Binomial convolutions and hypergeometric identities

By means of formal power series calculus, some new recurrences on the generating functions for the generalized Abel and Gould coefficients are derived from the Gould's work (1956–1961), which yield equivalently several convolution formulas of binomial coefficients. Alternatively, some of these can be verified either through a pair of relations due to Gould and Hsu (1973), from which some strange hypergeometric evaluations including one of Gessel and Stanton (1982) may be produced mechanically. By associating the binomial convolutions investigated in this paper with Whipple's transform (1926) on very well-poised series, a new family of the₇ F ₆-hypergeometric identities are established in a unified way.