# Terminating Basic Hypergeometric Representations and Transformations for the Askey–Wilson Polynomials

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Theorem**

**1.**

**Corollary**

**1.**

**Proof.**

**Corollary**

**2.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Remark**

**1.**

**Remark**

**2.**

## 3. The Askey–Wilson Polynomials

#### 3.1. The Askey–Wilson Polynomial Representations

**Theorem**

**3.**

**Proof.**

**Remark**

**3.**

#### 3.2. Terminating 4-Parameter Symmetric Transformations

**Corollary**

**3.**

**Proof.**

**Remark**

**4.**

#### 3.3. Terminating 4-Parameter Symmetric Interchange Transformations

**Corollary**

**4.**

**Proof.**

**Corollary**

**5.**

**Proof.**

**Remark**

**5.**

**Corollary**

**6.**

**Proof.**

**Remark**

**6.**

**Corollary**

**7.**

**Proof.**

**Remark**

**7.**

**Remark**

**8.**

**Corollary**

**8.**

**Proof.**

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Cohl, H.S.; Costas-Santos, R.S.; Ge, L.
Terminating Basic Hypergeometric Representations and Transformations for the Askey–Wilson Polynomials. *Symmetry* **2020**, *12*, 1290.
https://doi.org/10.3390/sym12081290

**AMA Style**

Cohl HS, Costas-Santos RS, Ge L.
Terminating Basic Hypergeometric Representations and Transformations for the Askey–Wilson Polynomials. *Symmetry*. 2020; 12(8):1290.
https://doi.org/10.3390/sym12081290

**Chicago/Turabian Style**

Cohl, Howard S., Roberto S. Costas-Santos, and Linus Ge.
2020. "Terminating Basic Hypergeometric Representations and Transformations for the Askey–Wilson Polynomials" *Symmetry* 12, no. 8: 1290.
https://doi.org/10.3390/sym12081290