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

Published electronically:
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 summation, Bailey's transformations, and Bailey's summation.

**1.**George E. Andrews,*Applications of basic hypergeometric functions*, SIAM Rev.**16**(1974), 441–484. MR**0352557****2.**Richard Askey and Mourad E. H. Ismail,*The very well poised ₆𝜓₆*, Proc. Amer. Math. Soc.**77**(1979), no. 2, 218–222. MR**542088**, 10.1090/S0002-9939-1979-0542088-2**3.**Richard Askey,*The very well poised ₆𝜓₆. II*, Proc. Amer. Math. Soc.**90**(1984), no. 4, 575–579. MR**733409**, 10.1090/S0002-9939-1984-0733409-8**4.**W. N. Bailey,*Series of hyerpergeometric type which are infinite in both directions*, Quart. J. Math.**7**(1936), 105-115.**5.**William Y. C. Chen and Zhi-Guo Liu,*Parameter augmentation for basic hypergeometric series. I*, Mathematical essays in honor of Gian-Carlo Rota (Cambridge, MA, 1996), Progr. Math., vol. 161, Birkhäuser Boston, Boston, MA, 1998, pp. 111–129. MR**1627355****6.**George Gasper and Mizan Rahman,*Basic hypergeometric series*, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 96, Cambridge University Press, Cambridge, 2004. With a foreword by Richard Askey. MR**2128719****7.**Mourad E. H. Ismail,*A simple proof of Ramanujan’s ₁𝜓₁ sum*, Proc. Amer. Math. Soc.**63**(1977), no. 1, 185–186. MR**0508183**, 10.1090/S0002-9939-1977-0508183-7**8.**F. Jouhet and M. Schlosser,*Another proof of Bailey's summation*, Aequationes Math.**70**(2005), 43-50.**9.**Michael Schlosser,*A simple proof of Bailey’s very-well-poised ₆𝜓₆ summation*, Proc. Amer. Math. Soc.**130**(2002), no. 4, 1113–1123 (electronic). MR**1873786**, 10.1090/S0002-9939-01-06175-5**10.**Michael Schlosser,*Abel-Rothe type generalizations of Jacobi’s triple product identity*, Theory and applications of special functions, Dev. Math., vol. 13, Springer, New York, 2005, pp. 383–400. MR**2132472**, 10.1007/0-387-24233-3_17**11.**L. J. Slater and A. Lakin,*Two proofs of the ₆Ψ₆ summation theorem*, Proc. Edinburgh Math. Soc. (2)**9**(1956), 116–121. MR**0084600**

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:
https://doi.org/10.1090/S0002-9939-05-08173-6

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

Published electronically:
December 5, 2005

Communicated by:
John R. Stembridge

Article copyright:
© Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.