On some congruences for the Bell numbers and for the Stirling numbers
Two congruences modulo a prime p for the Bell numbers B(n) and for the Stirling numbers S(n,k) (n\(\geq 0\), \(k\geq 1)\) of the second kind are shown. These numbers are defined by \[ \exp (e^ t- 1)=\sum^{\infty}_{n=0}B(n)t^ n/n!,\quad (e^ t-1)^ k/k!=\sum^{\infty}_{n=0}S(n,k)t^ n/n!. \] The mentioned congruences are the following: (A) \(B(p^ n)\equiv (B(p)-1)n+1\) (mod p) for \(n\geq 1.\) (B) S(m,k)\(\equiv S(n,k)\) (mod \(p^{N(k)})\), where \(0\leq m\leq n\), \(m\equiv n\) \((mod(p-1)p^ e)\), \(e\geq 0\), \(N(k)=Min(m,e+1)\), and it is assumed \(N(k)>0.\) The proofs of these congruences make use of the elementary properties of p-adic integers.
scientific article; zbMATH DE number 4214109