Pages from the computer files of R. William Gosper

Authors:
Mourad E. H. Ismail, Yu Takeuchi and Ruiming Zhang

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

MSC:
Primary 33B10; Secondary 33D15, 40A15

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

MathSciNet review:
1179588

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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.

**[1]**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****[2]**Richard Askey and Mourad Ismail,*Recurrence relations, continued fractions, and orthogonal polynomials*, Mem. Amer. Math. Soc.**49**(1984), no. 300, iv+108. MR**743545**, https://doi.org/10.1090/memo/0300**[3]**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****[4]**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**, https://doi.org/10.4153/CJM-1990-001-5**[5]**Ira Gessel and Dennis Stanton,*Strange evaluations of hypergeometric series*, SIAM J. Math. Anal.**13**(1982), no. 2, 295–308. MR**647127**, https://doi.org/10.1137/0513021**[6]**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**, https://doi.org/10.1090/S0002-9947-1983-0690047-7**[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****[8]**R. Wm. Gosper,*Some identities, for your amusement*, Ramanujan revisited (Urbana-Champaign, Ill., 1987) Academic Press, Boston, MA, 1988, pp. 607–609. MR**938981****[9]**-,*Material from Bill Gosper's computers & mathematics talk*, M. I. T., June 1989.**[10]**-, private communication, July 11, 1991.**[11]**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****[12]**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**, https://doi.org/10.1137/0518019**[13]**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****[14]**F. W. J. Olver,*Asymptotics and special functions*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Computer Science and Applied Mathematics. MR**0435697****[15]**L. J. Slater,*Hypergeometric series*, Cambridge Univ. Press, Cambridge, 1966.**[16]**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**

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

Article copyright:
© Copyright 1993
American Mathematical Society