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The behavior of the zero-balanced hypergeometric series $ {}\sb pF\sb {p-1}$ near the boundary of its convergence region

Authors: Megumi Saigo and H. M. Srivastava
Journal: Proc. Amer. Math. Soc. 110 (1990), 71-76
MSC: Primary 33C20
MathSciNet review: 1036991
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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].

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  • [1] P. Appell et J. Kampé de Fériet, Fonctions hypergéométriques et hypersphériques; Polynômes d'Hermite, Gauthier-Villars, Paris, 1926. MR 0038393 (12:397b)
  • [2] B. C. Berndt, Chapter 11 of Ramanujan's second notebook, Bull. London Math. Soc. 15 (1983), 273-320. MR 703753 (85a:01043)
  • [3] W. Bühring, The behavior at unit argument of the hypergeometric function $ _3{F_2}$, SIAM J. Math. Anal. 18 (1987), 1227-1234. MR 902328 (88j:33004)
  • [4] A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher transcendental functions, vol. I, McGraw-Hill, New York, 1953.
  • [5] R. J. Evans, Ramanujan's second notebook: Asymptotic expansions for hypergeometric series and related functions, in Ramanujan Revisited (G. E. Andrews et al., eds.), (Proc. of the Ramanujan Centenary Conference, Univ. of Illinois, June 1-5, 1987) Academic Press, New York, 1988, pp. 537-560. MR 938978 (89c:33003)
  • [6] R. J. Evans and D. Stanton, Asymptotic formulas for zero-balanced hypergeometric series, SIAM J. Math. Anal. 15 (1984), 1010-1020. MR 755861 (85i:33003)
  • [7] O. I. Marichev and S. L. Kalla, Behaviour of hypergeometric function $ _p{F_{p - 1}}\left( z \right)$ in the vicinity of unity, Rev. Técn. Fac. Ingr. Univ. Zulia 7 (1984), 1-8. MR 781315 (86e:33006)
  • [8] N. E. Nørlund, Hypergeometric functions, Acta Math. 94 (1955), 289-349. MR 0074585 (17:610d)
  • [9] S. Ramanujan, Notebooks of Srinivasa Ramanujan, vol. 2, Tata Inst. of Fundamental Research, Bombay, 1957. MR 0099904 (20:6340)
  • [10] M. Saigo, A certain boundary value problem for the Euler-Darboux equation. II and III, Math. Japon. 25 (1980), 211-220; 26 (1981), 103-119. MR 580227 (84a:35234)
  • [11] -, On a property of the Appell hypergeometric function $ {F_1}$, Math. Rep. College General Ed. Kyushu Univ. 12 (1980), 63-67.
  • [12] -, 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.
  • [13] -, On properties of the Lauricella hypergeometric function $ {F_D}$, Bull. Central Res. Inst. Fukuoka Univ. 104 (1988), 13-31.
  • [14] -, On properties of hypergeometric functions of three variables, $ {F_M}$ and $ {F_G}$, Rend. Circ. Mat. Palermo (2) 37 (1988), 449-468.
  • [15] M. 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), 1-10. MR 996550 (90c:33007)
  • [16] -, 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.
  • [17] H. M. Srivastava and P. W. Karlsson, Multiple Gaussian hypergeometric series, Wiley, New York, 1985. MR 834385 (87f:33015)
  • [18] H. M. Srivastava and M. Saigo, Multiplication of fractional calculus operators and boundary value problems involving the Euler-Darboux equation, J. Math. Anal. Appl. 121 (1987), 325-369. MR 872230 (88c:26008)

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Keywords: Hypergeometric functions, boundary value problems, Euler-Darboux equation, analytic continuation, asymptotic formula, Kampé de Fériet series
Article copyright: © Copyright 1990 American Mathematical Society

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