# The Chebyshev Difference Equation

^{1}

^{2}

^{2}Technologies of West Virginia, Fairmont, WV 26554, USA

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## Abstract

**:**

## 1. Introduction

## 2. Chebyshev Difference Equation

**Example**

**1.**

**Lemma**

**1.**

- 1.
- $\mathsf{\Theta}y\left(t\right)=t\Delta y(t-1)$, and
- 2.
- ${\mathsf{\Theta}}^{2}y\left(t\right)=t\Delta y(t-1)+{t}^{\underset{\_}{2}}{\Delta}^{2}y(t-2)$.

**Proof.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Corollary**

**2.**

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Corollary**

**3.**

**Example**

**2.**

## 3. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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$\mathit{n}=$ | ${\mathit{T}}_{\mathit{n}}=$ | ${\mathit{U}}_{\mathit{n}}=$ |
---|---|---|

0 | 1 | 1 |

1 | $t+1$ | $2t+2$ |

2 | $2{t}^{2}+2t+1$ | $4{t}^{2}+4t+3$ |

3 | $4{t}^{3}+5t+1$ | $8{t}^{3}+12t+4$ |

4 | $8{t}^{4}-16{t}^{3}+32{t}^{2}-8t+1$ | $16{t}^{4}-32{t}^{3}+68{t}^{2}-12t+5$ |

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**MDPI and ACS Style**

Cuchta, T.; Pavelites, M.; Tinney, R.
The Chebyshev Difference Equation. *Mathematics* **2020**, *8*, 74.
https://doi.org/10.3390/math8010074

**AMA Style**

Cuchta T, Pavelites M, Tinney R.
The Chebyshev Difference Equation. *Mathematics*. 2020; 8(1):74.
https://doi.org/10.3390/math8010074

**Chicago/Turabian Style**

Cuchta, Tom, Michael Pavelites, and Randi Tinney.
2020. "The Chebyshev Difference Equation" *Mathematics* 8, no. 1: 74.
https://doi.org/10.3390/math8010074