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Zbl 0798.33014
Lew, John S.
On the Darling-Mandelbrot probability density and the zeros of some incomplete gamma functions. (English)
[J] Constructive Approximation 10, No.1, 15-30 (1994). ISSN 0176-4276; ISSN 1432-0940/e

(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 $1-$.
[D.Kershaw (Lancaster)]

MSC 2000:: *33E99 Special functions
41A60 Asymptotic problems in approximation
60E99 Distribution theory in probability theory

Keywords: incomplete gamma function; Darling-Mandelbrot probability density; confluent hypergeometric function

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