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Landen inequalities for hypergeometric functions

Published online by Cambridge University Press:  22 January 2016

S.-L. Qiu and
M. Vuorinen
S.-L. Qiu
Affiliation:
President’s Office, Hangzhou Institute of Electronics Engineering, Hangzhou 310037, P. R. CHINA
M. Vuorinen
Affiliation:
Department of Mathematics, P.O. Box 4 (Yliopistonkatu 5), University of Helsinki, FIN-00014, FINLAND, vuorinen@csc.fi
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Abstract

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A generalization of the Landen identity, in the form of an inequality, is proved for hypergeometric functions. Some well-known asymptotic formulas are refined.

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 154 , 1999, pp. 31 - 56
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1999

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
[AlB] Almkvist, G. and Berndt, B., Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, π, and the Ladies Diary, Amer. Math. Monthly, 95 (1988), 585–608.Google 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.Google Scholar
[AVV1] Anderson, G. D., Vamanamurthy, M. K., and Vuorinen, M., Hypergeometric functions and elliptic integrals, in Current Topics in Analytic Function Theory (Srivastava, H. M. and Owa, S., eds.), World Scientific Publ. Co., Singapore – London (1992), pp. 48–85.Google Scholar
[AVV2] Anderson, G. D., Vamanamurthy, M. K., and Vuorinen, M., Conformal invariants, inequalities and quasiconformal mappings, J. Wiley, 1997.Google Scholar
[Ao] Aomoto, K., Hypergeometric functions – the past, today, and … (from the complex analytic point of view), Sugaku Expositions, 9 (1996), 99–116.Google Scholar
[Ask1] Askey, R., Ramanujan and hypergeometric and basic hypergeometric series, Ramanujan Internat. Symposium on Analysis, December 26–28, (1987), (Thakare, N. K., eds.), 1–83, Pune, India, Russian Math. Surveys 451 (1990), 37–86.Google Scholar
[Ask2] Askey, R., Handbooks of Special Functions, A Century of Mathematics in America, Part III, (Duren, P., eds.), Amer. Math. Soc., 1989.Google Scholar
[Be1] Berndt, B. C., Ramanujan’s Notebooks, Vol. I, Springer-Verlag, Berlin – Heidelberg – New York, 1985.Google Scholar
[Be2] Berndt, B. C., Ramanujan’s Notebooks, Vol. II, Springer-Verlag, Berlin – Heidelberg – New York, 1989.Google Scholar
[Be3] Berndt, B. C., Ramanujan’s Notebooks, Vol. III, Springer-Verlag, Berlin – Heidelberg – New York, 1991.Google Scholar
[Be4] Berndt, B. C., Ramanujan’s Notebooks, Vol. IV, Springer-Verlag, Berlin – Heidelberg – New York, 1993.Google Scholar
[BH] Beukers, F. and Heckman, G., Monodromy for the hypergeometric function nFn−1 , Invent. Math., 95 (1989), 325–354.Google Scholar
[BB] Borwein, J. M. and Borwein, P. B., The arithmetic-geometric mean and fast computation of elementary functions, SIAM Rev. 26 (1984), 351–366.CrossRefGoogle Scholar
[CG] Carlson, B. C. and Gustafson, J. L., Asymptotic approximations for symmetric elliptic integrals, SIAM J. Math. Anal., 25 (1994), 288–303.CrossRefGoogle Scholar
[CC] Chudnovsky, D. V. and Chudnovsky, G. V., Hypergeometric and modular function identities, and new rational approximations to a continued fraction expansions of classical constants and functions, in A Tribute to Emil Grosswald – Number Theory and Related Analysis (Knopp, M. and Sheingorn, M., eds.), Contemporary Math. Vol 143 (1993), pp. 117–162.Google Scholar
[DM] Deligne, P. and Mostow, G. D., Commensurabilities among lattices in PU(1, n), Annals of Mathematics Studies, Princeton University Press, Princeton, New Jersey, 1993.Google Scholar
[Du] Dutka, J., The early history of the hypergeometric function, Archieve for History of Exact Sciences, 31 (1984), 15–34.Google Scholar
[E] Evans, R. J., Ramanujan’s second notebook: Asymptotic expansions for hypergeometric series and related functions, in Ramanujan Revisited, Proc. of the Ramanujan Centenary Conf. at the Univ. of Illinois at Urbana-Champaign (Andrews, G. E., Askey, R. A., Berndt, B. C., Ramanathan, R. G., and Rankin, R. A., eds.), Academic Press, Boston (1988), pp. 537–560.Google Scholar
[GKZ] Gel’fand, I. M., Kapranov, M. M., and Zelevinsky, A. V., Generalized Euler Integrals and A-hypergeometric functions, Adv. Math., 84 (1990), 255–271.Google Scholar
[K] Kühnau, R., Eine Methode, die positivität einer Funktion zu prüfen, Zeitschrift f. angew. Math. u. Mech., 74 (1994), 140–142.Google Scholar
[PV] Ponnusamy, S. and Vuorinen, M., Asymptotic expansions and inequalities for hypergeometric functions, Mathematika, 44 (1997), 278–301.Google Scholar
[PBM] Prudnikov, A. P., Brychkov, Yu. A., and Marichev, O. I., Integrals and Series, Vol. 3: More Special Functions, transl. from the Russian by Gould, G. G., Gordon and Breach Science Publishers, New York, 1988.Google Scholar
[QV] Qiu, S.-L. and Vamanamurthy, M. K., Sharp estimates for complete elliptic integrals, SIAM J. Math. Anal., 27 (1996), 823–834.CrossRefGoogle Scholar
[Var] Varadarajan, V. S., Linear meromorphic differential equations: A modern point of view, Bull. Amer. Math. Soc., 33 (1996), 1–42.Google Scholar
[Va] Varchenko, A., Multidimensional hypergeometric functions and their appearance in conformal field theory, algebraic K-theory, algebraic geometry, etc., Proc. Internat. 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, London, l958.Google Scholar
[WZ] Wilf, H. S. and Zeilberger, D., An algorithmic proof theory for hypergeometric (ordinary and “q”) multisum/integral identities, Invent. Math., 108 (1992), 575–633.Google Scholar