# NEW RESULTS FOR THE SERIES 2F2(x) WITH AN APPLICATION

Choi, Junesang;Rathie, Arjun Kumar

• Published : 2014.01.31
• 31 4

#### Abstract

The well known quadratic transformation formula due to Gauss: $$(1-x)^{-2a}{_2F_1}${{a,b;}\\\hfill{21}{2b;}}\;-\frac{4x}{(1-x)^2}$={_2F_1}${{a,a-b+\frac{1}{2};}\\\hfill{65}{b+\frac{1}{2};}}\;x^2$$$ plays an important role in the theory of (generalized) hypergeometric series. In 2001, Rathie and Kim have obtained two results closely related to the above quadratic transformation for $_2F_1$. Our main objective of this paper is to deduce some interesting known or new results for the series $_2F_1(x)$ by using the above Gauss's quadratic transformation and its contiguous relations and then apply our results to provide a list of a large number of integrals involving confluent hypergeometric functions, some of which are (presumably) new. The results established here are (potentially) useful in mathematics, physics, statistics, engineering, and so on.

#### Keywords

Gamma function;hypergeometric function;generalized hypergeometric function;Gauss's quadratic transformation formula for $_2F_1$;Watson's summation theorem for $_3F_2(1)$

#### References

1. W. N. Bailey, Product of generalized hypergeometric series, Proc. London Math. Soc. (ser. 2) 28 (1928), 242-254.
2. 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.
3. A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of Integral Transforms. Vol. II, McGraw-Hill Book Company, New York, Toronto and London, 1954.
4. C. F. Gauss, Disquisitiones Generales Circa Seriem Infinitam $[\frac{{\alpha}{\beta}}{1{\cdot}{\gamma}}]x+[\frac{{\alpha}({\alpha}+1){\beta}({\beta}+1)}{1{\cdot}2{\cdot}{\gamma}({\gamma}+1)}]x^2+[\frac{{\alpha}({\alpha}+1)({\alpha}+2){\beta}({\beta}+1)({\beta}+2)}{1{\cdot}2{\cdot}{\gamma}({\gamma}+1)({\gamma}+2)}]x^3+etc$, Pars Prior. Comm. Soc. Regia Sci. Gottingen Rec. 2 (1813), 3-46. Reprinted in Gesammelte Werke (Gottingen), Bd. 3, pp. 123-163, 1866.
5. Y. S. Kim, M. A. Rakha, and A. K. Rathie, Generalization of Kummer's second theorem with applications, Comput. Math. Math. Phys. 50 (2010), no. 3, 387-402. https://doi.org/10.1134/S0965542510030024
6. Y. S. Kim, M. A. Rakha, and A. K. Rathie, Extensions of certain classical summation theorems for the series $_2F_1$, $_3F_2$ and $_4F_3$ with applications in Ramanujan's summations, Int. J. Math. Math. Sci. 2010 (2010), Article Id. 309503, 26 pages.
7. J. L. Lavoie, Some summation formulas for the series $_3F_2$(1), Math. Comp. 49 (1987), no. 179, 269-274.
8. J. L. Lavoie, F. Grondin, and A. K. Rathie, Generalizations of Watson's theorem on the sum of a $_3F_2$, Indian J. Math. 34 (1992), no. 1, 23-32.
9. 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
10. J. L. Lavoie, F. Grondin, A. K. Rathie, and K. Arora, Generalizations of Dixon's theorem on the sum of a $_3F_2$, Math. Comp. 62 (1994), no. 205, 267-276.
11. S. Lewanowicz, Generalized Watson's summation formula for $_3F_2$(1), J. Comput. Appl. Math. 86 (1997), no. 2, 375-386. https://doi.org/10.1016/S0377-0427(97)00170-2
12. W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, Third Enlarged Edition, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berucksichtingung der Anwendungsgebiete, Bd. 52, Springer-Verlag, New York, 1966.
13. M. Milgram, On hypergeometric $_3F_2$(1), Arxiv: math. CA/ 0603096, 2006.
14. A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, Vol. 4, Direct Laplace Transforms, Gordon and Breach Science Publishers, New York, Reading, Paris, Montreux, Tokyo and Melbourne, 1992.
15. M. A. Rakha and A. K. Rathie, Generalizations of classical summation theorems for the series $_2F_1$ and $_3F_2$ with applications, Integral Transforms Spec. Funct. 22 (2011), no. 11, 823-840. https://doi.org/10.1080/10652469.2010.549487
16. E. D. Rainville, Special Functions, Macmillan Company, New York, 1960; Reprinted by Chelsea Publishing Company, Bronx, New York, 1971.
17. A. K. Rathie and Y. S. Kim, On two results contiguous to a quadratic transformation formula due to Gauss, Far East J. Math. Sci. 3 (2001), no. 1, 51-58.
18. L. J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, Cambridge, London, and New York, 1966.
19. H. M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, Boston and London, 2001.
20. 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.
21. R. Vidunas, A generalization of Kummer's identity, Rocky Mountain J. Math. 32 (2002), no. 2, 919-935. https://doi.org/10.1216/rmjm/1030539701

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