# 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
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). …
##### 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. …
##### 8: 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 $\lim_{\lambda\to\infty}\frac{C^{(\lambda)}_{n}\left(x\right)}{C^{(\lambda)}_{n% }\left(1\right)}=x^{n},$
##### 9: 18.15 Asymptotic Approximations
The case $M=1$ of (18.15.1) goes back to Darboux. …
###### §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. …
##### 10: 18.7 Interrelations and Limit Relations
###### Hermite
Equations (18.7.13)–(18.7.20) are special cases of (18.2.22)–(18.2.23). …
###### Laguerre $\to$Hermite
See §18.11(ii) for limit formulas of Mehler–Heine type.