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

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

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...

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

Legendre polynomials are associated Legendre polynomials, Legendre functions, Legendre functions of the second kind, big q-Legendre polynomials, and associated...

\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...

B_{n,k}(\mu '_{1},\ldots ,\mu '_{n-k+1}).} Hermite polynomials can be expressed in terms of Bell polynomials as He n ⁡ ( x ) = B n ( x , − 1 , 0 , … ,...

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

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

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...

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...

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

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...

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

orthogonal polynomials: 4 ϕ {\displaystyle \phi } 3 Askey–Wilson | q-Racah 3 ϕ {\displaystyle \phi } 2 Continuous dual q-Hahn | Continuous q-Hahn | Big q-Jacobi...

conjecture on the Mahler measure of non-cyclotomic polynomials The mean value problem: given a complex polynomial f {\displaystyle f} of degree d ≥ 2 {\displaystyle...

theorem — continuous functions can be approximated uniformly by polynomials, or certain other function spaces Approximation by polynomials: Linear approximation...

friend M. Hermite well proposes to call them, their Canonical forms." In the same period, usage is attested by Hesse ("Normalform"), Hermite ("forme canonique")...

Here H2n denotes the Hermite polynomial of degree 2n. Pólya’s theorem. If φ {\displaystyle \varphi } is a real-valued, even, continuous function which satisfies...

expressions. If P and Q are monic polynomials of degree m and n with respective roots p1, …, pm and q1, …, qn, we have ∏ i = a b P ( i ) Q ( i ) = ( ∏ j = 1...

given by the product of a plane wave in y {\displaystyle y} and Hermite polynomials attenuated by the gaussian function in x {\displaystyle x} , which...

of a modified Laguerre equation (not Hermite's equation!), the solution of which involves the Laguerre polynomials as F n ( u ) = ( − 1 ) n π ℏ L n ( 4...