ON THE DARLING-MANDELBROT PROBABILITY DENSITY AND THE ZEROS OF SOME INCOMPLETE GAMMA-FUNCTIONS

Recently, Mandelbrot has encountered and numerically investigated a pr obability density p(D)(t) on the nonnegative reals, where 0 < D < 1. T his density has Fourier transform 1/f(D(-is), where f(D)(z) = -Dz(D)ga mma(-D, z) and gamma(because) is an incomplete gamma function. Previou sly, Darling had met this density, but had not studied its form. We ex press f(D)(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 p(D)(t) asymptotical ly as t --> 0+ and + infinity, then note some implications as D --> 0 and 1-.