ISRN Algebra

Volume 2011 (2011), Article ID 248519, 10 pages

http://dx.doi.org/10.5402/2011/248519

## Certain Transformation Formulae for Polybasic Hypergeometric Series

Department of Mathematics, Motilal Nehru National Institute of Technology, Allahabad 211 004, India

Received 4 August 2011; Accepted 21 August 2011

Academic Editors: A. Kiliçman, H. Rosengren, and A. Salemi

Copyright © 2011 Pankaj Srivastava and Mohan Rudravarapu. 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

Making use of Bailey's transformation and certain known summations of truncated series, an attempt has been made to establish transformation formulae involving polybasic hypergeometric series.

#### 1. Introduction

The remarkable contribution in the field of hypergeometric and basic hypergeometric series mainly due to Bailey [1] has appeared in Proceeding of London Mathematical society in 1947. The key result of the paper later on recognized as Bailley's transformation is as follows: where are functions of only, such that the series for exists. Bailey's paper [2] published in the London Mathematical society in 1949, that strengthened the importance of Bailey’s transformation. The main result of the paper [2] was recognized as Bailey’s lemma during the 20th century. Making use of celebrated transformation, Bailey [1, 2] developed a number of transformations for both ordinary and basic hypergeometric series, and later on he successfully used these transformations to obtain a number of identities of the Rogers-Ramanujan type. The extensive use of Bailey transformation appeared in the papers of Slater [3, 4] and these papers were published in 1951 and 1952, respectively. Slater established 130 identities of the Rogers-Ramanujan type in [3, 4]. The platform provided by Bailey and Slater motivated a number of mathematicians namely Agarwal [5, 6], Andrews [7–9], Andrews and Warner [10], Bressoud et al. [11, 12], Denis et al. [13], Joshi and Vyas [14], Schilling and Warnaar [15], Singh [16], Srivastava [17], Verma and Jain [18, 19] and due to the contribution of these mathematicians, literatures of ordinary and basic hypergeometric series were enriched. In the present paper, making use of certain known summations of truncated series, an attempt has been made to establish transformation formulae involving poly-basic hypergeometric series.

#### 2. Definitions and Notations

For real or complex , put Let be defined by

For arbitrary parameters and , so that the generalized basic hypergeometric series is defined by: where and for convergence.

The truncated basic hypergeometric series is defined by

The polybasic hypergeometric series is defined by (cf. Gasper and Rahman [20, (3.9.1) page 85]): where for convergence.

The other notations appearing in this paper have their usual meaning. We will use the following summation formulae in our analysis: see [5, App.II(8)] see [5, App.II(8)] see [5, App.II(25)] provided , see [20, App.II(II.34)] see [20, App.II(II.35)] which is , case of [20, App. II (II. 36)].

#### 3. Main Results

In this section we have established the following main results.

#### 4. Proof of Main Results

Taking in (1.1), Bailey's transformation takes the following form:

*Proof of Result (3.1). *Taking and in (4.1) and (4.2), respectively, and making use of (2.7), we get
Putting these values in (4.3), we get the following transformation:
which on simplification gives the result (3.1).

*Proof of Result (3.2). *Taking and in (4.1) and (4.2), respectively, and making use of (2.8) and (2.7), we get
Substituting these values in (4.3), we get the following transformation for :
which on simplification gives result (3.2).

*Proof of Result (3.3). *Taking ///;, where and in (4.1) and (4.2), respectively, and making use of (2.9) and (2.7), we get
Substituting these values in (4.3), we get the following transformation for :
which on simplification gives result (3.3).

*Proof of Result (3.4). *Taking and in (4.1) and (4.2), respectively and making use of (2.10) and (2.7), we get
Putting these values in (4.3), we get the following transformation for :
which on simplification gives result (3.4).

*Proof of Result (3.5). *Taking / (// and in (4.1) and (4.2), respectively, and making use of (2.11) and (2.7), we get
Putting these values in (4.3), we get the following transformation:
which on simplification gives result (3.5).

*Proof of Result (3.6). *Taking /((( and in (4.1) and (4.2), respectively, and making use of (2.12) and (2.7), we get
Putting these values in (4.3), we get the following transformation:
which on simplification gives result (3.6).

