ON AN EXTENSION OF A QUADRATIC TRANSFORMATION FORMULA DUE TO GAUSS

International Journal of Mathematical Modelling & Computations

مقاله 2، دوره 1، 3 (SUMMER)، فروردین 2011، صفحه 171-174   

نویسندگان

M. A. Rakha¹؛ A. K. Rathie²؛ P. Chopra³

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

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

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

چکیده

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.

کلیدواژه‌ها

Gauss hypergeometric function؛ 2F1 Hypergeometric function؛ Contiguous function relation؛ Linear recurrence relation