On a quadratic transformation due to Exton and its generalization

Abstract

In 2003, Exton established numerous quadratic transformation formulas. The aim of this short note is to provide generalization of one of the quadratic transformation formulas.

Keywords

• Quadratic transformation,Bailey’s transform,Kummer’s summation theorem,generalizations

References

1. [1] M. Abramowitz and I.A. Stegun (Eds.) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1972.
2. [2] W.N. Bailey, Products of generalized hypergeometric series, Proc. Lond. Math. Soc. s2-28 (1), 242-254, 1928.
3. [3] J. Choi and A.K. Rathie, Quadratic transformations involving hypergeometric functions of two and higher order, East Asian Math. J. 22, 71-77, 2006.
4. [4] W. Chu, Telescopic approach to a formula of ${}_2F_1$-series by Gosper and Ebisu, Proc. Japan Acad. Ser. A Math. Sci. 93 (3), 13-15, 2017.
5. [5] A. Ebisu, On a strange evaluation of the hypergeometric series by Gosper, Ramanujan J. 32 (1), 101-108, 2013.
6. [6] H. Exton, Quadratic transformations involving hypergeometric functions of higher order, Ganita 54, 13-15, 2003.
7. [7] I. Gessel and D. Stanton, Strange evaluations of hypergeometric series, SIAM J. Math. Anal. 13 (2), 295-308, 1982.
8. [8] R.Wm. Gosper, Private Communication to Richard Askey, Dec. 21, 1977.

9. [9] Y.S. Kim, A.K. Rathie and R.B. Paris, A note on a hypergeometric transformation formula due to Slater with application, Math. Aeterna, 5 (1), 217-223, 2015.
10. [10] G.V. Milovanović, R.K. Parmar and A.K. Rathie, A study of generalized summation theorems for the series ${}_2F_1$ with an applications to Laplace transforms of convolution type integrals involving Kummer’s functions ${}_1F_1$, Appl. Anal. Discrete Math. 12 (1), 257-272, 2018.
11. [11] T. Pogany and A.K. Rathie, Extension of a quadratic transformation due to Exton, Appl. Math. Comput. 215, 423-426, 2009.
12. [12] M.A. Rakha and A.K. Rathie, Generalizations of classical summation theorems for the series ${}_{2}F_{1}$ and ${}_3F_{2}$ with applications, Integral Transforms Spec. Funct. 229 (11), 823-840, 2011.
13. [13] L.J. Slater, Generalized Hypergeometric Functions, Cambridge Univ. Press, Cambridge, 1966.
14. [14] H.M. Srivastava and H.L. Manocha, A Treatise on Generating Functions, Ellis Horwood Ltd., Chichester, 1984.