A Tauberian theorem for the statistical generalized Nörlund-Euler summability method

Abstract

Abstract Let (pn) and (qn) be any two non-negative real sequences with R n : = ∑ k = 0 n p k q n - k ≠ 0 ( n ∈ 𝕅 ) {{\rm{R}}_{\rm{n}}}: = \sum\limits_{{\rm{k}} = 0}^{\rm{n}} {{{\rm{p}}_{\rm{k}}}{{\rm{q}}_{{\rm{n}} - {\rm{k}}}}} \ne 0\,\,\,\,\left( {{\rm{n}} \in {\rm\mathbb{N}}} \right) With E n 1 {\rm{E}}_{\rm{n}}^1 − we will denote the Euler summability method. Let (xn) be a sequence of real or complex numbers and set N p , q n E n 1 : = 1 R n ∑ k = 0 n p k q n - k 1 2 k ∑ v = 0 k ( v k ) x v {\rm{N}}_{{\rm{p}},{\rm{q}}}^{\rm{n}}{\rm{E}}_{\rm{n}}^1: = {1 \over {{{\rm{R}}_{\rm{n}}}}}\sum\limits_{{\rm{k}} = 0}^{\rm{n}} {{{\rm{p}}_{\rm{k}}}{{\rm{q}}_{{\rm{n - k}}}}{1 \over {{2^{\rm{k}}}}}\sum\limits_{{\rm{v}} = 0}^{\rm{k}} {\left( {_{\rm{v}}^{\rm{k}}} \right){{\rm{x}}_{\rm{v}}}} } for n ∈ ℕ. In this paper, we present necessary and sufficient conditions under which the existence of the st− limit of (xn) follows from that of st - N p , q n E n 1 {\rm{st - N}}_{{\rm{p}},q}^{\rm{n}}{\rm{E}}_{\rm{n}}^1 − limit of (xn). These conditions are one-sided or two-sided if (xn) is a sequence of real or complex numbers, respectively.