Revision as of 18:40, 24 May 2016

The Chebyshev polynomials of the second kind are orthogonal polynomials defined by $$U_n(x) = \sin(n \mathrm{arcsin}(x)).$$

Properties

1. REDIRECT Chebyshev differential equation

Theorem

The following formula holds for $n \in \{0,1,2,\ldots\}$: $$U_n(x) = (n+1){}_2F_1 \left( -n,n+2 ; \dfrac{3}{2}; \dfrac{1-x}{2} \right),$$ where $U_n$ denotes a Chebyshev polynomial of the second kind and ${}_2F_1$ denotes hypergeometric 2F1.

Proof

References

Theorem

The following formula holds for $n \in \{1,2,3,\ldots\}$: $$U_n(x)=\sqrt{1-x^2}C_{n-1}^1(x),$$ where $U_n$ denotes a Chebyshev polynomial of the second kind and $C_{n-1}^1$ denotes a Gegenbauer C polynomial.

Proof

References

Laguerre $L_n^{(\alpha)}$

Bateman $F_n$

Bernoulli $B$

Bessel poly

Chebyshev T

Chebyshev U

Dickson

Euler $E$

Gegenbauer $C$

Hermite (phys)

Hermite (prob)

Jacobi $P^{(\alpha,\beta)}$

Laguerre $L$

Legendre $P$

Bernoulli $B$

Chebyshev $T$

Fibonacci

Lucas