In 1997, Masanobu Kaneko defined poly-Bernoulli numbers, which bear much the same relation to polylogarithms as Berunoulli numbers do to logarithms. In 2008, Chet Brewbaker described a counting problem whose solution can be identified with the poly-Bernoulli numbers with negative index, the lonesum matrices.The main aim of this paper is to give formulae for the number of acyclic orientations of a complete bipartite graph, or of a complete bipartite graph with one edge added or removed.Our formula shows that the number of acyclic orientations of K n 1 ,n 2 is equal to the poly-Bernoulli number B (−n 2 ) n 1. We also give a simple bijective identification of acyclic orientations and lonesum matrices.We make some remarks on the context of our result, which are expanded in another paper.

This paper presents a family of rapidly convergent summation formulas for various finite sums of the form ⌊x⌋ k=0 f (k), where x is a positive real number.

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This paper presents a family of rapidly convergent summation formulas for various finite sums of analytic functions. These summation formulas are obtained by applying a series acceleration transformation involving Stirling numbers of the first kind to the asymptotic, but divergent, expressions for the corresponding sums coming from the Euler-Maclaurin summation formula. While it is well-known that the expressions obtained from the Euler-Maclaurin summation formula diverge, our summation formulas are all very rapidly convergent and thus computationally efficient.

The paper presents a new formula for the characteristic function of the prime numbers in form of a finite double product using only roots of unity.