In this paper we obtain a general reciprocity relation (Theorem 2.1) for a class of double series, from which we deduce several results including two alternating double series for f(3) (where ζ(s) is the Riemann zeta function), which complement a result of L. J. Mordell. Later in the paper, we obtain another reciprocity relation for the double series and also extend our investigations to multiple series.

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Let p*(n) denote the number of product partitions, that is, the number of ways of expressing a natural number n > 1 as the product of positive integers ≥ 2, the order of the factors in the product being irrelevant, with p*(1) = 1. For any integer if d is an ith power, and = 1, otherwise, and let . Using a suitable generating function for p*(n) we prove that