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)). … ►

Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, normalizations, leading coefficients, and parameter constraints. …

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Name	$p_n(x)$	$(a,b)$	$w(x)$	$h_n$	$k_n$	${\tilde{k}}_n/k_n$	Constraints
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Hermite	$H_n\left(x\right)$	$(-\mathrm{\infty },\mathrm{\infty })$	${\mathrm{e}}^{-x^2}$	${\pi }^{\frac{1}{2}}{2}^{n}n!$	$2^n$	$0$
Hermite	${\mathrm{𝐻𝑒}}_n\left(x\right)$	$(-\mathrm{\infty },\mathrm{\infty })$	${\mathrm{e}}^{-\frac{1}{2}{x}^2}$	${(2\pi )}^{\frac{1}{2}}n!$	$1$	$0$

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►For exact values of the coefficients of the Jacobi polynomials $P_n^{(\alpha ,\beta )}\left(x\right)$, the ultraspherical polynomials $C_n^{(\lambda )}\left(x\right)$, 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 $P_n\left(x\right)$ ($={𝖯}_n\left(x\right)$) 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

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7.10.1 $$\frac{d^{n+1}\mathrm{erf}z}{{dz}^{n+1}}={(-1)}^n\frac{2}{\sqrt{\pi }}{H}_n\left(z\right){\mathrm{e}}^{-z^2},$$ $n=0,1,2,\mathrm{\dots }$.

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{df}{dx}$: derivative, $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.1.19

Permalink:

http://dlmf.nist.gov/7.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §7.10 and Ch.7

►For the Hermite polynomial $H_n\left(z\right)$ see §18.3. …

4: 28.9 Zeros

… ►For $q\to \mathrm{\infty }$ the zeros of ${\mathrm{ce}}_{2n}(z,q)$ and ${\mathrm{se}}_{2n+1}(z,q)$ approach asymptotically the zeros of ${\mathrm{𝐻𝑒}}_{2n}\left(q^{1/4}(\pi -2z)\right)$, and the zeros of ${\mathrm{ce}}_{2n+1}(z,q)$ and ${\mathrm{se}}_{2n+2}(z,q)$ approach asymptotically the zeros of ${\mathrm{𝐻𝑒}}_{2n+1}\left(q^{1/4}(\pi -2z)\right)$. Here ${\mathrm{𝐻𝑒}}_n\left(z\right)$ denotes the Hermite polynomial of degree $n$ (§18.3). …

5: 18.38 Mathematical Applications

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Integrable Systems

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18.38.1 $$V_n(x)=2n{H}_{n+1}\left(x\right)H_{n-1}\left(x\right)/{(H_n\left(x\right))}^2,$$

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.38.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.38(ii), §18.38(ii), §18.38 and Ch.18

►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. …

6: 18.5 Explicit Representations

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Table 18.5.1: Classical OP’s: Rodrigues formulas (18.5.5).

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$p_n(x)$	$F(x)$	${\kappa }_n$
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$H_n\left(x\right)$	$1$	${(-1)}^n$
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$H_0\left(x\right)=1,$

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$H_1\left(x\right)=2x,$

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${\mathrm{𝐻𝑒}}_0\left(x\right)=1,$

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${\mathrm{𝐻𝑒}}_1\left(x\right)=x,$

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7: 18.8 Differential Equations

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Table 18.8.1: Classical OP’s: differential equations $A(x)f^{\prime \prime }(x)+B(x)f^{\prime }(x)+C(x)f(x)+{\lambda }_{n}f(x)=0$.

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$f(x)$	$A(x)$	$B(x)$	$C(x)$	${\lambda }_n$
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$H_n\left(x\right)$	$1$	$-2x$	$0$	$2n$
${\mathrm{e}}^{-\frac{1}{2}{x}^2}{H}_n\left(x\right)$	$1$	$0$	$-x^2$	$2n+1$
${\mathrm{𝐻𝑒}}_n\left(x\right)$	$1$	$-x$	$0$	$n$

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8: 18.18 Sums

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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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9: 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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10: 18.7 Interrelations and Limit Relations

§18.7 Interrelations and Limit Relations

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18.7.11 $${\mathrm{𝐻𝑒}}_n\left(x\right)=2^{-\frac{1}{2}n}{H}_n\left(2^{-\frac{1}{2}}x\right),$$

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, ${\mathrm{𝐻𝑒}}_n\left(x\right)$: Hermite polynomial, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.18

Referenced by:

§18.5(iii)

Permalink:

http://dlmf.nist.gov/18.7.E11

Encodings:

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See also:

Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18

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18.7.12 $$H_n\left(x\right)=2^{\frac{1}{2}n}{\mathrm{𝐻𝑒}}_n\left(2^{\frac{1}{2}}x\right).$$

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, ${\mathrm{𝐻𝑒}}_n\left(x\right)$: Hermite polynomial, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.19

Referenced by:

§18.7(i)

Permalink:

http://dlmf.nist.gov/18.7.E12

Encodings:

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See also:

Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18

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18.7.19 $$H_{2n}\left(x\right)={(-1)}^n{2}^{2n}n!L_n^{(-\frac{1}{2})}\left(x^2\right),$$

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $!$: factorial (as in $n!$), $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.40

Referenced by:

§18.16(v), §18.18(iii), §18.18(iv), §18.18(viii), §18.7(ii)

Permalink:

http://dlmf.nist.gov/18.7.E19

Encodings:

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See also:

Annotations for §18.7(ii), §18.7 and Ch.18

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