TY - JOUR
AU - Insuk Kim
PY - 2020/03/25
Y2 - 2020/04/04
TI - A new class of double integrals involving Generalized Hypergeometric Function 3F2
JF - Tamkang Journal of Mathematics
JA - Tamkang J. Math.
VL - 51
IS - 1
SE - Papers
DO - 10.5556/j.tkjm.51.2020.3020
UR - https://journals.math.tku.edu.tw/index.php/TKJM/article/view/3020
AB - 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.
ER -