Generalized hypergeometric functions at unit argument

Bühring, Wolfgang

doi:10.1090/S0002-9939-1992-1068116-2

Generalized hypergeometric functions at unit argument
HTML articles powered by AMS MathViewer

by Wolfgang Bühring

Proc. Amer. Math. Soc. 114 (1992), 145-153

DOI: https://doi.org/10.1090/S0002-9939-1992-1068116-2

PDF | Request permission

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

Handbook of mathematical functions

W. N. Bailey, Generalized hypergeometric series, Cambridge Tracts in Mathematics and Mathematical Physics, No. 32, Stechert-Hafner, Inc., New York, 1964. MR 0185155
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

Ramanujan’s notebooks

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

Transformation formulas for terminating Saalschützian hypergeometric series of unit argument

The behavior at unit argument of the generalized hypergeometric function

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

The special functions and their approximations

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, 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. MR 730319
Megumi Saigo and H. M. Srivastava, The behavior of the zero-balanced hypergeometric series ${}_pF_{p-1}$ near the boundary of its convergence region, Proc. Amer. Math. Soc. 110 (1990), no. 1, 71–76. MR 1036991, DOI 10.1090/S0002-9939-1990-1036991-1
Lucy Joan Slater, Generalized hypergeometric functions, Cambridge University Press, Cambridge, 1966. MR 0201688