A Tauberian theorem for the generalized Nörlund summability method

Abstract

Let (p n) and (q n) be any two non-negative real sequences, with Rn:= ∑k=0npkqn-k≠ 0 n ∈ N). Let k = 0 ∞ a be a series of real or complex numbers with partial sums (s n) and set tnp,q:= 1/Rn ∑k=0npkqn-ksk for n ∈N. In this paper, we present the necessary and sufficient conditions under which the existence of the limit lim n &inf;sn =L follows from that of lim n → ∞. These conditions are one-sided or two-sided if (s n) is a sequence of real or complex numbers, respectively.