ON AN EXTENSION OF A QUADRATIC TRANSFORMATION FORMULA DUE TO GAUSS

Subject Areas : International Journal of Mathematical Modelling & Computations

M. A. Rakha ¹ , A. K. Rathie ² , P. Chopra ³

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

Received: 2016-04-13 Accepted : 2016-04-13 Published : 2011-06-22

Keywords: Gauss hypergeometric function, 2F1 Hypergeometric function, Contiguous function relation, Linear recurrence relation,

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.