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
Full-text PDF Free Access
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Additional Information
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.
References
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References
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- R. G. Buschman, Contiguous relations for Appell functions, Indian J. Math. 29 vol 2 (1987), 165–171. MR 919894 (89d:33006)
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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
ORCID:
0000-0002-6050-9015
Email:
jl.lopez@unavarra.es
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.
