# Definition:Chebyshev Polynomials

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## Definition

### Chebyshev Polynomials of the First Kind

The **Chebyshev polynomials of the first kind** are defined as polynomials such that:

\(\ds \map {T_n} {\cos \theta}\) | \(=\) | \(\ds \map \cos {n \theta}\) |

### Chebyshev Polynomials of the Second Kind

The **Chebyshev polynomials of the second kind** are defined as polynomials such that:

\(\ds \map {U_n} {\cos \theta} \sin \theta\) | \(=\) | \(\ds \map \sin { (n + 1) \theta}\) |

## Also known as

The **Chebyshev polynomials** can also be seen as **Tchebyshev polynomials**.

Other transliterations exist.

Some sources define only the **Chebyshev polynomials of the first kind**, referring to them merely as **Chebyshev polynomials**.

## Also see

- Existence of Chebyshev Polynomials of the First Kind where its existence is demonstrated.
- Existence of Chebyshev Polynomials of the Second Kind where its existence is demonstrated.

## Source of Name

This entry was named for Pafnuty Lvovich Chebyshev.