# Hermite polynomials

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##### 1: 18.3 Definitions
###### §18.3 Definitions
Table 18.3.1 provides the definitions of Jacobi, Laguerre, and Hermite polynomials via orthogonality and normalization (§§18.2(i) and 18.2(iii)). … For exact values of the coefficients of the Jacobi polynomials $P^{(\alpha,\beta)}_{n}\left(x\right)$, the ultraspherical polynomials $C^{(\lambda)}_{n}\left(x\right)$, the Chebyshev polynomials $T_{n}\left(x\right)$ and $U_{n}\left(x\right)$, the Legendre polynomials $P_{n}\left(x\right)$, the Laguerre polynomials $L_{n}\left(x\right)$, and the Hermite polynomials $H_{n}\left(x\right)$, see Abramowitz and Stegun (1964, pp. 793–801). …
##### 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: 28.9 Zeros
For $q\to\infty$ the zeros of $\mathrm{ce}_{2n}\left(z,q\right)$ and $\mathrm{se}_{2n+1}\left(z,q\right)$ approach asymptotically the zeros of $\mathit{He}_{2n}\left(q^{1/4}(\pi-2z)\right)$, and the zeros of $\mathrm{ce}_{2n+1}\left(z,q\right)$ and $\mathrm{se}_{2n+2}\left(z,q\right)$ 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). …
##### 5: 18.38 Mathematical Applications
###### Integrable Systems
18.38.1 $V_{n}(x)=\ifrac{2nH_{n+1}\left(x\right)H_{n-1}\left(x\right)}{(H_{n}\left(x% \right))^{2}},$
with $H_{n}\left(x\right)$ as in §18.3, satisfies the Toda equation …
###### Random Matrix Theory
Hermite polynomials (and their Freud-weight analogs (§18.32)) play an important role in random matrix theory. …
##### 9: 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 $V\left(n+\tfrac{1}{2},z\right)=\sqrt{2/\pi}e^{\frac{1}{4}z^{2}}(-i)^{n}\mathit% {He}_{n}\left(iz\right)=\sqrt{2/\pi}e^{\frac{1}{4}z^{2}}(-i)^{n}2^{-\frac{1}{2% }n}H_{n}\left(iz/\sqrt{2}\right),$ $n=0,1,2,\dots$.
##### 10: 18.4 Graphics Figure 18.4.7: Monic Hermite polynomials h n ⁡ ( x ) = 2 - n ⁢ H n ⁡ ( x ) , n = 1 , 2 , 3 , 4 , 5 . Magnify