ON EXTENDED MULTIDIMENSIONAL SCHLOMILCH SERIES

We extend an earlier investigation of a class of m-dimensional lattice sums which were phase modulated by a continuous twist parameter tau = (tau(l), ..., tau(m)) and involved Bessel functions J(nu)(x), with argument x = 2 sigma q (q = \q\ = root q(l)(2) +...+ q(m)(2); qj = integer(For All)j is an element of m) such that 0 less than or equal to nu < pi tau. This investigation-motivated partly by the Henkel-Weston conjectures and partly by the propositions of Ortner and Wagner on the solution of the hyperbolic differential operators-generalizes previous analyses to include an m-dimensional shift parameter a and an arbitrary scalar b, so that x is now equal to 2 nu root\q+a\(2)+b(2); at the same time, it extends the range of nu from 0 to infinity. While these generalizations are broad enough to make this a worthwhile study in its own right, the main interest here lies in applying this special technique to such diverse physical problems as finite-sized ferromagnetic or quantum fluid model systems undergoing phase transitions, the Casimir effect and topological mass generation, etc. Consequently, there arise important connections to the zeta function regularization and to heat kernel techniques.