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Convergent expansions of the confluent hypergeometric functions in terms of elementary functions


Authors: Blanca Bujanda, José L. López and Pedro J. Pagola
Journal: Math. Comp. 88 (2019), 1773-1789
MSC (2010): Primary 33C15; Secondary 41A58
DOI: https://doi.org/10.1090/mcom/3389
Published electronically: November 5, 2018
MathSciNet review: 3925484
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Abstract: We consider the confluent hypergeometric function $ M(a,b;z)$ for $ z\in \mathbb{C}$ and $ \Re b>\Re a>0$, and the confluent hypergeometric function $ U(a,b;z)$ for $ b\in \mathbb{C}$, $ \Re a>0$, and $ \Re z>0$. We derive two convergent expansions of $ M(a,b;z)$; one of them in terms of incomplete gamma functions $ \gamma (a,z)$ and another one in terms of rational functions of $ e^z$ and $ z$. We also derive a convergent expansion of $ U(a,b;z)$ in terms of incomplete gamma functions $ \gamma (a,z)$ and $ \Gamma (a,z)$. The expansions of $ M(a,b;z)$ hold uniformly in either $ \Re z\ge 0$ or $ \Re z\le 0$; the expansion of $ U(a,b;z)$ holds uniformly in $ \Re z>0$. The accuracy of the approximations is illustrated by means of some numerical experiments.


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Additional Information

Blanca Bujanda
Affiliation: Departamento de Estadística, Informática y Mathemáticas and InaMat, Universidad Pública de Navarra, Pamplona, Spain
Email: blanca.bujanda@unavarra.es

José L. López
Affiliation: Departamento de Estadística, Informática y Mathemáticas and InaMat, Universidad Pública de Navarra, Pamplona, Spain
Email: jl.lopez@unavarra.es

Pedro J. Pagola
Affiliation: Departamento de Estadística, Informática y Mathemáticas, Universidad Pública de Navarra, Pamplona, Spain
Email: pedro.pagola@unavarra.es

DOI: https://doi.org/10.1090/mcom/3389
Keywords: Confluent hypergeometric functions, convergent expansions, uniform expansions
Received by editor(s): October 25, 2017
Received by editor(s) in revised form: May 2, 2018
Published electronically: November 5, 2018
Additional Notes: This research was supported by the Spanish Ministry of Economía y Competitividad, projects MTM2014-53178-P and TEC2013-45585-C2-1-R.
Article copyright: © Copyright 2018 American Mathematical Society