Definition:Chebyshev Polynomials/Second Kind

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Definition

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

\(\ds \map {U_n} {\cos \theta} \sin \theta\)

\(=\)

\(\ds \map \sin {\paren {n + 1} \theta}\)

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.

Sources

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