# Definition:Chebyshev Polynomials/First Kind

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## Definition

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

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

## Recursive Definition

- $\map {T_n} x = \begin{cases}

1 & : n = 0 \\ x & : n = 1 \\ 2 x \map {T_{n - 1} } x - \map {T_{n - 2} } x & : n > 1 \end{cases}$

## 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.
- Definition:Chebyshev Polynomials of the Second Kind

## Source of Name

This entry was named for Pafnuty Lvovich Chebyshev.

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**Tchebyshev polynomials** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**Tchebyshev polynomials**