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The asymptotic expansion of a generalised incomplete gamma function. (English) Zbl 1041.33002
The generalization has the form $\Gamma_p(a,z)=\int_z^\infty t^{a-1} F_{2p}(t)\,dt$, where $p=1,2,3,\ldots$ and $$ F_{2p}(t)=\sum_{k=0}^\infty (-1)^k {z^{k/p}\ \Gamma((2k+1)/(2p))\over k!\ \Gamma(k+1/2)}. $$ Because $F_2(t)=e^{-t}$, the function $\Gamma_1(a,z)$ is the standard incomplete gamma function. It is shown that the large-$z$ asymptotics of $\Gamma_p(a,z)$ in the sector $\vert \text{arg}\,z\vert <p\pi$ consists of $p$ exponential expansions. In $\operatorname{Re} z>0$, all these expansions are recessive at infinity and form a sequence of increasingly subdominant exponential contributions. A numerical example is included for $p=3$ and $a=1$.

33B20Incomplete beta and gamma functions
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
Full Text: DOI
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