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).

2: 12.7 Relations to Other Functions

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§12.7(i) Hermite Polynomials

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3: 12.11 Zeros

… ►Lastly, when $a=-n-\frac{1}{2}$, $n=1,2,\mathrm{\dots }$ (Hermite polynomial case) $U(a,x)$ 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. … ►

Hermite Polynomials

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Laguerre Polynomials

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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. … ►

Hermite Polynomials

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7: 18.17 Integrals

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Hermite

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Hermite

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Hermite

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Hermite

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Hermite

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8: 18.6 Symmetry, Special Values, and Limits to Monomials

… ►For Jacobi, ultraspherical, Chebyshev, Legendre, and Hermite polynomials, see Table 18.6.1. ►

Laguerre

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Table 18.6.1: Classical OP’s: symmetry and special values.

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$p_n(x)$	$p_n(-x)$	$p_n(1)$	$p_{2n}(0)$	$p_{2n+1}^{\prime }(0)$
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§18.6(ii) Limits to Monomials

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18.6.4 $$\underset{\lambda \to \mathrm{\infty }}{lim}\frac{C_n^{(\lambda )}\left(x\right)}{C_n^{(\lambda )}\left(1\right)}=x^n,$$

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Symbols:

$C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.18(iv), §18.6(ii)

Permalink:

http://dlmf.nist.gov/18.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.6(ii), §18.6 and Ch.18

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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 $U(-\frac{1}{2}{\mu }^2,\mu t\sqrt{2})$, 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

§18.7 Interrelations and Limit Relations

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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.