International Journal of Mathematics and Mathematical Sciences

Volume 2011 (2011), Article ID 132081, 7 pages

http://dx.doi.org/10.1155/2011/132081

## On a New Summation Formula for Basic Bilateral Hypergeometric Series and Its Applications

Department of Studies in Mathematics, University of Mysore, Manasagangotri, Mysore 570006, India

Received 7 December 2010; Accepted 24 January 2011

Academic Editor: J. Dydak

Copyright © 2011 D. D. Somashekara et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We have obtained a new summation formula for bilateral basic hypergeometric series by the method of parameter augmentation and demonstrated its various uses leading to some development of etafunctions, -gamma, and -beta function identities.

#### 1. Introduction

The summation formulae for hypergeometric series form a very interesting and useful component of the theory of (basic) hypergeometric series. The -binomial theorem of Cauchy [1] is perhaps the first identity in the class of the summation formulae, which can be stated as where For more details about the -binomial theorem and about the identities which fall in this sequel, one may refer to the book by Gasper and Rahman [2]. Another famous identity in the sequel is the Ramanujan's summation formula [3] There are a number of proofs of the summation formula (1.3) in the literature. For more details, one refers to the book by Berndt [4] and a recent paper of Johnson [5].

In this paper, we derive a new summation formula for basic bilateral hypergeometric series using the summation formula (1.3) by the method of parameter augmentation. We then use the formula to derive the -analogue of Gauss summation formula and to obtain a number of etafunction, -gamma, and -beta function identities, which complement the works of Bhargava and Somashekara [6], Bhargava et al. [7], Somashekara and Mamta [8], Srivastava [9], and Bhargava and Adiga [10].

First, we recall that -difference operator and the -shift operator are defined by respectively. In [11], Chen and Liu have constructed an operator as and thereby they defined the operator as Then, we have the following operator identities [12, Theorem 1]:

Further, the Dedekind etafunction is defined by where , and Im .

Jackson [13] defined the -analogue of the gamma function by In his paper on the -gamma and -beta function, Askey [14] has obtained -analogues of several classical results about the gamma function. Further, he has given the definition for -beta function as In fact, he has shown that

In Section 2, we prove our main result. In Section 3, we deduce the well-known -analogue of the Gauss summation formula and some etafunction, -gamma, and -beta function identities.

#### 2. Main Result

Theorem 2.1. *If , then
*

*Proof. *Ramanujan's summation formula (1.3) can be written as
This is the same as
On applying to both sides with respect to , we obtain
Multiplying (2.4) throughout by , we obtain
which yields (2.1).

#### 3. Some Applications of the Main Identity

The following identity is the well-known -analogue of the Gauss summation formula.

Corollary 3.1 (see [15]). *If , , then
*

*Proof. *Putting , , , and in (2.1), we obtain (3.1).

Corollary 3.2. *If , then
*

*Proof. *Putting , , , and then changing to in (2.1), we obtain
Simplifying the right hand side and then using (1.8), we obtain (3.2).

Similarly, putting , , , , and then changing to , we obtain (3.3). Putting , , , and then changing to , we obtain (3.4). Putting , , , , and then changing to , we obtain (3.5). Putting , , , , and then changing to , we obtain (3.6). Finally, putting , , , and then changing to , we obtain (3.7).

Corollary 3.3. *If , , , and , then
*

*Proof. *Putting , , and in (2.1), we get
On using (1.9), (1.10), and (1.11), we obtain (3.9).

Corollary 3.4. *If , , , and , then
*

*Proof. *Putting , , in (2.1), and then using (1.9), (1.10), and (1.11), we obtain (3.11).

Corollary 3.5. *If , , and , then
*

*Proof. *Letting in (3.9), we obtain (3.12).

Corollary 3.6. *If , , and , then
*

*Proof. *Letting 1 in (3.11), we obtain (3.13).

Corollary 3.7. *If , , , and , then
*

*Proof. *Putting , , and in (2.1), we obtain (3.14).

#### Acknowledgment

The authors would like to thank the referees for their valuable suggestions which considerably improved the quality of the paper.

