An equivalent formulation of the fact that Hermite polynomials are an orthogonal basis for consists in introducing Hermite " functions " ( see below ), and in saying that the Hermite functions are an orthonormal basis for.
22.
Many authors, particularly probabilists, use an alternate definition of the Hermite polynomials, with a weight function of e ^ {-x ^ 2 / 2 } instead of e ^ {-x ^ 2 }.
23.
The most widely used orthogonal polynomials are the classical orthogonal polynomials, consisting of the Hermite polynomials, the Laguerre polynomials, the Jacobi polynomials together with their special cases the Gegenbauer polynomials, the Chebyshev polynomials, and the Legendre polynomials.
24.
Recognizing that H 0 = 1, these rules can be followed to sequentially derive all N th order Hermite polynomials ( from N = 1 toward infinity ), and can be computer-coded relatively easily for practical applications .-->
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When interpolating a polynomial it can help to use a basis other than simple powers of x so I thought the Hermite polynomial might be useful for Hermite interpolation in the same way as the Newton basis polynomials are used in the Divided differences algorithm.
26.
The probabilists'Hermite polynomials " He " have similar formulas, which may be obtained from these by replacing the power of 2 with the corresponding power of ( " 2 ) " x ", and multiplying the entire sum by.
27.
It is a notable quantum system to solve for; since the solutions are exact ( but complicated in terms of Hermite polynomials ), and it can describe or at least approximate a wide variety of other systems, including basis of perturbation methods in quantum mechanics.
28.
For half-integer values of " a ", these ( that is, " U " and " V " ) can be re-expressed in terms of Hermite polynomials; alternatively, they can also be expressed in terms of Bessel functions.
29.
The work is focused on the properties of this distribution for instance a necessary condition on the parameters and their maximum likelihood estimators ( MLE ), the analysis of the probability generating function ( PGF ) and how it can be expressed in terms of the coefficients of ( modified ) Hermite polynomials.
30.
Where w _ 0, w, R, and \ zeta ( z ) are the waist, spot size, radius of curvature, and Gouy phase shift as given for a Gaussian beam; E _ 0 is a normalization constant; and H _ k is the th physicist's Hermite polynomial.
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