Sums of products of Kronecker's double series

Abstract

Closed expressions are obtained for sums of products of Kronecker's double series of the form ∑ ( n j 1 , … , j N ) B j 1 ( x 1 ′ , x 1 ; τ ) ⋯ B j N ( x N ′ , x N ; τ ) , where the summation ranges over all nonnegative integers j 1 , … , j N with j 1 + ⋯ + j N = n . Corresponding results are derived for functions which are an elliptic analogue of the periodic Euler polynomials. As corollaries, we reproduce the formulas for sums of products of Bernoulli numbers, Bernoulli polynomials, Euler numbers, and Euler polynomials, which were given by K. Dilcher.