Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Generalized hypergeometric functions at unit argument


Author: Wolfgang Bühring
Journal: Proc. Amer. Math. Soc. 114 (1992), 145-153
MSC: Primary 33C20
DOI: https://doi.org/10.1090/S0002-9939-1992-1068116-2
MathSciNet review: 1068116
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The analytic continuation near $ z = 1$ of the hypergeometric function $ _{p + 1}{F_p}\left( z \right)$ is obtained for arbitrary $ p = 2,3, \ldots ,$, including the exceptional cases when the sum of the denominator parameters minus the sum of the numerator parameters is equal to an integer.


References [Enhancements On Off] (What's this?)

  • [1] M. Abramowitz and I. A. Stegun, Handbook of mathematical functions, Dover, New York, 1965.
  • [2] W. N. Bailey, Generalized hypergeometric series, Stechert-Hafner, New York, 1964. MR 0185155 (32:2625)
  • [3] B. C. Berndt, Chapter 11 of Ramanujan's second notebook, Bull. London Math. Soc. 15 (1983), 273-320. MR 703753 (85a:01043)
  • [4] -, Ramanujan's notebooks, Part II, Springer-Verlag, New York, 1989.
  • [5] 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)
  • [6] -, Transformation formulas for terminating Saalschützian hypergeometric series of unit argument (to appear)
  • [7] -, The behavior at unit argument of the generalized hypergeometric function (to appear)
  • [8] R. J. Evans, Ramanujan's second notebook: Asymptotic expansions for hypergeometric series and related functions, Ramanujan Revisited (G. E. Andrews et al. eds.), Proc. of the Ramanujan Centenary Conference, 1987, Academic Press, New York, 1988, pp. 537-560. MR 938978 (89c:33003)
  • [9] 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)
  • [10] Y. L. Luke, The special functions and their approximations, vol. 1, Academic Press, New York, 1969.
  • [11] N. Nørlund, Hypergeometric functions, Acta Math. 94 (1955), 289-349. MR 0074585 (17:610d)
  • [12] S. Ramanujan, Notebooks, vol. 2, Tata Institute of Fundamental Research, Bombay, 1957. MR 0099904 (20:6340)
  • [13] M. Saigo, On properties of the Appell hypergeometric functions $ {F_2}$ and $ {F_3}$ and the generalized Gauss function $ _3{F_2}$, Bull. Central Research Institute Fukuoka University 66 (1983), 27-32. MR 730319 (85m:33006)
  • [14] M. Saigo and H. M. Srivastava, The behavior of the zero-balanced hypergeometric series $ _p{F_{p - 1}}$ near the boundary of its convergence region, Proc. Amer. Math. Soc. 110 (1990), 71-76. MR 1036991 (91d:33005)
  • [15] L. J. Slater, Generalized hypergeometric functions, Cambridge Univ. Press, Cambridge, 1966. MR 0201688 (34:1570)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 33C20

Retrieve articles in all journals with MSC: 33C20


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1068116-2
Keywords: Special functions, hypergeometric functions, hypergeometric series, continuation formulas, analytic continuation
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society