ON AN EXTENSION OF A QUADRATIC TRANSFORMATION FORMULA DUE TO GAUSS

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  • M. A. Rakha1,
  • A. K. Rathie2,
  • P. Chopra3,
  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

Revised: 13-04-2016

Accepted: 13-04-2016

Published in Issue 22-06-2011

How to Cite

Rakha, M. A., Rathie, A. K., & Chopra, P. (2011). ON AN EXTENSION OF A QUADRATIC TRANSFORMATION FORMULA DUE TO GAUSS. International Journal of Mathematical Modelling & Computations, 1(3), 171-174. http://oiccpress.com/ijm2c/article/view/11046

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.
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