Search results
Results from the WOW.Com Content Network
The Chebyshev polynomials of the second kind are defined by the recurrence relation: = = + = (). Notice that the two sets of recurrence relations are identical, except for () = vs. () =.
These are the roots of the Chebyshev polynomials of the second kind with degree . For nodes over an arbitrary interval [ a , b ] {\displaystyle [a,b]} an affine transformation can be used as above. The Chebyshev nodes of the second kind are also referred to as Chebyshev-Lobatto points or Chebyshev extreme points. [ 3 ]
The two most common types of discrete Chebyshev transforms use the grid of Chebyshev zeros, the zeros of the Chebyshev polynomials of the first kind () and the grid of Chebyshev extrema, the extrema of the Chebyshev polynomials of the first kind, which are also the zeros of the Chebyshev polynomials of the second kind (). Both of these ...
where is the kth Chebyshev polynomial of the 2nd kind. Since + =, we get that () ... problem will be the zeros of the nth Chebyshev polynomial of the second kind, ...
Because of this, expansion of functions in terms of Chebyshev polynomials is sometimes used for polynomial approximations in computer math libraries. Some authors use versions of these polynomials that have been shifted so that the interval of orthogonality is [0, 1] or [−2, 2].
Hermite polynomials were defined by Pierre-Simon Laplace in 1810, [1] [2] though in scarcely recognizable form, and studied in detail by Pafnuty Chebyshev in 1859. [3] Chebyshev's work was overlooked, and they were named later after Charles Hermite , who wrote on the polynomials in 1864, describing them as new. [ 4 ]
Specifically, for α = −1, the Dickson polynomials of the first kind are Fibonacci polynomials, and Dickson polynomials of the second kind are Lucas polynomials. By the composition rule above, when α is idempotent, composition of Dickson polynomials of the first kind is commutative. The Dickson polynomials with parameter α = 0 give monomials.
Somewhat more general Laguerre polynomial sequences are orthogonal with respect to gamma distributions. The Chebyshev polynomials of the first kind are orthogonal with respect to the measure . The Chebyshev polynomials of the second kind are orthogonal with respect to the Wigner semicircle distribution.