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Generalized hypergeometric functions at unit argument
Author:
Wolfgang Bühring
Journal:
Proc. Amer. Math. Soc. 114 (1992), 145-153
MSC:
Primary 33C20
MathSciNet review:
1068116
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Abstract: The analytic continuation near of the hypergeometric function is obtained for arbitrary , including the exceptional cases when the sum of the denominator parameters minus the sum of the numerator parameters is equal to an integer.
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- 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
, 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)
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- 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)
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- 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
and and the generalized Gauss function , 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
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)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002-9939-1992-1068116-2
PII:
S 0002-9939(1992)1068116-2
Keywords:
Special functions,
hypergeometric functions,
hypergeometric series,
continuation formulas,
analytic continuation
Article copyright:
© Copyright 1992 American Mathematical Society
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