S(n,k;\alpha,\beta,r)(x-r| \beta)_k. \] For example, assume that \(r\in {\mathbb Z}_p\), \(\beta\in {\mathbb Z}_p^{\times}\). If \(\alpha\in p{\mathbb Z}_p\) then \[ k!S(m,k;\alpha,\beta,r)\equiv k!S(n,k;\alpha,\beta,r)\pmod {p^A {\mathbb Z}_p} \] holds with \(A=\min\{m,n,a+1\}\) whenever \(m\equiv n\pmod {\varphi(q)p^a}\), while if \(\alpha\in {\mathbb Z}_p^{\times}\), then \[ k!S(m,k;\alpha,\beta,r)\equiv 0\pmod {m!{\mathbb Z}_p} \] holds for all \(m\geq 0\). Similar congruences for degenerate Eulerian polynomials are obtained in Section 4.
scientific article; zbMATH DE number 1974998
scientific article