Pages from the computer files of R. William Gosper

Ismail, Mourad E. H.; Takeuchi, Yu; Zhang, Ruiming

doi:10.1090/S0002-9939-1993-1179588-8

Pages from the computer files of R. William Gosper
HTML articles powered by AMS MathViewer

by Mourad E. H. Ismail, Yu Takeuchi and Ruiming Zhang

Proc. Amer. Math. Soc. 119 (1993), 747-760

DOI: https://doi.org/10.1090/S0002-9939-1993-1179588-8

PDF | Request permission

Abstract:

We give proofs of summation theorems and continued fraction evaluations conjectured by R. W. Gosper. We also give two new elementary proofs of a theorem of Gosper whose original proof uses matrix methods. One proof uses iteration of two term recurrence relation. The latter proof is also used to give elementary proofs of three other identities due to Gosper.

References

Waleed A. Al-Salam and Mourad E. H. Ismail, Orthogonal polynomials associated with the Rogers-Ramanujan continued fraction, Pacific J. Math. 104 (1983), no. 2, 269–283. MR 684290
Richard Askey and Mourad Ismail, Recurrence relations, continued fractions, and orthogonal polynomials, Mem. Amer. Math. Soc. 49 (1984), no. 300, iv+108. MR 743545, DOI 10.1090/memo/0300
George Gasper and Mizan Rahman, Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, vol. 35, Cambridge University Press, Cambridge, 1990. With a foreword by Richard Askey. MR 1052153
George Gasper and Mizan Rahman, An indefinite bibasic summation formula and some quadratic, cubic and quartic summation and transformation formulas, Canad. J. Math. 42 (1990), no. 1, 1–27. MR 1043508, DOI 10.4153/CJM-1990-001-5
Ira Gessel and Dennis Stanton, Strange evaluations of hypergeometric series, SIAM J. Math. Anal. 13 (1982), no. 2, 295–308. MR 647127, DOI 10.1137/0513021
Ira Gessel and Dennis Stanton, Applications of $q$-Lagrange inversion to basic hypergeometric series, Trans. Amer. Math. Soc. 277 (1983), no. 1, 173–201. MR 690047, DOI 10.1090/S0002-9947-1983-0690047-7
William Gosper, Strip mining in the abandoned orefields of nineteenth century mathematics, Computers in mathematics (Stanford, CA, 1986) Lecture Notes in Pure and Appl. Math., vol. 125, Dekker, New York, 1990, pp. 261–284. MR 1068539
R. Wm. Gosper, Some identities, for your amusement, Ramanujan revisited (Urbana-Champaign, Ill., 1987) Academic Press, Boston, MA, 1988, pp. 607–609. MR 938981

Material from Bill Gosper’s computers & mathematics talk

R. William Gosper, Mourad E. H. Ismail, and Ruiming Zhang, On some strange summation formulas, Illinois J. Math. 37 (1993), no. 2, 240–277. MR 1208821
Mourad E. H. Ismail and Fuad S. Mulla, On the generalized Chebyshev polynomials, SIAM J. Math. Anal. 18 (1987), no. 1, 243–258. MR 871835, DOI 10.1137/0518019
William B. Jones and Wolfgang J. Thron, Continued fractions, Encyclopedia of Mathematics and its Applications, vol. 11, Addison-Wesley Publishing Co., Reading, Mass., 1980. Analytic theory and applications; With a foreword by Felix E. Browder; With an introduction by Peter Henrici. MR 595864
F. W. J. Olver, Asymptotics and special functions, Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0435697

Hypergeometric series

E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions; Reprint of the fourth (1927) edition. MR 1424469, DOI 10.1017/CBO9780511608759