Terminating Zero-Balanced Hypergeometric Series Using Divided Differences

Abstract

There is a close relationship between divided differences and hypergeometric series, and recent studies have shown that divided differences can be used effectively to derive terminating hypergeometric identities. In this paper, we apply this approach to terminating zero-balanced hypergeometric series. Using explicit product evaluations arising from divided differences, we give alternative proofs of zero-balanced ${}_3F_2(1)$ and ${}_4F_3(1)$ summation formulas and then extend the argument to the general terminating ${}_{r+1}F_r(1)$ case. We also combine the Lagrange representation with the Leibniz rule for divided differences to derive a finite convolution transformation for a terminating ${}_{r+2}F_{r+1}(1)$ series. Thus, the divided-difference approach not only reproduces the known zero-balanced summation formulas but also yields a finite convolution transformation for terminating hypergeometric series, together with its zero-balanced specialization.