On a new class of integrals involving generalized hypergeometric functions

Adem Kilicman ; Shantha Kumari Kurumujji ; Arjun K. Rathie

Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 17 (2024) no. 2, pp. 266-271

Résumé

In the theory of hypergeometric and generalized hypergeometric series, classical summation theorems such as those of Gauss, Gauss second, Bailey and Kummer for the series ${}_2F_1$; Watson, Dixon, Whipple and Saalshüz play a key role. Applications of the above mentioned summation theorems are well known. In our present investigation, we aim to evaluate twenty five new class of integrals involving generalized hypergeometric function in the form of a single integral of the form: $$\int_0^1 x^{c-1}(1-x)^{c-1}{}_3F_2\left[\begin{array}{c}a, ~b, ~c+\frac{1}{2} \\ \frac{1}{2}(a+b+i+1),~ 2c+j \end{array}; 4x(1-x)\right] dx$$ for $i,j = 0, \pm 1, \pm 2.$ \indent The results are established with the help of the generalizations of the classical Watson's summation theorem obtained earlier by Lavoie et al.[lavoie1992]. Fifty interesting integrals in the form of two integrals (twenty five each) have also been given as special cases of our main findings.

Détail

Keywords: generalized hypergeometric function, Watsons theorem, definite integral, beta integral.