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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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
Section
Papers