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 y c be reals greater than 1 and P(a,b,c) = 2 r~a 2 k ~b ~c-r=\ k=\ l=\ Then P(a, b, c) + P(a, c, b) + P(b, c, a) + P(b, a, c) + (b)(c + a) + (c)S(a + b) + 2{( + * + c), where f denotes the Riemann zeta function. In particular, P(2,2,2) = 31 6 /15,12O.