Generating functions for unification of the multidimensional Bernstein polynomials and their applications

Şimşek, Yılmaz

doi:10.1002/mma.4746

Cited by 9 publications

(3 citation statements)

References 25 publications

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“…With the help of the method used by the second author (cf. previous works 25,26,29 ), we derive combinatoric sum which is given below. Integrating both sides of the equation given in (77) from 0 to 1, and after some algebraic operations, the following combinatoric sum is obtained: Corollary 21.…”

Section: Identities and Relations Including Chebyshev Polynomials Bernstein Polynomials And Trigonometric Polynomialsmentioning

confidence: 99%

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Computational formulas and identities for new classes of Hermite‐based Milne–Thomson type polynomials: Analysis of generating functions with Euler's formula

Kilar

Şimşek

2021

Math Methods in App Sciences

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The aim of this paper is to construct generating functions for a new family of polynomials, which are called parametric Hermite-based Milne-Thomson type polynomials. Many properties of these polynomials with their generating functions are investigated. These generating functions give us generalization of some well-known generating functions for special polynomials such as Hermite type polynomials, Milne-Thomson type polynomials, and Apostol type polynomials. Using the Euler formula, functional equation method for generating function, and differential operator technique, we give relations among parametric Hermite-based Milne-Thomson type polynomials, the Bernoulli numbers, the Euler numbers, the Chebyshev polynomials, the Bernstein basis functions, homogeneous harmonic polynomials, and parametric kinds of Apostol type polynomials. Moreover, some computational formulas for these polynomials are derived. Finally, using Wolfram Mathematica version 12.0, some of these polynomials and their generating functions are illustrated by their plots under the special conditions. Potential relationships and connections of this paper's results with the results of previous and future research are pointed out.

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Section: Identities and Relations Including Chebyshev Polynomials Bernstein Polynomials And Trigonometric Polynomialsmentioning

confidence: 99%

“…Let x ∈ [0, 1] and k = 0, 1, … n. The Bernstein basis functions are defined by means of the following generating functions (cf. previous literature 16,25,26,29,32,39 ):…”

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confidence: 96%

Computational formulas and identities for new classes of Hermite‐based Milne–Thomson type polynomials: Analysis of generating functions with Euler's formula

Kilar

Şimşek

2021

Math Methods in App Sciences

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“…Some cases are given in [6][7][8]. These change of bases transformations can also be studied for the cases between the Chebyshev polynomials of fourth kind and the q-Bernstein polynomials in [9], degenerate Bernstein polynomials in [10], and the multidimensional Bernstein polynomials in [11]. This paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

Change of Basis Transformation from the Bernstein Polynomials to the Chebyshev Polynomials of the Fourth Kind

Rababah

Hijazi

2019

Mathematics

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Degenerate Bernstein polynomials

Kim

Ким

2018

RACSAM

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Here we consider the degenerate Bernstein polynomials as a degenerate version of Bernstein polynomials, which are motivated by Simsek's recent work 'Generating functions for unification of the multidimensional Bernstein polynomials and their applications' ([15,16]) and Carlitz's degenerate Bernoulli polynomials. We derived thier generating function, symmetric identities, recurrence relations, and some connections with generalized falling factorial polynomials, higher-order degenerate Bernoulli polynomials and degenerate Stirling numbers of the second kind.2010 Mathematics Subject Classification. 11B83.

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Generating functions for unification of the multidimensional Bernstein polynomials and their applications

Cited by 9 publications

References 25 publications

Computational formulas and identities for new classes of Hermite‐based Milne–Thomson type polynomials: Analysis of generating functions with Euler's formula

Computational formulas and identities for new classes of Hermite‐based Milne–Thomson type polynomials: Analysis of generating functions with Euler's formula

Change of Basis Transformation from the Bernstein Polynomials to the Chebyshev Polynomials of the Fourth Kind

Degenerate Bernstein polynomials

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