ON AN EXTENSION OF A QUADRATIC TRANSFORMATION FORMULA DUE TO GAUSS

Authors

1 Faculty of Science - Suez Canal University - Ismailia Egypt Department of Mathematics and Statistics

2 Rajasthan Technical University, Village: TULSI, Post-Jakhmund, Dist. BUNDI-323021, Rajasthan State India Vedant College of Engineering and Technology

3 Marudhar Engineering College, NH-11, Jaipur Road, Raisar, BIKANER-334 001, Rajasthan State India Department of Mathematics

Abstract

The aim of this research note is to prove the following new transformation formula
\begin{equation*} (1-x)^{-2a}\,_{3}F_{2}\left[\begin{array}{ccccc} a, & a+\frac{1}{2}, & d+1 & & \\ & & & ; & -\frac{4x}{(1-x)^{2}} \\ & c+1, & d & & \end{array}\right] \\ =\,_{4}F_{3}\left[\begin{array}{cccccc} 2a, & 2a-c, & a-A+1, & a+A+1 & & \\ & & & & ; & -x \\ & c+1, & a-A, & a+A & & \end{array} \right], \end{equation*}
where $A^2=a^2-2ad+cd$ after the equation.
For d=c, we get a known quadratic transformations due to Gauss. The result is derived with the help of the generalized Gauss's summation theorem available in the literature.

Keywords