Congruences for Bernoulli, Euler, and Stirling numbers (Q1125378)

The author uses properties of \(p\)-adic integrals and measures to obtain congruences for higher-order Bernoulli polynomials and numbers. The Bernoulli polynomials of order \(w\), denoted by \(B_n^{(w)}(x)\), are defined by the generating function \(t^w(e^t-1)^{-w}e^{xt}\), and, as usual, \(B_n^{(w)}=B_n^{(w)}(0)\) are the corresponding Bernoulli numbers. The results are typically new generalizations of the classical Kummer congruences. For example, it is shown that \[ \frac{B_{m+w}^{(w)}}{(m+1)_w} \equiv \frac{B_{n+w}^{(w)}}{(n+1)_w} \pmod {p^C\mathbb{Z}_p}, \] provided \(0 \leq m \leq n\) and \(m \equiv n \pmod {(p-1)p^a}\), where the notation \((m+1)_w\) stands for \((m+w)!/m!\). Here the exponent \(a\) is assumed to exceed an explicitly given bound \(M\) depending on \(m\) and \(w\), and \(C\) is another explicit number depending on \(a\) and \(M\). Moreover, a lower bound is given for the \(p\)-adic ordinal of \(B_m^{(w)}\). When \(w\) is fixed and \(m \to\infty\), this bound is asymptotically better than what was known previously. Similar results are proved for the polynomial \(B_m^{(w)}(x)\) at any \(p\)-adic integer \(x\). The same arguments are also applied to the study of higher-order Euler polynomials \(H_n^{(w)}(u,x)\) and of weighted Stirling numbers \(S(n,w,x)\) of the second kind. The polynomials \(H_n^{(w)}(u,x)\) are defined by the generating function \((1-u)^w(e^t-u)^{-w}e^{xt}\), where \(u\) is a fixed algebraic number \(\neq 1\) and \(w\) is a \(p\)-adic integer. The numbers \(S(n,w,x)\) (for \(n \geq w\)) have \((e^t-1)^we^{xt}/w!\) as their generating function, with \(w\) again a nonnegative rational integer. The results proved are analogues of those obtained for Bernoulli numbers.