Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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The Appell's function $ F{}_2$ for large values of its variables

Authors: Esther Garcia and José L. López
Journal: Quart. Appl. Math. 68 (2010), 701-712
MSC (2000): Primary 41A60; Secondary 33C65
DOI: https://doi.org/10.1090/S0033-569X-2010-01186-3
Published electronically: September 15, 2010
MathSciNet review: 2761511
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Abstract: The second Appell's hypergeometric function $ F_2(a,b,b',c,c';x,y)$ has a Mellin convolution integral representation in the region $ \Re(x+y)<1$ and $ a>0$. We apply a recently introduced asymptotic method for Mellin convolution integrals to derive three asymptotic expansions of $ F_2(a,b,b',c,c';x,y)$ in decreasing powers of $ x$ and $ y$ with $ x/y$ bounded. For certain values of the real parameters $ a$, $ b$, $ b'$, $ c$ and $ c'$, two of these expansions involve logarithmic terms in the asymptotic variables $ x$ and $ y$. Some coefficients of these expansions are given in terms of the Gauss hypergeometric function $ {}_3F_2$ and its derivatives.

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Additional Information

Esther Garcia
Affiliation: Departamento de Matemáticas, IES Tubalcaín, 50500-Tarazona, Zaragoza, Spain
Email: estlabec@yahoo.es

José L. López
Affiliation: Departamento de Matemática e Informática, Universidad Pública de Navarra, 31006-Pamplona, Spain
Email: jl.lopez@unavarra.es

DOI: https://doi.org/10.1090/S0033-569X-2010-01186-3
Keywords: Second Appell hypergeometric function, asymptotic expansions, Mellin convolution integrals.
Received by editor(s): February 25, 2009
Published electronically: September 15, 2010
Additional Notes: The first author was supported by the “Gobierno de Navarra”, ref. 2301/2008.
The second author was supported by the “Dirección General de Ciencia y Tecnología", REF. MTM2007-63772, and the “Gobierno de Navarra", ref. 2301/2008.
Article copyright: © Copyright 2010 Brown University
The copyright for this article reverts to public domain 28 years after publication.

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