Mathematics of Computation

Author(s): Vincent Y. B. Chen; William Y. C. Chen; Nancy S. S. Gu.
Journal: Math. Comp. 77 (2008), 1057-1074.
MSC (2000): Primary 33D15; Secondary 33F10
Posted: October 24, 2007
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Abstract: Chu has recently shown that the Abel lemma on summation by parts reveals the telescoping nature of Bailey's ${}_6\psi_6$ bilateral summation formula. We present a systematic approach to compute Abel pairs for bilateral and unilateral basic hypergeometric summation formulas by using the

-Gosper algorithm. It is demonstrated that Abel pairs can be derived from Gosper pairs. This approach applies to many classical summation formulas.

1.: S. A. Abramov, P. Paule, and M. Petkovšek, -Hypergeometric solutions of -difference equations, Discrete Math. 180 (1998), 3-22. MR 1603685 (99f:39001)
2.: G. E. Andrews, On Ramanujan's summation of $_{1}\psi_1(a,b,z)$ , Proc. Amer. Math. Soc. 22 (1969), 552-553. MR 0241703 (39:3042)
3.: G. E. Andrews, On a transformation of bilateral series with applications, Proc. Amer. Math. Soc. 25 (1970), 554-558. MR 0257413 (41:2064)
4.: G. E. Andrews and R. Askey, A simple proof of Ramanujan's summation of the $_{1}\psi_1$ , Aequationes Math. 18 (1978), 333-337. MR 522519 (80h:33004)
5.: W. N. Bailey, Series of hypergeometric type which are infinite in both directions, Quart. J. Math. (Oxford) 7 (1936), 105-115.
6.: B. C. Berndt, Ramanujan's theory of theta-functions, In: Theta Functions from the Classical to the Modern, M. R. Murty, ed., CRM Proceedings and Lecture Notes, 1, Amer. Math. Soc., Providence, 1993, pp. 1-63. MR 1224050 (94m:11054)
7.: H. Böing and W. Koepf, Algorithm for -hypergeometric summation in computer algebra, J. Symbolic Comput. 28 (1999), 777-799. MR 1750546 (2001j:33019)
8.: W. Y. C. Chen, Q. H. Hou, and Y. P. Mu, Nonterminating basic hypergeometric series and the -Zeilberger algorithm, arXiv:math.CO/0509281.
9.: W. C. Chu, Bailey's very well-poised $_6\psi_6$ -series identity, J. Combin. Theory, Ser. A 113 (2006), 966-979 . MR 2244127
10.: G. Gasper and M. Rahman, Basic Hypergeometric Series, second ed., Cambridge University Press, Cambridge, 2004. MR 2128719 (2006d:33028)
11.: W. Hahn, Über Polynome, die gleichzeitig zwei verschiedenen Orthogonalsystemen angehören, Math. Nachr. 2 (1949), 263-278. MR 0032858 (11:356f)
12.: M. E. H. Ismail, A simple proof of Ramanujan's $_{1}\psi_1$ sum, Proc. Amer. Math. Soc. 63 (1977), 185-186. MR 0508183 (58:22695)
13.: M. Jackson, On Lerch's transcendant and the basic bilateral hypergeometric series $_{2}\psi_2$ , J. London Math. Soc. 25 (1950), 189-196. MR 0036882 (12:178f)
14.: T. H. Koornwinder, On Zeilberger's algorithm and its -analogue, J. Comput. Appl. Math. 48 (1993), 91-111. MR 1246853 (95b:33011)
15.: M. Mohammed and D. Zeilberger, Sharp upper bounds for the orders of the recurrences outputted by the Zeilberger and -Zeilberger Algorithms, J. Symbolic Comput. 39 (2005), 201-207. MR 2169800
16.: P. Paule, Short and easy computer proofs of the Roger-Ramanujan identities and of identities of similar type, Electron. J. Combin. 1 (1994), R10. MR 1293400 (95g:11099)
17.: P. Paule and M. Schorn, A Mathematica version of Zeilberger's algorithm for proving binomial coefficients identities, J. Symbolic Comput. 20 (1995), 673-698. MR 1395420 (97j:39006)
18.: A. Riese, A generalization of Gosper's algorithm to bibasic hypergeometric summation, Electron. J. Combin. 3 (1996), R19. MR 1394550 (97h:33031)
19.: M. Schlosser, Abel-Rothe type generalizations of Jacobi's triple product identity, in ``Theory and Applications of Special Functions. A Volume Dedicated to Mizan Rahman" (M. E. H. Ismail and E. Koelink, eds.), Dev. Math. 13 (2005), 383-400. MR 2132472 (2006b:33032)
20.: M. Schlosser, Inversion of bilateral basic hypergeometric series, Electron. J. Combin. 10 (2003), R10. MR 1975760 (2004d:33017)
21.: H. S. Shukla, A note on the sums of certain bilateral hypergeometric series, Proc. Cambridge Phil. Soc. 55 (1959), 262-266. MR 0104839 (21:3590)
22.: H. S. Wilf and D. Zeilberger, An algorithmic proof theory for hypergeometric (ordinary and ``") multisum/integral identities, Invent. Math. 108 (1992), 575-633. MR 1163239 (93k:33010)

Vincent Y. B. Chen
Affiliation: Center for Combinatorics, LPMC, Nankai University, Tianjin 300071, P. R. China
Email: ybchen@mail.nankai.edu.cn

William Y. C. Chen
Affiliation: Center for Combinatorics, LPMC, Nankai University, Tianjin 300071, P. R. China
Email: chen@nankai.edu.cn

Nancy S. S. Gu
Affiliation: Center for Combinatorics, LPMC, Nankai University, Tianjin 300071, P. R. China
Email: gu@nankai.edu.cn

DOI: 10.1090/S0025-5718-07-01968-0
PII: S 0025-5718(07)01968-0
Keywords: The Abel lemma, Abel pairs, basic hypergeometric series, the $q$-Gosper algorithm, Gosper pairs
Received by editor(s): July 26, 2006
Received by editor(s) in revised form: August 2, 2006.
Posted: October 24, 2007
Additional Notes: This work was supported by the 973 Project on Mathematical Mechanization, the Ministry of Education, the Ministry of Science and Technology, and the National Science Foundation of China.
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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