Transformation formulae for multiple series

In this paper we prove a general transformation formula for a triple series of complex terms. We deduce a transformation formula for double series and discuss some applications. Furthermore, some reciprocity relations of the following type are obtained: Let a, b, c be reals greater than 1 and

∑∞ ∑r ∑k P (a,b,c) = r− a k−b l−c r=1 k=1 l=1

Then

P(a,b,c)+ P (a,c,b)+ P(b,c,a)+ P(b,a,c) + P (c,a,b)+ P(c,b,a) = ζ(a)ζ(b)ζ(c)+ ζ(a)ζ(b +c) + ζ(b)ζ(c+ a)+ ζ(c)ζ(a +b)+ 2ζ(a+ b +c),

where ζ denotes the Riemann zeta function. In particular, P(2,2,2) = 31π⁶∕15,120

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Vol. 113, No. 2, 1984

R. Sitaramachandra Rao and M. V. Subba Rao

Abstract