# Definition:Chebyshev Polynomials/Second Kind

< Definition:Chebyshev Polynomials(Redirected from Definition:Chebyshev Polynomials of the Second Kind)

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## Definition

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

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

## Recursive Definition

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

## 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 Second Kind where its existence is demonstrated.
- Definition:Chebyshev Polynomials of the First Kind

## Source of Name

This entry was named for Pafnuty Lvovich Chebyshev.