• Positive Linear Operators Generated by Analytic Functions

• # Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/117/04/0485-0493

• # Keywords

Szász–Mirakyan operator; positive operator; limit 𝑞-Bernstein operator; 𝑞-integers; Poisson distribution; totally positive sequence.

• # Abstract

Let 𝜑 be a power series with positive Taylor coefficients $\{a_k\}^\infty_{k=0}$ and non-zero radius of convergence $r\leq\infty$. Let $\xi_x,\,0\leq x &lt; r$ be a random variable whose values $\alpha_k, k=0,1,\ldots,$ are independent of 𝑥 and taken with probabilities $a_kx^k/\varphi(x), k=0,1,\ldots$

The positive linear operator $(A_\varphi f)(x):=E[f(\xi_x)]$ is studied. It is proved that if $E(\xi_x)=x,E(\xi^2_x)=qx^2+bx+c,\, q, b, c\in R, q&gt;0$, then $A_\varphi$ reduces to the Szász–Mirakyan operator in the case $q=1$, to the limit 𝑞-Bernstein operator in the case $0 &lt; q &lt; 1$, and to a modification of the Lupaş operator in the case $q&gt;1$.

• # Author Affiliations

1. Department of Mathematics, Atilim University, 06836 Incek, Ankara, Turkey

• # Proceedings – Mathematical Sciences

Volume 131, 2021
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019