#### References

- W. N. Bailey, “Some identities in combinatory analysis,”
*Proceedings of the London Mathematical Society*, vol. 49, no. 2, pp. 421–425, 1947. View at Publisher · View at Google Scholar - W. N. Bailey, “Identities of the Rogers-Ramanujan type,”
*Proceedings of the London Mathematical Society*, vol. 50, pp. 1–10, 1948. View at Publisher · View at Google Scholar - L. J. Slater, “A new proof of Rogers's transformations of infinite series,”
*Proceedings of the London Mathematical Society*, vol. 53, pp. 460–475, 1951. View at Publisher · View at Google Scholar - L. J. Slater, “Further identies of the Rogers-Ramanujan type,”
*Proceedings of the London Mathematical Society*, vol. 54, pp. 147–167, 1952. View at Publisher · View at Google Scholar - R. P. Agarwal, “Generalized hypergeometric series and its applications to the theory of combinatorial analysis and partition theory,” unpublished monograph.
- R. P. Agarwal,
*Resonance Of Ramanujan's Mathematics, Vol .I*, New Age International, New Delhi, India, 1996. - G. E. Andrews, “A general theory of identities of the Rogers-Ramanujan type,”
*Bulletin of the American Mathematical Society*, vol. 80, pp. 1033–1052, 1974. View at Publisher · View at Google Scholar - G. E. Andrews, “An analytic generalization of the Rogers-Ramanujan identities for odd moduli,”
*Proceedings of the National Academy of Sciences of the United States of America*, vol. 71, pp. 4082–4085, 1974. View at Publisher · View at Google Scholar - G. E. Andrews, “Bailey's transform, lemma, chains and tree,” in
*Special Functions 2000: Current Perspective and Future Directions, Tempe, AZ*, vol. 30 of*NATO Science Series. Series II, Mathematics, Physics and Chemistry*, pp. 1–22, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001. View at Publisher · View at Google Scholar - G. E. Andrews and S. O. Warnaar, “The Bailey transform and false theta functions,”
*The Ramanujan Journal*, vol. 14, no. 1, pp. 173–188, 2007. View at Publisher · View at Google Scholar - D. M. Bressoud, “Some identities for terminating
*q*-series,”*Mathematical Proceedings of the Cambridge Philosophical Society*, vol. 89, no. 2, pp. 211–223, 1981. View at Publisher · View at Google Scholar - D. M. Bressoud, M. E. H. Ismail, and D. Stanton, “Change of base in Bailey pairs,”
*The Ramanujan Journal*, vol. 4, no. 4, pp. 435–453, 2000. View at Publisher · View at Google Scholar - R. Y. Denis, S. N. Singh, and S. P. Singh, “On certain transformation formulae for abnormal
*q*-series,”*South East Asian Journal of Mathematics and Mathematical Sciences*, vol. 1, no. 3, pp. 7–19, 2003. View at Google Scholar - C. M. Joshi and Y. Vyas, “Extensions of Bailey's transform and applications,”
*International Journal of Mathematics and Mathematical Sciences*, no. 12, pp. 1909–1923, 2005. View at Publisher · View at Google Scholar - A. Schilling and S. O. Warnaar, “A higher-level Bailey lemma,”
*International Journal of Modern Physics B*, vol. 11, no. 1-2, pp. 189–195, 1997. View at Publisher · View at Google Scholar - U. B. Singh, “A note on a transformation of Bailey,”
*The Quarterly Journal of Mathematics*, vol. 45, no. 177, pp. 111–116, 1994. View at Publisher · View at Google Scholar - P. Srivastava, “A note on Bailey's transform,”
*South East Asian Journal of Mathematics and Mathematical Sciences*, vol. 2, no. 2, pp. 9–14, 2004. View at Google Scholar - A. Verma and V. K. Jain, “Transformations between basic hypergeometric series on different bases and identities of Rogers-Ramanujan type,”
*Journal of Mathematical Analysis and Applications*, vol. 76, no. 1, pp. 230–269, 1980. View at Publisher · View at Google Scholar - A. Verma and V. K. Jain, “Transformations of nonterminating basic hypergeometric series, their contour integrals and applications to Rogers-Ramanujan identities,”
*Journal of Mathematical Analysis and Applications*, vol. 87, no. 1, pp. 9–44, 1982. View at Publisher · View at Google Scholar - G. Gasper and M. Rahman,
*Basic Hypergeometric Series*, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, New York, NY, USA, 1991.