The behavior of the zero-balanced hypergeometric series ${}_ pF_ {p-1}$ near the boundary of its convergence region
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- by Megumi Saigo and H. M. Srivastava
- Proc. Amer. Math. Soc. 110 (1990), 71-76
- DOI: https://doi.org/10.1090/S0002-9939-1990-1036991-1
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Abstract:
For a zero-balanced generalized hypergeometric function $_p{F_{p - 1}}\left ( z \right )$, the authors prove a formula exhibiting its behavior near the boundary point $z = 1$ of the region of convergence of the series defining it. The result established here provides an interesting extension of a formula which appeared in one of Ramanujan’s celebrated Notebooks; it also serves to solve the problem posed by R. J. Evans [5].References
- Paul Appell, Analyse mathématique à l’usage des candidats au certificat de mathématiques générales et aux grandes écoles. Tome I. Analyse des courbes, surfaces et fonctions usuelles, intégrales simples, Gauthier-Villars, Paris, 1951 (French). 6th ed. MR 0038393
- Bruce C. Berndt, Chapter 11 of Ramanujan’s second notebook, Bull. London Math. Soc. 15 (1983), no. 4, 273–320. MR 703753, DOI 10.1112/blms/15.4.273
- Wolfgang Bühring, The behavior at unit argument of the hypergeometric function $_3F_2$, SIAM J. Math. Anal. 18 (1987), no. 5, 1227–1234. MR 902328, DOI 10.1137/0518089 A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher transcendental functions, vol. I, McGraw-Hill, New York, 1953.
- Ronald J. Evans, Ramanujan’s second notebook: asymptotic expansions for hypergeometric series and related functions, Ramanujan revisited (Urbana-Champaign, Ill., 1987) Academic Press, Boston, MA, 1988, pp. 537–560. MR 938978
- Ronald J. Evans and Dennis Stanton, Asymptotic formulas for zero-balanced hypergeometric series, SIAM J. Math. Anal. 15 (1984), no. 5, 1010–1020. MR 755861, DOI 10.1137/0515078
- O. I. Marichev and S. L. Kalla, Behaviour of hypergeometric function $_pF_{p-1}(z)$ in the vicinity of unity, Rev. Técn. Fac. Ingr. Univ. Zulia 7 (1984), no. 2, 1–8 (English, with Spanish summary). MR 781315
- N. E. Nørlund, Hypergeometric functions, Acta Math. 94 (1955), 289–349. MR 74585, DOI 10.1007/BF02392494
- Srinivasa Ramanujan, Notebooks. Vols. 1, 2, Tata Institute of Fundamental Research, Bombay, 1957. MR 0099904
- Megumi Saigo, A certain boundary value problem for the Euler-Darboux equation. II, Math. Japon. 25 (1980), no. 2, 211–220. MR 580227 —, On a property of the Appell hypergeometric function ${F_1}$, Math. Rep. College General Ed. Kyushu Univ. 12 (1980), 63-67. —, On properties of the Appell hypergeometric functions ${F_2}$ and ${F_3}$ and the generalized Gauss function $_3{F_2}$, Bull. Central Res. Inst. Fukuoka Univ. 66 (1983), 27-32. —, On properties of the Lauricella hypergeometric function ${F_D}$, Bull. Central Res. Inst. Fukuoka Univ. 104 (1988), 13-31. —, On properties of hypergeometric functions of three variables, ${F_M}$ and ${F_G}$, Rend. Circ. Mat. Palermo (2) 37 (1988), 449-468.
- Megumi Saigo and H. M. Srivastava, The behaviors of the Appell double hypergeometric series $F_4$ and certain Lauricella triple hypergeometric series near the boundaries of their convergence regions, Fukuoka Univ. Sci. Rep. 19 (1989), no. 1, 1–10. MR 996550 —, The behavior of the Lauricella hypergeometric series $F_D^{\left ( n \right )}$ in $n$ variables near the boundaries of its convergence region, Univ. of Victoria Report No. DM-480-IR, 1988, pp. 1-23.
- H. M. Srivastava and Per W. Karlsson, Multiple Gaussian hypergeometric series, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chichester; Halsted Press [John Wiley & Sons, Inc.], New York, 1985. MR 834385
- H. M. Srivastava and Megumi Saigo, Multiplication of fractional calculus operators and boundary value problems involving the Euler-Darboux equation, J. Math. Anal. Appl. 121 (1987), no. 2, 325–369. MR 872230, DOI 10.1016/0022-247X(87)90251-4
Bibliographic Information
- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 110 (1990), 71-76
- MSC: Primary 33C20
- DOI: https://doi.org/10.1090/S0002-9939-1990-1036991-1
- MathSciNet review: 1036991