On the relation of Cesàro summability to generalized (F_α,q) summability

Abstract

Standard special cases of the sequence-to-function F (α, q)-transform of Meir admit of a more general, essentially more refined, characterization as the F ^k (α, q)-transform of the sequel, adapted from one of Faulhaber's. The theorem proved, viz., that (F ^kα, q) summability for sequences, corresponding to the latter transform, includes Cesáro summability of a positive integer order with a certain rapidity, applies to the standard special cases of (F _{α, q}) and (F ^k, _{α, q}) summabilities which are, in famiiar notation, summabilities(E _υ), (T _α), (S _β), (V _α) and (B _{α, γ}) whose further special case (B_{1, 1}) is Borel summability. The special cases of (V_½) and Borel summabilities go back to Hyslop.