Partial sums of hypergeometric series of unit argument

Bühring, Wolfgang

doi:10.1090/S0002-9939-03-07010-2

Partial sums of hypergeometric series of unit argument
HTML articles powered by AMS MathViewer

by Wolfgang Bühring PDF

Proc. Amer. Math. Soc. 132 (2004), 407-415 Request permission

Abstract:

The asymptotic behaviour of partial sums of generalized hypergeometric series of unit argument is investigated.

References

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
Bruce C. Berndt, Ramanujan’s notebooks. Part II, Springer-Verlag, New York, 1989. MR 970033, DOI 10.1007/978-1-4612-4530-8
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
Wolfgang Bühring, Generalized hypergeometric functions at unit argument, Proc. Amer. Math. Soc. 114 (1992), no. 1, 145–153. MR 1068116, DOI 10.1090/S0002-9939-1992-1068116-2
W. Bühring, Asymptotic expansions for ratios of products of gamma functions, Internat. J. Math. Math. Sci. 2003:18 (2003), 1167–1171.
W. Bühring and H. M. Srivastava, Analytic continuation of the generalized hypergeometric series near unit argument with emphasis on the zero-balanced series, Approximation Theory and Applications (Th. M. Rassias, ed.), Hadronic Press, Palm Harbor, 1998, pp. 17–35 (updated version: e-print math.CA/0102032 on http://arXiv.org).
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
Yudell L. Luke, The special functions and their approximations, Vol. I, Mathematics in Science and Engineering, Vol. 53, Academic Press, New York-London, 1969. MR 0241700
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
Anand Kumar Srivastava, Asymptotic behaviour of certain zero-balanced hypergeometric series, Proc. Indian Acad. Sci. Math. Sci. 106 (1996), no. 1, 39–51. MR 1392457, DOI 10.1007/BF02837185