On a Generalization of Kummer’s Second-Type F 1 1 and F 2 2

Abstract

The aim of this paper is to establish a general form of Kummer’s second-type summation theory. By defining a new form for the divided of the Pochhammer symbol $\left({\left(d+i\right)}_n/{\left(d\right)}_n\right)$ , we can develop a general form of Kummer’s second-type summation theorem as $e^{-\left(x/2\right)}{}_2{{\displaystyle F}}_2\left[\begin{array}{ccc}a,& i+d;& \\ & & x\\ b+l,& d;& \end{array}\right]$ in the form of a sum of $e^{-\left(x/2\right)}{}_1{{\displaystyle F}}_1\left[\begin{array}{ccc}a;& & \\ & & x\\ b+i;& & \end{array}\right]$ for $i,l=\mathrm{0,1,2},\dots ,$ Then, some properties of the generalized Kummer’s second-type summation theorem can yield a number of known and novel results.