Original scientific paper

On an extension of a quadratic transformation formula due to Kummer

Medhat A. Rakha ; athematics Department, Faculty of Science, Suez Canal University, Ismailia 41 522, Egypt
Navratna Rathie ; Department of Mathematics, Rajasthan Technical University, Kota-324 010, Rajasthan,India
Purnima Chopra ; Department of Mathematics, Marudhar Engineering College Raisar, NH-11, Jaipur Road,Bikaner-334 001, Rajasthan, India

Abstract

The aim of this research note is to prove the following new transformation formula
\begin{eqnarray*}
(1-x)^{-2a}&\,_{3}F_{2}&
\left[ \begin{array}{ccccc}%
a, & a+\frac{1}{2}, & d+1 & & \\
& & & ; & \frac{x^{2}}{(1-x)^{2}}\\
& c+\frac{3}{2}, & d & &
\end{array} \right] \\\\
=&\,_{4}F_{3}&\left[
\begin{array}{cccccc}%
2a, & c, & 2d+\frac{1}{2}A+\frac{1}{2}, & 2d-\frac{1}{2}A+\frac{1}{2} & & \\
& & & & ; & 2x\\
& 2c+2, & 2d+\frac{1}{2}A-\frac{1}{2}, & 2d-\frac{1}{2}A-\frac{1}{2} & &
\end{array}
\right]
\end{eqnarray*}
valid for $\left\vert x\right\vert <\frac{1}{2}$ and if $\left\vert x\right\vert =\frac{1}{2}$,
then $\operatorname{Re}(c-2a)>0$, where $A=\sqrt{{ 16d^{2}-16cd-8d+1}}$. For $d=c+\frac{1}{2}$,
we get quadratic transformations due to Kummer. The result is derived with the help of the
generalized Gauss's summation theorem available in the literature.

Keywords

quadratic transformation formula; Kummer transformation; Gauss summation theorem

Hrčak ID:

43997

URI

https://hrcak.srce.hr/43997

Publication date:

9.12.2009.

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