Hermite polynomials

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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)). … ►

Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, standardizations, 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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2: 37.6 Plane with Weight Function ${\mathrm{e}}^{-x^2-y^2}$

… ►For the Hermite polynomial $H_n\left(x\right)$ in one variable see Table 18.3.1. … ►

§37.6(i) Complex Circular Hermite Polynomials

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§37.6(ii) Real Circular Hermite Polynomials

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$$S_{m,n}^{(\mathrm{c})}(r\cos \theta ,r\sin \theta )={(-1)}^{n}n!L_n^{(m-n)}(2r^2-1)r^{m-n}\cos \left((m-n)\theta \right),$$ $m\ge n$,

… ►The polynomials $S_{m,n-m}^{(\mathrm{c})}$ ($\frac{1}{2}n\le m\le n$) and $S_{m,n-m}^{(\mathrm{s})}$ () form together an orthogonal basis of ${𝒱}_n$. …

3: 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). …

4: 37.17 Hermite Polynomials on ${ℝ}^d$

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§37.17(i) Product Hermite Polynomials

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37.17.2 $$H_{𝝂}\left(𝐱\right)=H_{{\nu }_1}\left(x_1\right)\mathrm{\cdots }{H}_{{\nu }_d}\left(x_d\right),$$ $|𝝂|=n$,

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Defines:

$H_{𝝂}\left(𝐱\right)$: Hermite product polynomial

Symbols:

$H_n\left(x\right)$: Hermite polynomial, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space, $|𝐱|$: $l1$ norm, $𝝂$: multi-index of $d$-dimensions and $x$: real variable

Referenced by:

§37.17(v)

Permalink:

http://dlmf.nist.gov/37.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §37.17(i), §37.17 and Ch.37

… ►

§37.17(ii) Spherical Hermite Polynomials

… ►The polynomials $S_{Y,k,n}$ are called spherical Hermite polynomials. … ►

37.17.9 $$⟨S_{Y_1,k,n},S_{Y_2,k,n}⟩=\frac{{\left(\frac{1}{2}d\right)}_{n-k}}{k!}{⟨Y_1,Y_2⟩}_{{𝕊}^{d-1}},$$ $0\le k\le \lfloor \frac{1}{2}n\rfloor$, $Y_1,Y_2\in {\mathscr{H}}_{n-2k}^{0,d}$.

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\in$: element of, $!$: factorial (as in $n!$), $\lfloor x\rfloor$: floor of $x$ , $S_{Y,k,n}\left(𝐱\right)$: spherical Hermite polynomial, $n$: nonnegative integer, $k$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , ${\mathscr{H}}_n^{0,d}$: space of spherical harmonics of degree $n$ on ${𝕊}^{d-1}$ , ${⟨f,g⟩}_{{𝕊}^{d-1}}$: inner product on $(d-1)$-dimensional sphere, ${𝕊}^{d-1}$: unit sphere of dimension $d-1$ and $⟨f,g⟩$: inner product for $d$-variable Hermite polynomials

Source:

Dunkl and Xu (2014, p. 140)

Permalink:

http://dlmf.nist.gov/37.17.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §37.17(ii), §37.17 and Ch.37

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5: 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 of $f$ with respect to $x$ , $\mathrm{erf}z$: 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. …

6: 18.36 Miscellaneous Polynomials

… ►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 ►

18.36.8 $${\widehat{H}}_0\left(x\right)=2^{3/2}{\pi }^{-1/4},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, ${\widehat{H}}_n\left(x\right)$: exceptional Hermite polynomial and $x$: real variable

Referenced by:

§18.39(i)

Permalink:

http://dlmf.nist.gov/18.36.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §18.36(vi), §18.36(vi), §18.36 and Ch.18

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18.36.9 $${\widehat{H}}_{n+3}\left(x\right)=\frac{(4x^2+2)H_{n+1}\left(x\right)+8x{H}_n\left(x\right)}{{\pi }^{1/4}\sqrt{2^{n+1}(n+3)n!}}=\frac{𝒲\left\{H_1\left(x\right),H_2\left(x\right),H_{n+3}\left(x\right)\right\}}{{\pi }^{1/4}\sqrt{2^{n+7}(n+1)(n+2)(n+3)!}},$$ $n=0,1,\mathrm{\dots }$,

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, ${\widehat{H}}_n\left(x\right)$: exceptional Hermite polynomial, $!$: factorial (as in $n!$), $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.39(i)

Permalink:

http://dlmf.nist.gov/18.36.E9

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.36(vi), §18.36(vi), §18.36 and Ch.18

… ►In §18.39(i) it is seen that the functions, $\sqrt{w(x)}{\widehat{H}}_{n+3}\left(x\right)$, 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. …

7: 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). …

8: 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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12	$H_n\left(x\right)$	$1$	$-2x$	$0$	$2n$
13	${\mathrm{e}}^{-\frac{1}{2}{x}^2}{H}_n\left(x\right)$	$1$	$0$	$-x^2$	$2n+1$
14	${\mathrm{𝐻𝑒}}_n\left(x\right)$	$1$	$-x$	$0$	$n$

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9: 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:

pMML, png, TeX

See also:

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

►

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:

pMML, png, TeX

See also:

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

… ►

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:

pMML, png, TeX

See also:

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

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18.7.20 $$H_{2n+1}\left(x\right)={(-1)}^n{2}^{2n+1}n!x{L}_n^{(\frac{1}{2})}\left(x^2\right).$$

ⓘ

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:

22.5.41

Referenced by:

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

Permalink:

http://dlmf.nist.gov/18.7.E20

Encodings:

pMML, png, TeX

See also:

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

…

10: 18.6 Symmetry, Special Values, and Limits to Monomials

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

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

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$p_n(x)$	$p_n(-x)$	$p_n(1)$	$p_{2n}(0)$	$p_{2n+1}^{\prime }(0)$
…
$H_n\left(x\right)$	${(-1)}^n{H}_n\left(x\right)$		${(-1)}^n{\left(n+1\right)}_n$	$2{(-1)}^n{\left(n+1\right)}_{n+1}$
${\mathrm{𝐻𝑒}}_n\left(x\right)$	${(-1)}^n{\mathrm{𝐻𝑒}}_n\left(x\right)$		${(-\frac{1}{2})}^n{\left(n+1\right)}_n$	${(-\frac{1}{2})}^n{\left(n+1\right)}_{n+1}$

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