Abstract and Applied Analysis

On an Extension of Kummer's Second Theorem

Medhat A. Rakha, Mohamed M. Awad, and Arjun K. Rathie

Full-text: Open access

Abstract

The aim of this paper is to establish an extension of Kummer's second theorem in the form    e - x / 2 F 2 2 [ a , 2 + d ; x 2 a + 2 , d ; ] = F 1 0 [ - ; x 2 / 16 a + 3 / 2 ; ] + ( ( a / d - 1 / 2 ) / ( a + 1 ) ) x F 1 0 [ - ; x 2 / 16 a + 3 / 2 ; ] + ( c x 2 / 2 ( 2 a + 3 ) ) F 1 0 [ - ; x 2 / 16 a + 5 / 2 ; ] , where   c = 1 / a + 1 1 / 2 - a / d + a / d ( d + 1 ) ,   d 0 , - 1 , - 2 , . For d = 2 a , we recover Kummer's second theorem. The result is derived with the help of Kummer's second theorem and its contiguous results available in the literature. As an application, we obtain two general results for the terminating F 2 3 ( 2 ) series. The results derived in this paper are simple, interesting, and easily established and may be useful in physics, engineering, and applied mathematics.

Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 128458, 6 pages.

Dates
First available in Project Euclid: 26 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393448855

Digital Object Identifier
doi:10.1155/2013/128458

Mathematical Reviews number (MathSciNet)
MR3039181

Zentralblatt MATH identifier
1277.33008

Citation

Rakha, Medhat A.; Awad, Mohamed M.; Rathie, Arjun K. On an Extension of Kummer's Second Theorem. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 128458, 6 pages. doi:10.1155/2013/128458. https://projecteuclid.org/euclid.aaa/1393448855


Export citation

References

  • Y. S. Kim, M. A. Rakha, and A. K. Rathie, “Generalizations of Kummer's second theorem with application,” Journal of Computational Mathematics and Mathematical Physics, vol. 50, no. 3, pp. 387–402, 2010.
  • Y. S. Kim, M. A. Rakha, and A. K. Rathie, “Extensions of certain classical summation theorems for the series $_{2}$${F}_{1}$, $_{3}$${F}_{2}$, and $_{4}$${F}_{3}$ with applications in Ramanujan's summations,” International Journal of Mathematics and Mathematical Sciences, vol. 2010, Article ID 309503, 26 pages, 2010.
  • M. A. Rakha and A. K. Rathie, “Generalizations of classical summation theorems for the series $_{2}$${F}_{1}$ and $_{3}$${F}_{2}$ with applications,” Integral Transforms and Special Functions, vol. 22, no. 11, pp. 823–840, 2011.
  • W. N. Bailey, “Products of generalized hypergeometric series,” Proceedings of the London Mathematical Society, vol. 28, no. 1, pp. 242–250, 1928.
  • E. E. Kummer, “Über die hypergeometridche Reihe,” Journal für Die Reine Und Angewandte Mathematik, vol. 15, pp. 39–83, 1836.
  • J. Choi and A. K. Rathie, “Another proof of Kummer's second theorem,” Communications of the Korean Mathematical Society, vol. 13, no. 4, pp. 933–936, 1998.
  • E. D. Rainville, Special Functions, The Macmillan Company, New York, NY, USA, 1960.
  • A. K. Rathie and V. Nagar, “On Kummer's second theorem involving product of generalized hypergeometric series,” Le Matematiche, vol. 50, no. 1, pp. 35–38, 1995.
  • J.-L. Lavoie, F. Grondin, and A. K. Rathie, “Generalizations of Watson's theorem on the sum of a $_{3}$${F}_{2}$,” Indian Journal of Mathematics, vol. 34, no. 1, pp. 23–32, 1992.
  • A. K. Rathie and T. K. Pogány, “New summation formula for $_{3}$${F}_{2}(1/2)$ and a Kummer-type II transformation of $_{2}$${F}_{2}(x)$,” Mathematical Communications, vol. 13, no. 1, pp. 63–66, 2008.
  • M. A. Rakha, “A note on Kummer-type II transformation for the generalized hypergeometric function $_{2}$${F}_{2}$,” Mathematical Notes, vol. 19, no. 1, pp. 154–156, 2012.
  • Y. S. Kim, J. Choi, and A. K. Rathie, “Two results for the terminating $_{3}$${F}_{2}(2)$ with applications,” Bulletin of the Korean Mathematical Society, vol. 49, no. 3, pp. 621–633, 2012.