On the Darling-Mandelbrot probability density and the zeros of some incomplete gamma functions

Abstract

Recently, Mandelbrot has encountered and numerically investigated a probability density pd(t) on the nonnegative reals, where, 0<D<1. This density has Fourier transform 1/fd(-is), where fd(z)=-Dzdγ(-D, z) and γ(·.·) is an incomplete gamma function. Previously, Darling had met this density, but had not studied its form. We express fd(z) as a confluent hypergeometric function, then locate and approximate its zeros, thereby improving some results of Buchholz. Via properties of Laplace transforms, we approximate pd(t) asymptotically as t→0+ and +∞, then note some implications as D→0+ and 1-. © 1994 Springer-Verlag New York Inc.