On Certain Hypergeometric Summation Theorems Motivated by the Works of Ramanujan, Chudnovsky and Borwein

Abstract

In the present paper, we obtain numerical values for Gaussian
hypergeometric summation theorems by giving particular values to the
parameters $a,~b$ and the argument $x$; three summation theorems for
${}_{2}F_{3}(\frac{1}{4},\frac{3}{4};\frac{1}{2},\frac{1}{2},1;x)$,
three summation theorems for
${}_{4}F_{3}(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{a+b}{b};1,1,\frac{a}{b};x)$,
two summation theorems for
${}_{4}F_{3}(\frac{1}{2},\frac{1}{3},\frac{2}{3},\frac{a+b}{b};1,1,\frac{a}{b};x)$,
four summation theorems for
${}_{4}F_{3}(\frac{1}{2},\frac{1}{6},\frac{5}{6},\frac{a+b}{b};1,1,\frac{a}{b};x)$
and ten summation theorems for
${}_{4}F_{3}(\frac{1}{2},\frac{1}{4},\frac{3}{4},\frac{a+b}{b};1,1,\frac{a}{b};x)$.