# Hermite

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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)). … For finite power series of the Jacobi, ultraspherical, Laguerre, and Hermite polynomials, see §18.5(iii) (in powers of $x-1$ for Jacobi polynomials, in powers of $x$ for the other cases). Explicit power series for Chebyshev, Legendre, Laguerre, and Hermite polynomials for $n=0,1,\ldots,6$ are given in §18.5(iv). …
##### 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
They are related to Hermite–Padé approximation and can be used for proofs of irrationality or transcendence of interesting numbers. …
###### Type III $X_{2}$-Hermite EOP’s
Hermite EOP’s are defined in terms of classical Hermite OP’s. 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 … 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: 12.1 Special Notation
Unless otherwise noted, primes indicate derivatives with respect to the variable, and fractional powers take their principal values. … Whittaker’s notation $D_{\nu}\left(z\right)$ is useful when $\nu$ is a nonnegative integer (Hermite polynomial case).
##### 6: 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) Hermite Polynomials
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 $V\left(n+\tfrac{1}{2},z\right)=\sqrt{2/\pi}e^{\frac{1}{4}z^{2}}(-i)^{n}\mathit% {He}_{n}\left(iz\right)=\sqrt{2/\pi}e^{\frac{1}{4}z^{2}}(-i)^{n}2^{-\frac{1}{2% }n}H_{n}\left(iz/\sqrt{2}\right),$ $n=0,1,2,\dots$.