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1. The Cesàro methods for ordinary [9, p. 17; 6, p. 96] and for absolute [9, p. 25] summation of infinite series can be generalized by the Riesz methods [7, p. 21; 12; 9, p. 52; 6, p. 86; 5, p. 2] and by “the generalized Cesàro methods“ introduced by Burkill  and Borwein and Russell . (Also cf. ; for another generalization, see .) These generalizations raise the question as to their equivalence.
We shall consider series
with complex terms an. Throughout, we will assume that
and we call (1) Riesz summable to a sum s relative to the type λ = (λn ) and to the order κ, or summable (R, λ, κ) to s briefly, if the Riesz means
(of the partial sums of (1)) tend to s as x → ∞.