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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 $p\sb D(t)$ on the nonnegative reals, where $0<D<1$. This density has Fourier transform $1/f\sb D(-is)$, where $f\sb D(z)=- Dz\sp D \gamma(-D,z)$ and $\gamma$ is an incomplete gamma function. Previously, Darling had met this density but had not studied its form. We express $f\sb D(z)$ as a confluent hypergeometric function, then locate and approximate its zeros, thereby improving some results of Buchholtz. Via properties of Laplace transforms, we approximate $p\sb D(t)$ asymptotically as $t\to 0+$ and $+\infty$, then note some implications as $D\to 0+$ and $1-$.
33E99Other special functions
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
60E99Distribution theory in probability theory
Full Text: DOI
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