Explicit expressions for the hypergeometric series $_2F_1(-n, a; 2a\pm j;2)$ and $_2F_1(-n, a; -2n \pm j;2)$ for positive integer $n$ and arbitrary integer $j$ are obtained with the help of generalizations of Kummer's second and third summation theorems obtained earlier by Rakha and Rathie. Results for $ j \leq 5$ derived previously using different methods are also obtained as special cases. Two applications are considered, where the first summation formula is applied to a terminating $_3F_2(2)$ series and the confluent hypergeometric function $_1F_1(x)$.
KIM, YONG SUP; RATHIE, ARJUNKUMAR; and PARIS, RICHARD B.
"Evaluations of some terminating hypergeometric $_2F_1(2)$ series with applications,"
Turkish Journal of Mathematics: Vol. 42:
5, Article 35.
Available at: https://journals.tubitak.gov.tr/math/vol42/iss5/35