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ON A NEW CLASS OF INTEGRALS INVOLVING GENERALIZED HYPERGEOMETRIC FUNCTION 3F2

  • Kim, Insuk (Department of Mathematics Education, Wonkwang University) ;
  • Shantha Kumari., K. (Department of Mathematics, A J Institute of Engineering and Technology) ;
  • Vyas, Yashoverdhan (Department of Mathematics, School of Engineering, Sir Padampat Singhania University)
  • Received : 2017.09.27
  • Accepted : 2017.11.02
  • Published : 2018.03.25

Abstract

The main aim of this research paper is to evaluate the general integral of the form $${\int_{0}^{1}}x^{c-1}(1-x)^{c+{\ell}}[1+{\alpha}x+{\beta}(1-x)]^{-2c-{\ell}-1}\atop {\times}_3F_2\left\[ {a,\;b,\;2c+{\ell}+1} \\ {\frac{1}{2}(a+b+i+1),\;2c+j\;;\frac{(1+{\alpha})x}{1+{\alpha}x+{\beta}(1-x)} }\right]dx$$ in the most general form for any ${\ell}{\in}\mathbb{Z}$; and $i, j=0,{\pm}1,{\pm}2$. The results are established with the help of generalized Watson's summation theorem due to Lavoie, et al. Fifty interesting general integrals have also been obtained as special cases of our main findings.

Keywords

Hypergeometric Summation Theorems;Watson's Theorem;Definite integrals

References

  1. Andrews, George E. and Askey, Richard and Roy, Ranjan, Special functions, Cambridge University Press, Cambridge, UK ; New York, NY, USA, Encyclopedia of mathematics and its applications, (1999).
  2. Bailey, Wilfrid Norman, Generalized hypergeometric series, The University Press, Cambridge Eng., Cambridge tracts in mathematics and mathematical physics, (1935)
  3. Choi, J. and Rathie, A. K., A new class of integrals involving generalized hypergeometric function, Far East J. Math. Sci., 102(7), pp. 1559-1570, (2017).
  4. Choi, J. and Rathie, A. K., A new class of double integrals involving generalized hypergeometric functions, Adv. Stud. Contemp. Math., 27(2), pp. 189-198, (2017).
  5. Lavoie, J. L. and Grondin, F. and Rathie, A. K., Generalizations of Watsons theorem on the sum of a $_3F_2$, Indian J. Math, 34 (2), pp. 23-32, (1992).
  6. Lavoie, J. L. and Grondin, F. and Rathie, A. K., Generalizations of Whipple's theorem on the sum of a $_3F_2(1)$, Journal of Computational and Applied Mathematics, Elsevier, 72(2), pp. 293-300, (1996). https://doi.org/10.1016/0377-0427(95)00279-0
  7. Lavoie, J. L. and Grondin, F. and Rathie, A. K. and Arora, K., Generalizations of Dixon's theorem on the sum of a $_3F_2(1)$, mathematics of computation, JSTOR, pp.267-276, (1994).
  8. MacRobert, TM, Beta-function formulae and integrals involving E-functions, Mathematische Annalen, Springer, 142(5), pp. 450-452, (1961). https://doi.org/10.1007/BF01450936
  9. Rainville, Earl David, Special functions, Macmillan, New York, (1960).