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On a new class of integrals involving generalized hypergeometric function \(_3F_2\). (English) Zbl 1393.33004

Summary: The main aim of this research paper is to evaluate the general integral of the form \[ \int_0^1 x^{c-1}(1-x)^{c+\ell} [1+\alpha x+\beta(1-x)]^{-2c-\ell-1} \times\, _3F_2 \biggl[ \begin{matrix} a,b,2c+\ell+1 \\ \frac{1}{2}(a+b+i+1), 2c+j \end{matrix} ; \frac{(1+\alpha)x}{1+\alpha x + \beta(1-x)} \biggr] dx \] 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 integals have also been obtained as special cases of our main findings.

MSC:

33C05 Classical hypergeometric functions, \({}_2F_1\)
33C20 Generalized hypergeometric series, \({}_pF_q\)
33C70 Other hypergeometric functions and integrals in several variables
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