Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Special functions, infinite divisibility and transcendental equations

Mourad E. H. Ismaila1 and C. Ping Maya2

a1 McMaster University, Hamilton, Ontario, Canada

a2 University of Alberta, Edmonton, Alberta, Canada

Abstract

We establish integral representations for quotients of Tricomi ψ functions and of several quotients of modified Bessel functions and of linear combinations of them. These integral representations are used to prove the complete monotonicity of the functions considered and to prove the infinite divisibility of a three parameter probability distribution. Limiting cases of this distribution are the hitting time distributions considered recently by Kent and Wendel. We also derive explicit formulas for the Kent–Wendel probability density functions.

(Received June 14 1978)

(Revised October 06 1978)