# Generalized Shifted Chebyshev Koornwinder’s Type Polynomials: Basis Transformations

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## Abstract

**:**

## 1. Introduction

#### The Generalized Shifted Chebyshev-I Koornwinder’s Type Polynomials

**Theorem**

**1**

**.**For ${K}_{0},{K}_{1}\ge 0,$ the rth degree generalized shifted Chebyshev-I polynomials ${\mathcal{T}}_{r}^{*({K}_{0},{K}_{1})}\left(x\right)$ have the next representation using Bernstein polynomial basis,

## 2. Results: Bases Transformations

#### 2.1. Generalized Shifted Chebyshev-I to Bernstein

**Theorem**

**2.**

**Proof.**

**Corollary**

**1.**

**Proof.**

#### 2.2. Bernstein to Generalized Shifted Chebyshev-I

**Theorem**

**3**

**.**Let ${\mathbb{B}}_{r}^{n}\left(x\right)$ be the nth degree Bernstein polynomial and ${\mathcal{T}}_{i}^{*({K}_{0},{K}_{1})}\left(x\right)$ be the ith degree generalized shifted Chebyshev-I polynomials. For $r,i=0,1,\dots ,n$, we obtain

**Proof.**

**Theorem**

**4.**

**Proof.**

## 3. Discussion

#### Applications

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**MDPI and ACS Style**

AlQudah, M.A.; AlMheidat, M.N.
Generalized Shifted Chebyshev Koornwinder’s Type Polynomials: Basis Transformations. *Symmetry* **2018**, *10*, 692.
https://doi.org/10.3390/sym10120692

**AMA Style**

AlQudah MA, AlMheidat MN.
Generalized Shifted Chebyshev Koornwinder’s Type Polynomials: Basis Transformations. *Symmetry*. 2018; 10(12):692.
https://doi.org/10.3390/sym10120692

**Chicago/Turabian Style**

AlQudah, Mohammad A., and Maalee N. AlMheidat.
2018. "Generalized Shifted Chebyshev Koornwinder’s Type Polynomials: Basis Transformations" *Symmetry* 10, no. 12: 692.
https://doi.org/10.3390/sym10120692