Proceedings of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6826(e) ISSN 0002-9939(p)

Semi-finite forms of bilateral basic hypergeometric series

Authors: William Y. C. Chen and Amy M. Fu
Journal: Proc. Amer. Math. Soc. 134 (2006), 1719-1725
MSC (2000): Primary 33D15
Posted: December 5, 2005
MathSciNet review: 2204284
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show that several classical bilateral summation and transformation formulas have semi-finite forms. We obtain these semi-finite forms from unilateral summation and transformation formulas. Our method can be applied to derive Ramanujan's $_{1}\psi _{1}$ summation, Bailey's $_{2}\psi _{2}$ transformations, and Bailey's $_{6}\psi _{6}$ summation.

References

1. G. E. Andrews, Applications of basic hypergeometric functions, SIAM, Rev. 16 (1974), 441-484. MR 0352557 (50:5044)
2. R. Askey and M. E. H. Ismail, The very well poised $_{6}\psi _{6}$ , Proc. Amer. Math. Soc. 77 (1979), 218-222. MR 0542088 (80m:33002)
3. R. Askey, The very well poised $_{6}\psi _{6}$ II, Proc. Amer. Math. Soc. 90 (1984), 575-579. MR 0733409 (85h:33001)
4. W. N. Bailey, Series of hyerpergeometric type which are infinite in both directions, Quart. J. Math. 7 (1936), 105-115.
5. W. Y. C. Chen and Z. G. Liu, Parameter augmentation for basic hypergeometric series, I, In: Mathematical Essays in Honor of Gian-Carlo Rota, Eds., B. E. Sagan and R. P. Stanley, Birkhäuser, Boston, 1998, pp. 111-129. MR 1627355 (99f:33020)
6. G. Gasper and M. Rahman, Basic Hypergeometric Series, $2^{\text{nd}}$ ed., Encyclopedia of Mathematics and Its Applications, Vol. 96, Cambridge University Press, Cambridge, 2004. MR 2128719
7. 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)
8. F. Jouhet and M. Schlosser, Another proof of Bailey's $_{6}\psi _{6}$ summation, Aequationes Math. 70 (2005), 43-50.
9. M. Schlosser, A simple proof of Bailey's very-well-poised $_{6}\psi _{6}$ summation, Proc. Amer. Math. Soc. 130 (2002), 1113-1123. MR 1873786 (2003b:33027)
10. 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, Springer, New York, 2005, pp. 383-400. MR 2132472
11. L. J. Slater and A. Lakin, Two proofs of the $_{6}\psi _{6}$ summation theorem, Proc. Edin. Math. Soc. 9 (1956), 116-121. MR 0084600 (18:888b)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 33D15

Retrieve articles in all journals with MSC (2000): 33D15

Additional Information

William Y. C. Chen
Affiliation: Center for Combinatorics, LPMC, Nankai University, Tianjin 300071, People's Republic of China
Email: chen@nankai.edu.cn

Amy M. Fu
Affiliation: Center for Combinatorics, LPMC, Nankai University, Tianjin 300071, People's Republic of China
Email: fu@nankai.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08173-6
PII: S 0002-9939(05)08173-6
Keywords: Bilateral hypergeometric summation, semi-finite forms, Ramanujan's ${}_{1}\psi _{1}$ summation, Bailey's ${}_{2}\psi _{2}$ transformations, Bailey's ${}_{6}\psi _{6}$ summation.
Received by editor(s): December 8, 2004
Received by editor(s) in revised form: January 11, 2005
Posted: December 5, 2005
Communicated by: John R. Stembridge
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|