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Abstract
We construct asymptotic expansions for the normalised incomplete gamma function Q(a, z) = Γ(a, z)/Γ(a) that are valid in the transition regions, including the case z ≈ a, and have simple polynomial coefficients. For Bessel functions, these type of expansions are well known, but for the normalised incomplete gamma function they were missing from the literature. A detailed historical overview is included. We also derive an asymptotic expansion for the corresponding inverse problem, which has importance in probability theory and mathematical statistics. The coefficients in this expansion are again simple polynomials, and therefore its implementation is straightforward. As a byproduct, we give the first complete asymptotic expansion as a → −∞ of the unique negative zero of the regularised incomplete gamma function γ* (a, x).
Original language  English 

Pages (fromto)  18051827 
Number of pages  21 
Journal  Mathematics of computation 
Volume  88 
Issue number  318 
Early online date  8 Nov 2018 
DOIs  
Publication status  Epub ahead of print  8 Nov 2018 
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Dive into the research topics of 'Asymptotic expansions for the incomplete gamma function in the transition regions'. Together they form a unique fingerprint.Projects
 1 Finished

Rigorous and Presentable Asymptotics for Special Functions and Orthogonal Polynomials
1/09/15 → 31/08/20
Project: Research
Profiles

Adri Olde Daalhuis
 School of Mathematics  Personal Chair of Asymptotics and Special Functions
Person: Academic: Research Active