A series evaluation technique based on a modified Abel lemma

Abstract

We introduce a technique for determining infinite series identities through something of a combination of the modified Abel lemma on summation by parts and a method of undetermined coefficients. We succeed in applying our technique in our proving a nontrivial variant of Gauss’ hypergeometric identity, giving us an evaluation for a family of $_{3}F_{2}(1)$ -series with three free parameters, and to establish a $_{3}F_{2}(-1)$ -variant of Kummer’s hypergeometric identity. Also, we apply the technique upon which this article is based to formulate a new and simplified proof of a remarkable series evaluation recently derived by Cantarini via the generalized Clebsch–Gordan integral.