# CERTAIN HYPERGEOMETRIC IDENTITIES DEDUCIBLE BY USING THE BETA INTEGRAL METHOD

• Choi, Junesang (Department of Mathematics Dongguk University) ;
• Rathie, Arjun K. (Department of Mathematics School of Mathematical and Physical Sciences Central University of Kerala Riverside Transit Campus) ;
• Srivastava, Hari M. (Department of Mathematics and Statistics University of Victoria)
• Published : 2013.09.30

#### Abstract

The main objective of this paper is to show how one can obtain eleven new and interesting hypergeometric identities in the form of a single result from the old ones by mainly employing the known beta integral method which was recently introduced and used in a systematic manner by Krattenthaler and Rao [6]. The results are derived with the help of a generalization of a well-known hypergeometric transformation formula due to Kummer. Several identities including one obtained earlier by Krattenthaler and Rao [6] follow as special cases of our main results.

#### Acknowledgement

Supported by : Natural Sciences and Engineering Research Council of Canada

#### References

1. W. N. Bailey, Generalized Hypergeometric Series, Cambridge Tracts in Mathematics and Mathematical Physics, Vol. 32, Cambridge University Press, Cambridge, London and New York, 1935; Reprinted by Stechert-Hafner Service Agency, New York and London, 1964.
2. B. C. Berndt, Ramanujan's Notebooks. Part II, Springer-Verlag, Berlin, Heidelberg and New York, 1989.
3. J. Choi and A. K. Rathie, Two formulas contiguous to a quadratic transformation due to Kummer with an application, Hacet. J. Math. Stat. 40 (2011), no. 6, 885-894.
4. J. Choi, A. K. Rathie, and H. M. Srivastava, A generalization of a formula due to Kummer, Integral Transforms Spec. Funct. 22 (2011), no. 11, 851-859. https://doi.org/10.1080/10652469.2011.588786
5. A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions. Vols. I, II, McGraw-Hill Book Company, New York, Toronto and London, 1953.
6. C. Krattenthaler and K. S. Rao, Automatic generation of hypergeometric identities by the beta integral method, J. Comput. Appl. Math. 160 (2003), no. 1-2, 159-173. https://doi.org/10.1016/S0377-0427(03)00629-0
7. E. E. Kummer, Uber die hypergeometrische Reihe $1+\frac{{\alpha\cdot\beta}}{{1{\cdot}}{\gamma}}x$+${\frac{{\alpha}({\alpha}+1){\cdot}{\beta}({\beta}+1)}{{1{\cdot}2{\cdot}{\gamma}({\gamma+1})}}x^2+{\cdot}{\cdot}{\cdot}$, J. Reine Angew. Math. 15 (1836), 39-83 and 127-172.
8. J.-L. Lavoie, F. Grondin, and A. K. Rathie, Generalizations of Whipple's theorem on the sum of a $_3F_2$, J. Comput. Appl. Math. 72 (1996), no. 2, 293-300. https://doi.org/10.1016/0377-0427(95)00279-0
9. E. D. Rainville, Special Functions, Macmillan Company, New York, 1960; Reprinted by Chelsea Publishing Company, Bronx, New York, 1971.
10. H. M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, London and New York, 2012.

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