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
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
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. AlSalam and Mourad
E. H. Ismail, Orthogonal polynomials associated with the
RogersRamanujan continued fraction, Pacific J. Math.
104 (1983), no. 2, 269–283. MR 684290
(85g:33007)
 [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
(85g:33008)
 [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
(91d:33034)
 [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
(90k:33032), http://dx.doi.org/10.4153/CJM19900015
 [5]
Ira
Gessel and Dennis
Stanton, Strange evaluations of hypergeometric series, SIAM J.
Math. Anal. 13 (1982), no. 2, 295–308. MR 647127
(83c:33002), http://dx.doi.org/10.1137/0513021
 [6]
Ira
Gessel and Dennis
Stanton, Applications of 𝑞Lagrange
inversion to basic hypergeometric series, Trans. Amer. Math. Soc. 277 (1983), no. 1, 173–201. MR 690047
(84f:33009), http://dx.doi.org/10.1090/S00029947198306900477
 [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
(91h:11154)
 [8]
R.
Wm. Gosper, Some identities, for your amusement, Ramanujan
revisited (UrbanaChampaign, Ill., 1987) Academic Press, Boston, MA,
1988, pp. 607–609. 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.
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
(95g:33025)
 [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
(88d:33009), http://dx.doi.org/10.1137/0518019
 [13]
William
B. Jones and Wolfgang
J. Thron, Continued fractions, Encyclopedia of Mathematics and
its Applications, vol. 11, AddisonWesley Publishing Co., Reading,
Mass., 1980. Analytic theory and applications; With a foreword by Felix E.
Browder; With an introduction by Peter Henrici. MR 595864
(82c:30001)
 [14]
F.
W. J. Olver, Asymptotics and special functions, Academic Press
[A subsidiary of Harcourt Brace Jovanovich, Publishers], New YorkLondon,
1974. Computer Science and Applied Mathematics. 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, 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
(97k:01072)
 [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)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC:
33B10,
33D15,
40A15
Retrieve articles in all journals
with MSC:
33B10,
33D15,
40A15
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199311795888
PII:
S 00029939(1993)11795888
Article copyright:
© Copyright 1993 American Mathematical Society
