The Pure and Applied Mathematics (한국수학교육학회지시리즈B:순수및응용수학)

Korean Society of Mathematical Education (한국수학교육학회)
권호 표지
DOI QR코드

DOI QR Code

APPELL'S FUNCTION F1 AND EXTON'S TRIPLE HYPERGEOMETRIC FUNCTION X9

  • Choi, Junesang (Department of Mathematics, Dongguk University) ;
  • Rathie, Arjun K. (Department of Mathematics, School of Mathematical & Physical Sciences, Central University of Kerala, Riverside Transit Campus)
  • Received : 2012.11.20
  • Accepted : 2013.01.14
  • Published : 2013.02.28
Download PDF
⟨ Previous Next ⟩

Abstract

In the theory of hypergeometric functions of one or several variables, a remarkable amount of mathematicians's concern has been given to develop their transformation formulas and summation identities. Here we aim at presenting explicit expressions (in a single form) of the following weighted Appell's function $F_1$: $$(1+2x)^{-a}(1+2z)^{-b}F_1\;\(c,\;a,\;b;\;2c+j;\;\frac{4x}{1+2x},\;\frac{4z}{1+2z}\)\;(j=0,\;{\pm}1,\;{\ldots},\;{\pm}5)$$ in terms of Exton's triple hypergeometric $X_9$. The results are derived with the help of generalizations of Kummer's second theorem very recently provided by Kim et al. A large number of very interesting special cases including Exton's result are also given.

Keywords

References

  1. P. Appell & J. Kampe de Feriet: Fonctions Hypergeometriques et Hyperspheriques; Polynomes d'Hermite. Gauthier - Villars, Paris, 1926.
  2. W.N. Bailey: Product of generalized hypergeometric series. Proc. London Math. Soc. 28 (1928), no. 2, 242-254.
  3. J. Choi: Notes on formal manipulations of double series. Commun. Korean Math. Soc. 18 (2003), no. 4, 781-789. https://doi.org/10.4134/CKMS.2003.18.4.781
  4. H. Exton: Hypergeometric functions of three variables. J. Indian Acad. Math. 4 (1982), 113-119.
  5. Y.S. Kim, M.A. Rakha & A.K. Rathie: Generalization of Kummer's second summation 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 & A.K. Rathie: Extensions of certain classical summation theorems for the series $_2F_1$ and $_3F_2$ with applications in Ramanujan's summations. Inter. J. Math. Math. Sci. Vol. 2010, Article ID 309503, 26 pages; doi:10.1155/2010/309503.
  7. J.L. Lavoie, F. Grondin & A.K. Rathie: Generalizations of Watson's theorem on the sum of a $_3F_2$. Indian J. Math. 34 (1992), 23-32.
  8. J.L. Lavoie, F. Grondin, A.K. Rathie & K. Arora: Generalizations of Dixon's theorem on the sum of a $_3F_2$. Math. Comput. 62 (1994), 267-276.
  9. J.L. Lavoie, F. Grondin & A.K. Rathie: Generalizations of Whipple's theorem on the sum of a $_3F_2$. J. Comput. Appl. Math. 72 (1996), 293-300. https://doi.org/10.1016/0377-0427(95)00279-0
  10. S. Lewanowicz: Generalized Watson's summation formula for $_3F_2(1)$. J. Comput. Appl. Math. 86 (1997), 375-386. https://doi.org/10.1016/S0377-0427(97)00170-2
  11. M. Milgram: On hypergeometric $_3F_2(1)$. Arxiv:math.CA/0603096, 2006.
  12. E.D. Rainville: Special Functions. Macmillan Company, New York, 1960; Reprinted by Chelsea Publishing Company, Bronx, New York, 1971.
  13. M.A. Rakha & A.K. Rathie. Generalizations of classical summation theorems for the series $_2F_1$ and $_3F_2$ with applications. Integral Transforms Spec. Func. 22 (2011), no. 11, 823-840. https://doi.org/10.1080/10652469.2010.549487
  14. A.K. Rathie & J. Choi: Another proof of Kummer's second theorem. Commun. Korean Math. Soc. 13 (1998), 933-936.
  15. H.M. Srivastava & J. Choi: Series Associated with the Zeta and Related Functions. Kluwer Academic Publishers, Dordrecht, Boston and London, 2001.
  16. H.M. Srivastava & P.W. Karlsson: Multiple Gaussian Hypergeometric Series. Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1985.