of polynomials orthogonal with respect to the normal distribution Continuous q-Hermite polynomials Continuous big q-Hermite polynomials Discrete q-Hermite...

In mathematics, the continuous q-Hermite polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek...

analysis, a cubic Hermite spline or cubic Hermite interpolator is a spline where each piece is a third-degree polynomial specified in Hermite form, that is...

In mathematics, the continuous big q-Hermite polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof...

orthogonal polynomials are the classical orthogonal polynomials, consisting of the Hermite polynomials, the Laguerre polynomials and the Jacobi polynomials. The...

mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number of...

_{n}\left(2x{\sqrt {\pi }}\right),} where Hen(x) are the "probabilist's" Hermite polynomials, defined as H e n ( x ) = ( − 1 ) n e 1 2 x 2 ( d d x ) n e − 1 2...

orthogonal polynomials are the most widely used orthogonal polynomials: the Hermite polynomials, Laguerre polynomials, Jacobi polynomials (including as...

Rogers–Szegő polynomials are a family of polynomials orthogonal on the unit circle introduced by Szegő (1926), who was inspired by the continuous q-Hermite polynomials...

\end{aligned}}} Cm p(η, ε) are the even Ince polynomials of order p and degree m where ε is the ellipticity parameter. The Hermite-Gaussian and Laguerre-Gaussian modes...

polynomial, commonly given by two explicit formulas, the Lagrange polynomials and Newton polynomials. The original use of interpolation polynomials was...

algebra A 2 ( 2 ) {\displaystyle A_{2}^{(2)}} . Rogers polynomials Continuous q-Hermite polynomials "A003114 - OEIS". Retrieved 2022-08-06. "A003106 - OEIS"...

formula. The partial or incomplete exponential Bell polynomials are a triangular array of polynomials given by B n , k ( x 1 , x 2 , … , x n − k + 1 ) =...

to P − Q = 0 {\displaystyle P-Q=0} . It follows that the study of algebraic equations is equivalent to the study of polynomials. A polynomial equation...

interpolation fits low-degree polynomials to small subsets of the values, for example, fitting nine cubic polynomials between each of the pairs of ten...

functions. A similar set of polynomials, based on a generating function, is the family of Euler polynomials. The Bernoulli polynomials Bn can be defined by a...

fundamental theorem of algebra, namely that every polynomial with real coefficients can be factored into polynomials with real coefficients of degree at most two...

shown, the zeros of the following polynomials are not expressible by sums, products, and radicals. For the latter polynomial, this fact is known as the Abel–Ruffini...

uncountably infinite. Since the polynomials with rational coefficients are countable, and since each such polynomial has a finite number of zeroes, the...

one or three real roots (which may not be distinct); all odd-degree polynomials with real coefficients have at least one real root. The graph of a cubic...

well-approximated by polynomials on [ − 1 , 1 ] {\displaystyle [-1,1]} , the associated orthogonal polynomials are Legendre polynomials, denoted by Pn(x)...

Rubio and Santhanam, 2005). For the continuous Fourier transform, the natural orthogonal eigenfunctions are the Hermite functions, so various discrete analogues...

Hermite functions. For unit variance, the n-th derivative of the Gaussian is the Gaussian function itself multiplied by the n-th Hermite polynomial,...

Laguerre polynomials. This example illustrates how the Hermite and Laguerre polynomials are linked through the Wigner map. Meanwhile, the Husimi Q function...

and the functions H n {\displaystyle {\mathcal {H}}_{n}} are the Hermite polynomials of order n {\displaystyle n} . The solution set may be generated...

streaming data involves the use of Hermite series based estimators. These estimators, based on Hermite polynomials, allow sequential estimation of the...

purpose (compare with Liouville number); the proof was given by Charles Hermite in 1873. The number e is one of only a few transcendental numbers for which...

⁠ p {\displaystyle p} ⁠ is not an integer. See also generalized Hermite polynomials. The expectation of ⁠ X {\displaystyle X} ⁠ conditioned on the event...

expression then follows from the integral relation between Hermite and Laguerre polynomials. The Wigner transformation is a general invertible transformation...

the Gaussian function ϕ {\displaystyle \phi } is given in terms of Hermite polynomial as ϕ ( n ) ( x ) = ( − 1 ) n σ n H e n ( x − μ σ ) ϕ ( x ) , {\displaystyle...