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

Full-text PDF

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]**W. Al-Salam and M. E. H. Ismail,*Orthogonal polynomials associated with the Rogers-Ramanujan continued fraction*, Pacific J. Math.**104**(1983), 269-283. 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), 1-27. MR**1043508 (90k:33032)****[5]**I. Gessel and D. Stanton,*Strange evaluations of hypergeometric series*, SIAM J. Math. Anal.**13**(1982), 185-308. MR**647127 (83c:33002)****[6]**-,*Applications of*-*Lagrange inversion to basic hypergeometric series*, Trans. Amer. Math. Soc.**277**(1983), 173-201. 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. 261-284. 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. 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. W. Gosper, M. E. H. Ismail, and R. Zhang,*On some strange summation formulas*, Illinois J. Math.**37**(1993), 240-277. MR**1208821 (95g:33025)****[12]**M. E. H. Ismail and F. S. Mulla,*On the generalized Chebyshev polynomials*, SIAM J. Math. Anal.**18**(1987), 243-258. 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)**

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

Article copyright:
© Copyright 1993
American Mathematical Society