# 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)
• Accepted : 2017.11.02
• Published : 2018.03.25
• 195 7

#### 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

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