On the <i>q</i>-Bernstein polynomials of the logarithmic function in the case <i>q</i> &gt; 1

In this paper, some qualitative approximation results for the (p, q)-Bernstein operators B n p,q (f ; x) are obtained for the Cauchy kernel 1 x-α with a pole α ∈ [0, 1] for q > p > 1. The main focus lies in the study of behavior of operators B n p,q (f ; x) for the function f m (x) = 1 x-p m q-m , x = p m q-m and f m (p m q-m) = a, a ∈ R and the extra parameter p provides flexibility for the approximation.

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“…x-α , α ∈ [0, 1]. The previously obtained results [27][28][29][30] lead to the following conclusions.…”

Section: Introductionmentioning

confidence: 54%

The convergence of $(p,q)$-Bernstein operators for the Cauchy kernel with a pole via divided difference

et al. 2019

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“…Even though a generalization of Bernstein polynomials associated with q-integers was suggested in 1987 (see ([60], Section 1)), the q-analogue of Bernstein polynomials (23) have been received as a standard definition and investigated in such diverse ways as (an extension of several variables [61]; other q-polynomials and operators [7,20,[62][63][64][65]; other types of Bernstein polynomials [19,66]; convergence and iterates [9,60,67]; monotonicity [48]; Cauchy kernel [68]; norm estimates [69]; unbounded function [70]; overview of the first decade [71]).…”

Section: The Q-bernstein Polynomials Expressed In Terms Of the Q-stirling Polynomials Of The Second Kindmentioning

confidence: 99%

Certain Identities Associated with (p,q)-Binomial Coefficients and (p,q)-Stirling Polynomials of the Second Kind

2020

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The q-Stirling numbers (polynomials) of the second kind have been investigated and applied in a variety of research subjects including, even, the q-analogue of Bernstein polynomials. The (p,q)-Stirling numbers (polynomials) of the second kind have been studied, particularly, in relation to combinatorics. In this paper, we aim to introduce new (p,q)-Stirling polynomials of the second kind which are shown to be fit for the (p,q)-analogue of Bernstein polynomials. We also present some interesting identities involving the (p,q)-binomial coefficients. We further discuss certain vanishing identities associated with the q-and (p,q)-Stirling polynomials of the second kind.

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“…The properties of Phillips' q-Bernstein polynomilas for q ∈ (0, 1) were treated for example in [6,15,16,[22][23][24], while those for q > 1 were developed for instance in [17][18][19][20].…”

Section: Introductionmentioning

confidence: 99%

On p-Adic Fermionic Integrals of q-Bernstein Polynomials Associated with q-Euler Numbers and Polynomials †

et al. 2018

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We study a q-analogue of Euler numbers and polynomials naturally arising from the p-adic fermionic integrals on Z p and investigate some properties for these numbers and polynomials. Then we will consider p-adic fermionic integrals on Z p of the two variable q-Bernstein polynomials, recently introduced by Kim, and demonstrate that they can be written in terms of the q-analogues of Euler numbers. Further, from such p-adic integrals we will derive some identities for the q-analogues of Euler numbers.

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On the q-Bernstein polynomials of the logarithmic function in the case q > 1

Abstract: TheThe aim of this paper is to present new results related to the

Cited by 5 publications

References 12 publications

The convergence of $(p,q)$-Bernstein operators for the Cauchy kernel with a pole via divided difference

The convergence of $(p,q)$-Bernstein operators for the Cauchy kernel with a pole via divided difference

Certain Identities Associated with (p,q)-Binomial Coefficients and (p,q)-Stirling Polynomials of the Second Kind

On p-Adic Fermionic Integrals of q-Bernstein Polynomials Associated with q-Euler Numbers and Polynomials †

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