Kummer congruences for universal Bernoulli numbers and related congruences for poly-Bernoulli numbers (Q2720361)

Let \(F(t)=t+\sum_{i=1}^\infty c_i\left(t^{i+1}/(i+1)\right)\), where \(c_1,c_2,\dots\) are indeterminates over \(\mathbb Q\). Universal Bernoulli numbers \(\widehat B_n\in\mathbb Q[c_1,c_2,\dots]\) are defined by the generating function \(t/G(t)\), where \(G(t)\) is the compositional inverse of \(F(t)\). The author proves a Kummer congruence mod \(p\) relating \(\widehat B_n/n\) to \(\widehat B_r/r\) for \(n\equiv r\pmod{p-1}\), \(1\leq r<p-1\). This extends and simplifies his previous work [\textit{A. Adelberg} J. Number Theory 84, 119-135 (2000; Zbl 0971.11004)] in which the result was proved under the restriction \(r\neq 1\). NEWLINENEWLINENEWLINEThe author also studies universal poly-Bernoulli numbers \(\widehat B_{n,k}\) whose generating functions are \(Li_k(G(t))/G(t)\). Here \(k\in\mathbb Z\) and \(Li_k(t)\) denotes the polylogarithm function of index \(k\). He proves universal von Staudt type congruences for \(\widehat B_{n,k}\) which strengthen and simplify known congruences for classical poly-Bernoulli numbers; see, e.g., the article by \textit{T. Arakawa} and \textit{M. Kaneko} [Comment. Math. Univ. St. Pauli 48, No. 2, 159-167 (1999; Zbl 0994.11009)].