New identities for a sum of products of the Kummer functions

In this note we present a summation formula for the Clausen series ${}_3F_{2}(1)$ with two integral parameter differences of opposite sign and apply it to give a generalization of a particular case of a very general reduction formula for a sum of products of the generalized hypergeometric functions discovered recently by Feng, Kuznetsov and Yang (J. Math. Anal. Appl. 443(2016), 116–122). Our generalization pertains to the case when the generalized hypergeometric function is reduced to the Kummer function ${}_1F_1$ and contains an additional integer shift in the bottom parameter of the Kummer function. In the ultimate section of the paper we prove another formula for a particular product difference of the Kummer functions in terms of a linear combination of these functions.