# CERTAIN SUMMATION FORMULAS FOR HUMBERT'S DOUBLE HYPERGEOMETRIC SERIES Ψ2 AND Φ2

• CHOI, JUNESANG (Department of Mathematics Dongguk University) ;
• RATHIE, ARJUN KUMAR (Department of Mathematics School of Mathematical and Physical Sciences Central University of Kerala, Riverside Transit Campus)
• Published : 2015.10.31

#### Abstract

The main objective of this paper is to establish certain explicit expressions for the Humbert functions ${\Phi}_2$(a, a + i ; c ; x, -x) and ${\Psi}_2$(a ; c, c + i ; x, -x) for i = 0, ${\pm}1$, ${\pm}2$, ..., ${\pm}5$. Several new and known summation formulas for ${\Phi}_2$ and ${\Psi}_2$ are considered as special cases of our main identities.

#### References

1. Yu. A. Brychkov, Handbook of Special Functions, Derivatives, Integrals, Series and Other Formulas, CRC Press, Taylar & Francis Group, A Chapman & Hall Book, 2008.
2. J. L. Burchnall and T. W. Chaundy, Expansions of Appell's double hypergeometric functions, Quart. J. Math. (Oxford Ser.) 11 (1940), 249-270.
3. J. L. Burchnall and T. W. Chaundy, Expansions of Appell's double hypergeometric functions. II, Quart. J. Math. (Oxford Ser.) 12 (1941), 112-128.
4. 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
5. V. V. Manako, A connection formula between double hypergeometric series ${\Psi}_2$ and ${\Phi}_3$, Integral Transforms Spec. Funct. 23 (2012), no. 7, 503-508. https://doi.org/10.1080/10652469.2011.607450
6. A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series. Vol. 3, Gordon and Breach Science Publishers, New York, 1990.
7. E. D. Rainville, Special Functions, Macmillan Company, New York, 1960; Reprinted by Chelsea Publishing Company, Bronx, New York, 1971.
8. A. K. Rathie, On representation of Humbert's double hypergeometric series in a series of Gauss's $_2F_1$ function, arXiv: 1312.0064v1, 2013.
9. 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.
10. H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1984.

#### Cited by

1. On new reduction formulas for the Humbert functions Ψ2, Φ2 and Φ3 vol.28, pp.5, 2017, https://doi.org/10.1080/10652469.2017.1297438