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

Let (p_n) and (q_n) be any two non-negative real sequences with

Rn:=∑k=0npkqn-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 En1{\rm{E}}_{\rm{n}}^1 − we will denote the Euler summability method. Let (x_n) be a sequence of real or complex numbers and set

Np,qnEn1:=1Rn∑k=0npkqn-k12k∑v=0k(vk)xv{\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 (x_n) follows from that of st-Np,qnEn1{\rm{st - N}}_{{\rm{p}},q}^{\rm{n}}{\rm{E}}_{\rm{n}}^1 − limit of (x_n). These conditions are one-sided or two-sided if (x_n) is a sequence of real or complex numbers, respectively.