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Boundary Value Problems
Published in Vladimir A. Dobrushkin, Applied Differential Equations with Boundary Value Problems, 2017
6. Using the exponential generating function g^(x,t)=exp(2xt-t2)=XHn(x)tnn! $ \hat{g}(x, t) = {\text{exp }}(2xt - t^{2} ) = X^{{H_{n} (x)\frac{{t^{n} }}{n!}}} $ , prove the recurrence for Hermite polynomials: Hn+1(x) = 2xHn(x) - 2nHn-1(x) .
The Hamiltonian Approach to Electrodynamics
Published in V. L. Ginzburg, Oleg Glebov, Applications of Electrodynamics in Theoretical Physics and Astrophysics, 2017
where q0 = (ħ/ω0)1/2, Hn(x) is a Hermite polynomial, and Cn=(πq02)−1/4(2nn!)−1/2 is a normalization factor. For instance, () ψ0(q)=1(π1/2q0)1/2exp(−q22q02).
Plane wave packets and beams
Published in G. Someda Carlo, Electromagnetic Waves, 2017
An interesting property that can be proven for the Hermite polynomial of any degree m is that its m zeros are all real. Therefore, the orders m1 and m2 in Eq. (5.115) have a simple physical meaning: they are the numbers of zeroes of the function E, Eq. (5.84), along the x and y axes, respectively. They are called the transverse wavenumbers of the Hermite-Gauss modes, which in turn (for reasons similar to those outlined in Section 5.5) are usually labelled TEMm1,m2 modes.
Degenerate 2D bivariate Appell polynomials: properties and applications
Published in Applied Mathematics in Science and Engineering, 2023
Shahid Ahmad Wani, Arundhati Warke, Javid Gani Dar
The expression (4) is a solution of the heat equation: In recent years, a number of generalizations of mathematical physics especially, special functions have seen a considerable evolution. The new advancement in the special functions theory provides the analytical basis for the solution of numerous mathematical physics problems, which have several wide-ranging applications. The significant advancement in the theory of generalized special functions is based on the introduction of multi-variable and multi-index special functions. The significance of special functions has been acknowledged in both pure mathematics and practical contexts. The need for multi- variable and multi-index special functions are realized to tackle the issues emerging in the theory of abstract algebra and partial differential equations. Hermite himself [3] first devised the notion of multiple-index, multiple-variable Hermite polynomials. The Hermite polynomials are found in physics, where they generate the eigenstates of the quantum harmonic oscillator and also appear in the solution of the Schrodinger equation for the harmonic oscillator. They are also used in the numerical analysis as Gaussian quadrature.
Generalized Hermite kernel function for support vector machine classifications
Published in International Journal of Computers and Applications, 2020
We find that which gives when the coefficient of tn is equated to zero. The recurrence relation (8), connecting three Hermite polynomials with consecutive indices can be used to calculate the Hermite polynomials step by step, starting from H0(x) = 1 and H1(x) = x. The orthogonal of Hermite polynomials Hn(x), n = 0, 1, 2, 3, … , on the interval , with respect to the weighting function can be stated as in [30]