Results old and new on the hyper-Bessel equation

Abstract

SynopsisWe consider a variety of integral representations, single and multiple, old and new, for solutions of the hyper-Bessel equation u(n) – zmu =0. In particular, we show how a very early multiple Laplace integral solution of Molins (1876) may be related to recent Mellin–Barnes integral representations given by the present authors by way of multiple integral solutions given by Saxton and the second author for an associated equation. Although both these multiple integral solutions may be found by elementary methods, it is not easy to find their asymptotic expansions for large z, and we show how these may conveniently be obtained from our earlier results [10].