<abstract xmlns="http://www.w3.org/1999/xhtml">Let (pn) and (qn) be any two non-negative real sequences with<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2019-0019_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mtext>R</mtext></mrow><mtext>n</mtext></msub><mo>:</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mtext>k</mtext><mo>=</mo><mn>0</mn></mrow><mtext>n</mtext></munderover><mrow><msub><mrow><mtext>p</mtext></mrow><mtext>k</mtext></msub><msub><mrow><mtext>q</mtext></mrow><mrow><mtext>n</mtext><mo>-</mo><mtext>k</mtext></mrow></msub></mrow><mo>≠</mo><mn>0</mn><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mrow><mo>(</mo><mrow><mtext>n</mtext><mo>∈</mo><mtext>𝕅</mtext></mrow><mo>)</mo></mrow></mrow></math><tex-math>{{\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)</tex-math></alternatives></disp-formula>With <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2019-0019_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mtext>E</mtext></mrow><mtext>n</mtext><mn>1</mn></msubsup></mrow></math><tex-math>{\rm{E}}_{\rm{n}}^1</tex-math></alternatives></inline-formula> − we will denote the Euler summability method. Let (xn) be a sequence of real or complex numbers and set<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2019-0019_eq_003.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mtext>N</mtext></mrow><mrow><mtext>p</mtext><mo>,</mo><mtext>q</mtext></mrow><mtext>n</mtext></msubsup><msubsup><mrow><mtext>E</mtext></mrow><mtext>n</mtext><mn>1</mn></msubsup><mo>:</mo><mo>=</mo><mfrac><mn>1</mn><mrow><msub><mrow><mtext>R</mtext></mrow><mtext>n</mtext></msub></mrow></mfrac><munderover><mo>∑</mo><mrow><mtext>k</mtext><mo>=</mo><mn>0</mn></mrow><mtext>n</mtext></munderover><mrow><msub><mrow><mtext>p</mtext></mrow><mtext>k</mtext></msub><msub><mrow><mtext>q</mtext></mrow><mrow><mtext>n</mtext><mo>-</mo><mtext>k</mtext></mrow></msub><mfrac><mn>1</mn><mrow><msup><mrow><mn>2</mn></mrow><mtext>k</mtext></msup></mrow></mfrac><munderover><mo>∑</mo><mrow><mtext>v</mtext><mo>=</mo><mn>0</mn></mrow><mtext>k</mtext></munderover><mrow><mrow><mo>(</mo><mrow><msubsup><mrow></mrow><mtext>v</mtext><mtext>k</mtext></msubsup></mrow><mo>)</mo></mrow><msub><mrow><mtext>x</mtext></mrow><mtext>v</mtext></msub></mrow></mrow></mrow></math><tex-math>{\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}}}} } </tex-math></alternatives></disp-formula>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 <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2019-0019_eq_004.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mrow><mtext>st</mtext><mo>-</mo><mtext>N</mtext></mrow></mrow><mrow><mtext>p</mtext><mo>,</mo><mi>q</mi></mrow><mtext>n</mtext></msubsup><msubsup><mrow><mtext>E</mtext></mrow><mtext>n</mtext><mn>1</mn></msubsup></mrow></math><tex-math>{\rm{st - N}}_{{\rm{p}},q}^{\rm{n}}{\rm{E}}_{\rm{n}}^1</tex-math></alternatives></inline-formula> − limit of (xn). These conditions are one-sided or two-sided if (xn) is a sequence of real or complex numbers, respectively.</abstract>

Let (pn) and (qn) be any two non-negative real sequences withRn:=∑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 (xn) be a sequence of real or complex numbers and setNp,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 (xn) follows from that of st-Np,qnEn1{\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.

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

Ilirias research Institute Kosovo and Department of Mathematics and the Computer Sciences

Acta Universitatis Sapientiae, Mathematica

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Let (pn) and (qn) be any two non-negative real sequences withRn:=∑k=0npkqn-k≠0    (n∈𝕅){{\rm{R}}_{\rm{n}}}: = \sum\limits_{{\rm{k}} = 0}^{\rm{n}}...

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

Published Online: Feb 27, 2020

Page range: 251 - 263

Received: Oct 15, 2018

DOI: https://doi.org/10.2478/ausm-2019-0019

Keywords
generalized Nörlund-Euler summability, one-sided and two-sided Tauberian conditions, statistical convergence

© 2019 Naim L. Braha, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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

Published Online: Feb 27, 2020

Page range: 251 - 263

Received: Oct 15, 2018

DOI: https://doi.org/10.2478/ausm-2019-0019

Keywordsgeneralized Nörlund-Euler summability, one-sided and two-sided Tauberian conditions, statistical convergence

© 2019 Naim L. Braha, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
generalized Nörlund-Euler summability, one-sided and two-sided Tauberian conditions, statistical convergence