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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 ${p}_{D}\left(t\right)$ on the nonnegative reals, where $0. This density has Fourier transform $1/{f}_{D}\left(-is\right)$, where ${f}_{D}\left(z\right)=-D{z}^{D}\gamma \left(-D,z\right)$ and $\gamma$ is an incomplete gamma function. Previously, Darling had met this density but had not studied its form. We express ${f}_{D}\left(z\right)$ as a confluent hypergeometric function, then locate and approximate its zeros, thereby improving some results of Buchholtz. Via properties of Laplace transforms, we approximate ${p}_{D}\left(t\right)$ asymptotically as $t\to 0+$ and $+\infty$, then note some implications as $D\to 0+$ and $1-$.
##### MSC:
 33E99 Other special functions 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 60E99 Distribution theory in probability theory
##### References:
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