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On the Darling-Mandelbrot probability density and the zeros of some incomplete gamma functions. (English) Zbl 0798.33014
(Author’s abstract). Recently, Mandelbrot has encountered and numerically investigated a probability density p D (t) on the nonnegative reals, where 0<D<1. This density has Fourier transform 1/f D (-is), where f D (z)=-Dz D γ(-D,z) and γ is an incomplete gamma function. Previously, Darling had met this density but had not studied its form. We express f D (z) as a confluent hypergeometric function, then locate and approximate its zeros, thereby improving some results of Buchholtz. Via properties of Laplace transforms, we approximate p D (t) asymptotically as t0+ and +, then note some implications as D0+ and 1-.
MSC:
33E99Other special functions
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
60E99Distribution theory in probability theory
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