Characterization of deferred type statistical convergence andP‐summability method for operators: Applications toq‐Lagrange–Hermite operator

Abstract

The present work considers two important convergence techniques, namely, deferred type statistical convergence and P-summability method in respect of linear positive operators. With regard to these techniques, following closely the ideas developed in the articles (Appl.

A q‐Erkuş–Srivastava polynomials operator

“…The purpose of this segment is to study the convergence of the operators L 𝛽 (1) n , … ,𝛽 (r) n n,m 1 , … ,m r ;q n via weighted A-statistical method. First, we establish the weighted A-statistical convergence of the sequence of positive linear operators L 𝛽 (1) n , … ,𝛽 (r) n n,m 1 , … ,m r ;q n using the Korovkin-type theorem given by Agrawal et al [35].…”

“…Proof. From the Korovkin-type theorem [35,Theorem 1] for b n = 0, and c n = n, it is sufficient to show that…”

“…First, we establish the weighted $$$ A $$$‐statistical convergence of the sequence of positive linear operators $$$ {L}_{n,{m}_1,\dots, {m}_r;{q}_n}^{\beta_n^{(1)},\dots, {\beta}_n^{(r)}} $$$ using the Korovkin‐type theorem given by Agrawal et al [35].…”

“…It is remarked that if we take m i = 1, ∀ i = 1, … , r, then the operators L 𝛽 (1) n , … ,𝛽 (r) n n,m 1 , … ,m r ;q (. ; x) include the q-Lagrange polynomials operators considered by Baxhaku et al [34] and if we assume m i = i, ∀ i = 1, … , r, then the operators defined by (1.9) reduce to q-Lagrange-Hermite polynomials operator studied by Agrawal et al [35].…”

“…It is remarked that if we take $$$ {m}_i=1,\forall i=1,\dots, r $$$, then the operators $$$ {L}_{n,{m}_1,\dots, {m}_r;q}^{\beta_n^{(1)},\dots, {\beta}_n^{(r)}}\left(. ;x\right) $$$ include the $$$ q $$$‐Lagrange polynomials operators considered by Baxhaku et al [34] and if we assume $$$ {m}_i=i,\forall i=1,\dots, r $$$, then the operators defined by () reduce to $$$ q $$$‐Lagrange–Hermite polynomials operator studied by Agrawal et al [35].…”