Congruences for the Stirling numbers and associated Stirling numbers
Copyright policy )Congruences for the Stirling numbers and associated Stirling numbers
Let s(n,k) and S(n,k) be the Stirling numbers of the first and second kind, respectively. The author proves that if \(k+n\) is odd, then \[ s(n,k)\equiv 0 (mod\left( \begin{matrix} n\\ 2\end{matrix} \right)),\quad S(n,k)\equiv 0 (mod\left( \begin{matrix} k+1\\ 2\end{matrix} \right)). \] He also shows how to find congruences (mod p), p prime, for the Stirling numbers and associated Stirling numbers, and he illustrates his method by finding the residues for \(p=2,3,5\). This paper extends the work of \textit{L. Carlitz} [Acta Arith. 10, 409-422 (1965; Zbl 0151.026)].
Stirling numbers of the first and second kind, congruences, Fibonacci and Lucas numbers and polynomials and generalizations, Congruences; primitive roots; residue systems, associated Stirling numbers
Stirling numbers of the first and second kind, congruences, Fibonacci and Lucas numbers and polynomials and generalizations, Congruences; primitive roots; residue systems, associated Stirling numbers
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