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), 747760
MSC:
Primary 33B10; Secondary 33D15, 40A15
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.
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Waleed
A. AlSalam and Mourad
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Richard
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Rahman, An indefinite bibasic summation formula and some quadratic,
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, private communication, July 11, 1991.
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William Gosper, Mourad
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37 (1993), no. 2, 240–277. MR 1208821
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E. H. Ismail and Fuad
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 [1]
 W. AlSalam and M. E. H. Ismail, Orthogonal polynomials associated with the RogersRamanujan continued fraction, Pacific J. Math. 104 (1983), 269283. MR 684290 (85g:33007)
 [2]
 R. Askey and M. E. H. Ismail, Recurrence relations, continued fractions and orthogonal polynomials, Mem. Amer. Math. Soc., vol. 49, no. 300, Amer. Math. Soc., Providence, RI, 1984. MR 743545 (85g:33008)
 [3]
 G. Gasper and M. Rahman, Basic hypergeometric series, Cambridge Univ. Press, Cambridge, 1990. MR 1052153 (91d:33034)
 [4]
 , An infinite bibasic summation formula and some quadratic transformation formulas, Canad. J. Math. 42 (1990), 127. MR 1043508 (90k:33032)
 [5]
 I. Gessel and D. Stanton, Strange evaluations of hypergeometric series, SIAM J. Math. Anal. 13 (1982), 185308. MR 647127 (83c:33002)
 [6]
 , Applications of Lagrange inversion to basic hypergeometric series, Trans. Amer. Math. Soc. 277 (1983), 173201. MR 690047 (84f:33009)
 [7]
 R. W. Gosper, Strip mining in the abandoned ore fields of nineteenth century mathematics, Computers in Mathematics (D. V. Chudnovsky and R. D. Jenks, eds.), Lecture Notes in Pure and Appl. Math., vol. 125, Marcel Dekker, New York, 1990, pp. 261284. MR 1068539 (91h:11154)
 [8]
 , Some identities for your amusement, Ramanujan Revisited (G. Andrews, R. Askey, B. Berndt, K. Ramanathan, and R. Rankin, eds.), Academic Press, New York, 1988, pp. 607609. MR 938981 (89d:05016)
 [9]
 , Material from Bill Gosper's computers & mathematics talk, M. I. T., June 1989.
 [10]
 , private communication, July 11, 1991.
 [11]
 R. W. Gosper, M. E. H. Ismail, and R. Zhang, On some strange summation formulas, Illinois J. Math. 37 (1993), 240277. MR 1208821 (95g:33025)
 [12]
 M. E. H. Ismail and F. S. Mulla, On the generalized Chebyshev polynomials, SIAM J. Math. Anal. 18 (1987), 243258. MR 871835 (88d:33009)
 [13]
 W. Jones and W. Thron, Continued fractions, analytic theory and applications, Cambridge Univ. Press, Cambridge, 1984. MR 595864 (82c:30001)
 [14]
 F. W. J. Olver, Asymptotics and special functions, Academic Press, New York, 1974. MR 0435697 (55:8655)
 [15]
 L. J. Slater, Hypergeometric series, Cambridge Univ. Press, Cambridge, 1966.
 [16]
 E. T. Whittaker and G. N. Watson, A course of modern analysis, fourth ed., Cambridge Univ. Press, Cambridge, 1966. MR 1424469 (97k:01072)
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DOI:
http://dx.doi.org/10.1090/S00029939199311795888
PII:
S 00029939(1993)11795888
Article copyright:
© Copyright 1993
American Mathematical Society
