ON THE ZETA-FUNCTION REGULARIZATION OF A 2-DIMENSIONAL SERIES OF EPSTEIN-HURWITZ TYPE

As a further step in the general program of zeta-function regularization of multiseries expressions, some original formulas are provided for the analytic continuation, to any value of s, of two-dimensional series of Epstein-Hurwitz type, namely, (Equation Presented), where the aj, are positive reals and the cj are not simultaneously nonpositive integers. They come out from a generalization to Hurwitz functions of the zeta-function regularization theorem of the author and Romeo [Phys. Rev. D 40, 436 (1989)] for ordinary zeta functions. For s = - k,0,2, with k = 1,2,3,..., the final results are, in fact, expressed in terms of Hurwitz zeta functions only. For general s they also involve Bessel functions. A partial numerical investigation of the different terms of the exact, algebraic equations is also carried out. As a by-product, the series Σn=0∞exp[ - a(n + c)2], a,c>0, is conveniently calculated in terms of them. © 1989 American Institute of Physics.