Abstract. We prove upper and lower bounds for the complementary incomplete gamma function Γ(a, z) with complex parameters a and z . Our bounds are refined within the circular hyperboloid of one sheet {(a, z) : |z| > c|a − 1|} with a real and z complex. Our results show that within the hyperboloid, |Γ(a, z)| is of order |z| a−1 e − Re(z) , and extends an upper estimate of Natalini and Palumbo to complex values of z . (2000): 33B20. Mathematics subject classification

No abstract

Duality relations are deduced for tails of multiple-zeta values using elementary methods. These formulas extend the classical duality formulas for multiple-zeta values.

A standard method for computing values of Bessel functions has been to use the well-known ascending series for small argument, and to use an asymptotic series for large argument; with the choice of the series changing at some appropriate argument magnitude, depending on the number of digits required. In a recent paper, D. Borwein, J. Borwein, and R. Crandall [D. Borwein, J.M. Borwein, R. Crandall, Effective Laguerre asymptotics, preprint at http://locutus.cs.dal.ca:8088/archive/00000334/] derived a series for an "exp-arc" integral which gave rise to an absolutely convergent series for the J and I Bessel functions with integral order. Such series can be rapidly evaluated via recursion and elementary operations, and provide a viable alternative to the conventional ascendingasymptotic switching. In the present work, we extend the method to deal with Bessel functions of general (non-integral) order, as well as to deal with the Y and K Bessel functions.

We characterize the Stirling numbers of the second kind S(n, k) modulo prime powers in terms of binomial coefficients. Our results are surprisingly simple when k is a multiple of the modulus.