#### References

- A.-L. Cauchy, “Mémoire sur les fonctions dont plusieurs valeurs sont liées entre elles par une équation linéaire, et sur diverses transformations de produits composés d'un nombre indéfini de facteurs,”
*Comptes Rendus de l'Académie des Sciences, Paris*, vol. 17, pp. 42–50, 1893, Oeuvres de Cauchy, 1re érie, T. VIII, Gauthier-Villars, Paris, France. View at Google Scholar - G. Gasper and M. Rahman,
*Basic Hypergeometric Series*, vol. 96 of*Encyclopedia of Mathematics and Its Applications*, Cambridge University Press, Cambridge, UK, 2nd edition, 2004. View at Publisher · View at Google Scholar - S. Ramanujan,
*Notebooks*, vol. 2, Tata Institute of Fundamental Research, Bombay, India, 1957. - B. C. Berndt,
*Ramanujan's Notebooks. Part III*, Springer, New York, NY, USA, 1991. - W. P. Johnson, “How Cauchy missed Ramanujan's ${}_{1}{\psi}_{1}$ summation,”
*American Mathematical Monthly*, vol. 111, no. 9, pp. 791–800, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Bhargava and D. D. Somashekara, “Some eta-function identities deducible from Ramanujan's ${}_{1}{\psi}_{1}$ summation,”
*Journal of Mathematical Analysis and Applications*, vol. 176, no. 2, pp. 554–560, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Bhargava, D. D. Somashekara, and S. N. Fathima, “A new transformation for basic bilateral hypergeometric series,”
*Mathematics Student*, vol. 74, no. 1–4, pp. 225–230, 2005. View at Google Scholar · View at Zentralblatt MATH - D. D. Somashekara and D. Mamta, “On some identities of Ramanujan found in his lost notebook,”
*Advanced Studies in Contemporary Mathematics*, vol. 16, no. 2, pp. 171–180, 2008. View at Google Scholar · View at Zentralblatt MATH - H. M. Srivastava, “Some convolution identities based upon Ramanujan’s bilateral sum,”
*Bulletin of the Australian Mathematical Society*, vol. 49, pp. 433–427, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. Bhargava and C. Adiga, “A basic bilateral series summation formula and its applications,”
*Integral Transforms and Special Functions*, vol. 2, no. 3, pp. 165–184, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - W. Y. C. Chen and Z.-G. Liu, “Parameter augmentation for basic hypergeometric series. I,” in
*Mathematical Essays in Honor of Gian-Carlo Rota (Cambridge, MA, 1996)*, B. E. Sagan and R. P. Stanley, Eds., vol. 161 of*Progress in Mathematics*, pp. 111–129, Birkhäuser, Boston, Mass, USA, 1998. View at Google Scholar · View at Zentralblatt MATH - Z.-G. Liu, “Some operator identities and $q$-series transformation formulas,”
*Discrete Mathematics*, vol. 265, no. 1–3, pp. 119–139, 2003. View at Publisher · View at Google Scholar · View at MathSciNet - F. H. Jackson, “On $q$- definite integrals,”
*Pure and Applied Mathematics Quarterly*, vol. 41, pp. 193–203, 1910. View at Google Scholar - R. Askey, “The $q$-gamma and $q$-beta functions,”
*Applicable Analysis*, vol. 8, no. 2, pp. 125–141, 1978. View at Publisher · View at Google Scholar - E. Heine, “Untersuchungen über die Reihe
$1+[(1-{q}^{\alpha})(1-{q}^{\beta})]/[(1-q)(1-{q}^{\gamma})]X+$$[(1-{q}^{\alpha})(1-{q}^{\alpha +1}){(1-{q}^{\beta})}^{\prime}{(1-{q}^{\beta +1})}^{\prime}]/[(1-q)(1-{q}^{2})(1-{q}^{\gamma})(1-{q}^{\gamma +1})]{X}^{2}+\cdots $,”
*Journal fur die Reine und Angewandte Mathematik*, vol. 34, pp. 285–328, 1847. View at Google Scholar