# A new class of double integrals involving Generalized Hypergeometric Function 3F2

• Insuk Kim Wonkwang University
Keywords: Generalized hypergeometric function, generalized Watson's summation theorem, generalization of Edwards's double integrals

### Abstract

The aim of this research paper is to evaluate fifty double integrals invoving generalized hypergeometric function (25 each) in the form of

\begin{align*}

\int_{0}^{1}\int_{0}^{1} & x^{c-1}y^{c+\al-1} (1-x)^{\al- 1}(1-y)^{\be-1}\, (1-xy)^{c+\ell-\al-\be+1}\;\\ &\times \;{}_3F_2 \left[\begin{array}{c}

a,\,\,\,\,\,b,\,\,\,\,\,2c+\ell+ 1 \\ \frac{1}{2}(a+b+i+1),\,\,2c+j \end{array}; xy\right]\,dxdy

\end{align*}

and

\begin{align*}

\int_{0}^{1}\int_{0}^{1} & x^{c+\ell}y^{c+\ell+\al} (1-x)^{\al-1}(1-y)^{\be-1}\, (1-xy)^{c- \al-\be}\;\\ &\times \;{}_3F_2 \left[\begin{array}{c}

a,\,\,\,\,\,b,\,\,\,\,\,2c+\ell+ 1 \\ \frac{1}{2}(a+b+i+1),\,\,2c+j \end{array}; 1-xy\right]\,dxdy

\end{align*}

in the most general form for any $\ell \in \mathbb{Z}$ and $i, j = 0, \pm 1, \pm2$.

The results are derived with the help of generalization of Edwards's well known double integral due to Kim, {\it et al.} and generalized classical Watson's summation theorem obtained earlier by Lavoie, {\it et al}.

More than one hundred ineteresting special cases have also been obtained.

### Author Biography

Insuk Kim, Wonkwang University
Department of Mathematics Education

### References

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{em Generalizations of Dixon's theorem on the sum of a ${}_{3}F_{2}$}, Math. Comp., 62,

-276, (1994).

bibitem{lav3} Lavoie, J.L., Grondin, F. and Rathie, A.K.,

{em Generalizations of Whipple's theorem on the sum of a ${}_{3}F_{2}$}, J. Comput. Appl. Math., 72,

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{em Generalizations of classical summation theorems for the series ${}_{2}F_{1}$ and ${}_{3}F_{2}$ with applications}, Integral Transforms Spec. Fun., 22(11), 823-840, (2011).

Published
2020-03-25
How to Cite
Kim, I. (2020). A new class of double integrals involving Generalized Hypergeometric Function 3F2. Tamkang Journal of Mathematics, 51(1). https://doi.org/10.5556/j.tkjm.51.2020.3020
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