Vol. 113, No. 2, 1984

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ISSN: 0030-8730
Transformation formulae for multiple series

R. Sitaramachandra Rao and M. V. Subba Rao

Vol. 113 (1984), No. 2, 471–479
DOI: 10.2140/pjm.1984.113.471
Abstract
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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) = 31π615,120.

Mathematical Subject Classification 2000
Primary: 11N99
Milestones
Received: 22 April 1981
Published: 1 August 1984
Authors
R. Sitaramachandra Rao
M. V. Subba Rao