# Hermite polynomial case

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##### 1: 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).
##### 3: 12.11 Zeros
Lastly, when $a=-n-\tfrac{1}{2}$, $n=1,2,\dots$ (Hermite polynomial case) $U\left(a,x\right)$ has $n$ zeros and they lie in the interval $[-2\sqrt{-a},2\sqrt{-a}\,]$. …
##### 4: 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)). This table also includes the following special cases of Jacobi polynomials: ultraspherical, Chebyshev, and Legendre. … 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). … Legendre polynomials are special cases of Legendre functions, Ferrers functions, and associated Legendre functions (§14.7(i)). …
##### 5: 13.18 Relations to Other Functions
Special cases are the error functions …
###### §13.18(v) Orthogonal Polynomials
Special cases of §13.18(iv) are as follows. …
##### 6: 13.6 Relations to Other Functions
Special cases are the error functions … and in the case that $b-2a$ is an integer we have …Note that (13.6.11_1) and (13.6.11_2) are special cases of (13.11.1) and (13.11.2), respectively … Special cases of §13.6(iv) are as follows. …
##### 7: 18.6 Symmetry, Special Values, and Limits to Monomials
For Jacobi, ultraspherical, Chebyshev, Legendre, and Hermite polynomials, see Table 18.6.1.
###### §18.6(ii) Limits to Monomials
18.6.4 $\lim_{\lambda\to\infty}\frac{C^{(\lambda)}_{n}\left(x\right)}{C^{(\lambda)}_{n% }\left(1\right)}=x^{n},$
##### 8: 18.15 Asymptotic Approximations
###### §18.15(v) Hermite
With $\mu=\sqrt{2n+1}$ the expansions in Chapter 12 are for the parabolic cylinder function $U\left(-\tfrac{1}{2}\mu^{2},\mu t\sqrt{2}\right)$, which is related to the Hermite polynomials via … For asymptotic approximations of Jacobi, ultraspherical, and Laguerre polynomials in terms of Hermite polynomials, see López and Temme (1999a). These approximations apply when the parameters are large, namely $\alpha$ and $\beta$ (subject to restrictions) in the case of Jacobi polynomials, $\lambda$ in the case of ultraspherical polynomials, and $|\alpha|+|x|$ in the case of Laguerre polynomials. …