CERTAIN HYPERGEOMETRIC IDENTITIES DEDUCIBLE BY USING THE BETA INTEGRAL METHOD

Title & Authors
CERTAIN HYPERGEOMETRIC IDENTITIES DEDUCIBLE BY USING THE BETA INTEGRAL METHOD
Choi, Junesang; Rathie, Arjun K.; Srivastava, Hari M.;

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.
Keywords
generalized hypergeometric function $\small{_pF_q}$;Gamma function;Pochhammer symbol;Beta integral method;Kummers formula;generalization of Kummers formula;
Language
English
Cited by
1.
FURTHER HYPERGEOMETRIC IDENTITIES DEDUCIBLE BY FRACTIONAL CALCULUS,;;

대한수학회논문집, 2014. vol.29. 3, pp.429-437
2.
TWO GENERAL HYPERGEOMETRIC TRANSFORMATION FORMULAS,;;

대한수학회논문집, 2014. vol.29. 4, pp.519-526
3.
Reduction Formulas for Srivastava's Triple Hypergeometric Series F(3)[x, y, z],;;;

Kyungpook mathematical journal, 2015. vol.55. 2, pp.439-447
4.
CERTAIN HYPERGEOMETRIC SERIES IDENTITIES DEDUCIBLE BY FRACTIONAL CALCULUS,;;

Advanced Studies in Contemporary Mathematics, 2016. vol.26. 2, pp.323-332
1.
Remark on certain transformations for multiple hypergeometric functions, Advances in Difference Equations, 2014, 2014, 1, 126
2.
FURTHER HYPERGEOMETRIC IDENTITIES DEDUCIBLE BY FRACTIONAL CALCULUS, Communications of the Korean Mathematical Society, 2014, 29, 3, 429
3.
TWO GENERAL HYPERGEOMETRIC TRANSFORMATION FORMULAS, Communications of the Korean Mathematical Society, 2014, 29, 4, 519
4.
Reduction Formulas for Srivastava's Triple Hypergeometric Series F(3)[x, y, z], Kyungpook mathematical journal, 2015, 55, 2, 439
5.
Certain transformations for multiple hypergeometric functions, Advances in Difference Equations, 2013, 2013, 1, 360
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.

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.

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.

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.