Extensions of the Kummer evaluation of a finite hypergeometric series

Kummerʹs evaluation of a finite hypergeometric series F ( a , b ; c ; - 1 ) ( c = b - a + 1 , b = - 2 m ) is extended in two directions to answer questions related to the positivity of Jost functions arising from one-dimensional Schrِdinger equations with exponentially decaying potentials.