# Hermite polynomials

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##### 1: 18.3 Definitions
###### §18.3 Definitions
The classical OP’s comprise the Jacobi, Laguerre and Hermite polynomials. … Table 18.3.1 provides the traditional definitions of Jacobi, Laguerre, and Hermite polynomials via orthogonality and standardization (§§18.2(i) and 18.2(iii)). …
##### 2: 18.41 Tables
For $P_{n}\left(x\right)$ ($=\mathsf{P}_{n}\left(x\right)$) see §14.33. Abramowitz and Stegun (1964, Tables 22.4, 22.6, 22.11, and 22.13) tabulates $T_{n}\left(x\right)$, $U_{n}\left(x\right)$, $L_{n}\left(x\right)$, and $H_{n}\left(x\right)$ for $n=0(1)12$. The ranges of $x$ are $0.2(.2)1$ for $T_{n}\left(x\right)$ and $U_{n}\left(x\right)$, and $0.5,1,3,5,10$ for $L_{n}\left(x\right)$ and $H_{n}\left(x\right)$. The precision is 10D, except for $H_{n}\left(x\right)$ which is 6-11S. … For $P_{n}\left(x\right)$, $L_{n}\left(x\right)$, and $H_{n}\left(x\right)$ see §3.5(v). …
##### 3: 7.10 Derivatives
###### §7.10 Derivatives
7.10.1 $\frac{{\mathrm{d}}^{n+1}\operatorname{erf}z}{{\mathrm{d}z}^{n+1}}=(-1)^{n}% \frac{2}{\sqrt{\pi}}H_{n}\left(z\right)e^{-z^{2}},$ $n=0,1,2,\dots$.
For the Hermite polynomial $H_{n}\left(z\right)$ see §18.3. …
##### 4: 18.36 Miscellaneous Polynomials
The type III $X_{2}$-Hermite EOP’s, missing polynomial orders $1$ and $2$, are the complete set of polynomials, with real coefficients and defined explicitly as
18.36.9 $\hat{H}_{n+3}\left(x\right)=\frac{(4x^{2}+2)H_{n+1}\left(x\right)+8xH_{n}\left% (x\right)}{{\pi}^{1/4}\sqrt{2^{n+1}(n+3)n!}}=\frac{\mathscr{W}\left\{H_{1}% \left(x\right),H_{2}\left(x\right),H_{n+3}\left(x\right)\right\}}{{\pi}^{1/4}% \sqrt{2^{n+7}(n+1)(n+2)(n+3)!}},$ $n=0,1,\dots$,
In §18.39(i) it is seen that the functions, $\sqrt{w(x)}\hat{H}_{n+3}\left(x\right)$, are solutions of a Schrödinger equation with a rational potential energy; and, in spite of first appearances, the Sturm oscillation theorem, Simon (2005c, Theorem 3.3, p. 35), is satisfied. …
##### 5: 28.9 Zeros
For $q\to\infty$ the zeros of $\operatorname{ce}_{2n}\left(z,q\right)$ and $\operatorname{se}_{2n+1}\left(z,q\right)$ approach asymptotically the zeros of $\mathit{He}_{2n}\left(q^{1/4}(\pi-2z)\right)$, and the zeros of $\operatorname{ce}_{2n+1}\left(z,q\right)$ and $\operatorname{se}_{2n+2}\left(z,q\right)$ approach asymptotically the zeros of $\mathit{He}_{2n+1}\left(q^{1/4}(\pi-2z)\right)$. Here $\mathit{He}_{n}\left(z\right)$ denotes the Hermite polynomial of degree $n$18.3). …
##### 8: 18.6 Symmetry, Special Values, and Limits to Monomials
For Jacobi, ultraspherical, Chebyshev, Legendre, and Hermite polynomials, see Table 18.6.1. …
##### 10: 12.7 Relations to Other Functions
###### §12.7(i) HermitePolynomials
12.7.2 $U\left(-n-\tfrac{1}{2},z\right)=D_{n}\left(z\right)=e^{-\frac{1}{4}z^{2}}% \mathit{He}_{n}\left(z\right)=2^{-n/2}e^{-\frac{1}{4}z^{2}}H_{n}\left(z/\sqrt{% 2}\right),$ $n=0,1,2,\dots$ ,
12.7.3 $V\left(n+\tfrac{1}{2},z\right)=\sqrt{2/\pi}e^{\frac{1}{4}z^{2}}(-i)^{n}\mathit% {He}_{n}\left(iz\right)=\sqrt{2/\pi}e^{\frac{1}{4}z^{2}}(-i)^{n}2^{-\frac{1}{2% }n}H_{n}\left(iz/\sqrt{2}\right),$ $n=0,1,2,\dots$.