A Tauberian theorem for the generalized Nörlund summability method

Let (pn) and (qn) be any two non-negative real sequences, with R n := ? k = 0 n n p k q n - k ? 0 (n ? N). Let ?? k=0 a k be a series of real or complex numbers with partial sums (sn), and set tn p , q := 1 Rn ?n k = 0 p k qn k sk for n ? N. In this paper, we present the necessary and sufficient conditions under which the existence of the limit limn›? sn = L follows from that of lim n›? tn p , q = L. These conditions are one-sided or two-sided if (sn) is a sequence of real or complex numbers, respectively. © 2018 Walter de Gruyter GmbH, Berlin/Boston.