Extended incomplete gamma functions with applications. (English) Zbl 1011.33002
In this paper the authors introduce the following functions
\[
\gamma (\alpha,x; b,\beta)= \int^x_0 t^{\alpha-1} e^{-t-bt^{-\beta}} dt,\;x\geq 0,\;b\geq 0,
\]
and
\[
\Gamma (\alpha,x;b, \beta)=\int^\infty_x t^{\alpha-1} e^{-t-bt^{-\beta}}dt,\;x\geq 0,\;b\geq 0,
\]
and call them extended incomplete gamma functions. Several properties of these functions such as a recurrence formula and a decomposition formula are given. The relationship between these extended incomplete gamma functions and some of the known functions including generalized incomplete gamma functions, astrophysical thermonuclear functions, exponential integral function, Dowson’s integral function, Abramowitz’s function and Goodwin and Stalon’s function are also exhibited. Note that a generalized inverse Gaussian density function closely related to these extended incomplete gamma functions is proposed. In addition, extended Meijer’s \(G\)-functions and Fox’s \(H\)-functions are defined. Numerous applications of these functions in various fields are also discussed.
Reviewer: Stamatis Koumandos (Nicosia)
MSC:
| 33B20 | Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals) |
| 33B15 | Gamma, beta and polygamma functions |
| 60E05 | Probability distributions: general theory |
Software:
Algorithm 597References:
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