Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-28T02:08:20.944Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic expansions and inequalities for hypergeometric function

Published online by Cambridge University Press:  26 February 2010

S. Ponnusamy  and
M. Vuorinen
S. Ponnusamy
Affiliation:
Department of Mathematics, Indian Institute of Technology, Institution of Engineers Building, Panbazar, Guwahati-781 001, India. e-maii: [email protected]
M. Vuorinen
Affiliation:
Department of Mathematics, P.O. Box 4, (Yliopistonkatu 5), FIN-000 14, University of Helsinki, Finland. e-mail: [email protected]
Get access

Abstract

Ramanujan's work on the asymptotic behaviour of the hypergeometric function has been recently refined to the zero-balanced Gaussian hypergeometric function F(a, b; a + b; x) as x→1.We extend these results for F(a, b; c; x) when a, b, c>0 and c<a + b.

MSC classification

Secondary: 33C05: Classical hypergeometric functions, $_2F_1$
Type
Research Article
Information
Mathematika , Volume 44 , Issue 2 , December 1997 , pp. 278 - 301
Copyright
Copyright © University College London 1997

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

As.Abramowitz, M. and Stegun, I. A., editors. Handbook of mathematical functions with formulas. Graphs and mathematical tables (Dover, New York, 1965).Google Scholar
A1.Alzer, H.. Some gamma function inequalities. Math. Comput., 60 (1993), 337–346.CrossRefGoogle Scholar
ABRVV.Anderson, G. D.Barnard, R. W.Richards, K. C.Vamanamurthy, M. K. and Vuorinen, M.. Inequalities for zero-balanced hypergeometric functions. Trans. Amer. Math. Soc., 347 (1995), 1713–1723.CrossRefGoogle Scholar
As.Askey, R.Ramanujan, S. and hypergeometric and basic hypergeometric series (Russian). Translated from English and with a remark by Atakishiev, N. M. and Suslov, S. K.. Uspekhi Mat. Nauk, 45 (1990), 33–76 222; translation in Russian Math. Surveys, 45 (1990), 37 86.Google Scholar
B.Berndt, B. C.. Ramanujan's Notebooks, Part II (Springer, Berlin, 1989).CrossRefGoogle Scholar
BK.Biernacki, M. and Krzyz, J.,. On the monotonicity of certain functionals in the theory of analytic functions. Ann. Univ. M. Curie-Sktodowska, 2 (1995), 134–145.Google Scholar
BI.Bustoz, J. and Ismail, M. E. H.. On gamma function inequalities. Math. Comput., 47 (1986), 659–667.CrossRefGoogle Scholar
Ev.Evans, R. J.. Ramanujan's second notebook: asymptotic expansions for hypergeometric series and related Junctions, In Ramanujan Revisited: Proc. of the Centena y Conference Univ. of Illinois at Urbana-Champaign, ed. by Andrews, G. E.Askey, R. A.Berndt, B. C.Ramanathan, R. G.Rankin, R. A. (Academic Press, Bosto, 1988), 537–560.Google Scholar
M.Mitrinovic, D. S.. Analytic inequalities, Die Grundlehren der math. Wissenschaften, Band 165 (Springer, 1970).Google Scholar
R.Rainville, E. D.. Special functions (Chelsea Publishing Company, New York, 1960).Google Scholar
Ra.Ramanujan, S.. The lost notebook and other unpublished papers (Springer, New York, 1988).Google Scholar
V.Varchenko, A.. Multidimensional hypergeometric functions and their appearance in conformal field theory, algebraic K-theory, algebraic geometry, etc. Proc. Internal. Congr. Math. (Kyoto, Japan, 1990), 281–300.Google Scholar
WW.Whittaker, E. T. and Watson, G. N.. A Course of Modern Analysis, 4th ed. (Cambridge Univ. Press, 1958).Google Scholar