Treffer: Universal Kummer congruences mod prime powers

Let \(c_1,c_2,\dots\) be indeterminates over \(\mathbb Q\) and let \(F(t) = t + \sum_{i=1}^\infty c_it^{i+1}/(i+1)\). The universal Bernoulli numbers \(\widehat B_n\) are elements of \({\mathbb Q}[c_1,\dots,c_n]\) defined by \(t/G(t) = \sum_{n=0}^\infty\widehat B_nt^n/n!\), where \(G(t)\) is the inverse of \(F(t)\) with respect to the operation of composition of formal power series. Let \(p\) be an odd prime and assume \(m \not\equiv 0,1 \!\!\pmod {p-1}\). The author proves the ``Kummer congruences'' \[ \widehat B_n/n \equiv c_{p-1}^l\widehat B_m/m \pmod {p^{N+1}{\mathbb Z}_p[c_1,\dots,c_n]} \] for \(N \geq 0\), \(l \geq 1\) and \(n = m+l(p-1)\), provided \(p^N\mid l\) and \(m \geq N+2\). He also proves that a similar result, with an additional (more complicated) term on the right hand side of the congruence, holds true for \(m \equiv 1 \!\!\pmod {p-1}\). These results generalize the Kummer congruences mod \(p\) proven previously by the author [Int. Math. J. 1, No. 1, 53--63 (2002; Zbl 0984.11012)]. The present proof is independent of the previous one and uses only elementary number theory. The result also contains the corresponding Kummer congruences for ordinary Bernoulli numbers as special cases. On the other hand it seems that the two other known types of usual Kummer congruences, i.e., the one with Euler factors and the other with delta operators, have no counterpart in this general setting.