Chebyshev polynomials [SEP] named after
Chebyshev polynomials
In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula and which can be defined recursively. One usually distinguishes between Chebyshev polynomials of the first kind which are denoted and Chebyshev polynomials of the second kind which are denoted . The letter is used because of the alternative transliterations of the name "Chebyshev" as "Tchebycheff", "Tchebyshev" (French) or "Tschebyschow" (German).
The Chebyshev polynomials or are polynomials
Gegenbauer polynomials
In mathematics, Gegenbauer polynomials or ultraspherical polynomials "C"("x") are orthogonal polynomials on the interval [−1,1] with respect to the weight function (1 − "x"). They generalize Legendre polynomials and Chebyshev polynomials, and are special cases of Jacobi polynomials. They are named after Leopold Gegenbauer.
Characterizations.
A variety of characterizations of the Gegenbauer polynomials are available.
- The polynomials can be defined in terms of their generating function :
- The polynomials satisfy the recurrence relation