Hermite polynomials

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21: 13.6 Relations to Other Functions

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Hermite Polynomials

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13.6.16 $$M(-n,\frac{1}{2},z^2)={(-1)}^n\frac{n!}{(2n)!}{H}_{2n}\left(z\right),$$

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $!$: factorial (as in $n!$), $n$: nonnegative integer and $z$: complex variable

A&S Ref:

13.6.17

Referenced by:

§13.6(v)

Permalink:

http://dlmf.nist.gov/13.6.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §13.6(v), §13.6(v), §13.6 and Ch.13

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13.6.17 $$M(-n,\frac{3}{2},z^2)={(-1)}^n\frac{n!}{(2n+1)!2z}{H}_{2n+1}\left(z\right),$$

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $!$: factorial (as in $n!$), $n$: nonnegative integer and $z$: complex variable

A&S Ref:

13.6.18 (as corrected in later editions)

Permalink:

http://dlmf.nist.gov/13.6.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §13.6(v), §13.6(v), §13.6 and Ch.13

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13.6.18 $$U(\frac{1}{2}-\frac{1}{2}n,\frac{3}{2},z^2)=2^{-n}{z}^{-1}{H}_n\left(z\right).$$

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, $U(a,b,z)$: Kummer confluent hypergeometric function, $n$: nonnegative integer and $z$: complex variable

A&S Ref:

13.6.38 (as corrected in later editions)

Referenced by:

§13.6(v)

Permalink:

http://dlmf.nist.gov/13.6.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §13.6(v), §13.6(v), §13.6 and Ch.13

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22: 18.21 Hahn Class: Interrelations

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Charlier $\to$ Hermite

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18.21.9 $$\underset{a\to \mathrm{\infty }}{lim}{(2a)}^{\frac{1}{2}n}{C}_n({(2a)}^{\frac{1}{2}}x+a;a)={(-1)}^n{H}_n\left(x\right).$$

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Symbols:

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

Referenced by:

§18.21(ii)

Permalink:

http://dlmf.nist.gov/18.21.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §18.21(ii), §18.21(ii), §18.21 and Ch.18

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18.21.13 $$n!\underset{\lambda \to \mathrm{\infty }}{lim}{\lambda }^{-n/2}{P}_n^{(\lambda )}(x{\lambda }^{1/2};\pi /2)=H_n\left(x\right).$$

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, $P_n^{(\lambda )}(x;\varphi )$: Meixner–Pollaczek polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.21(ii), §18.21(ii), §18.30(v), Erratum (V1.2.0) Section 18.21

Permalink:

http://dlmf.nist.gov/18.21.E13

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.21(ii), §18.21(ii), §18.21 and Ch.18

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►►

Figure 18.21.1: Askey scheme. The number of free real parameters is zero for Hermite polynomials. … Magnify

23: 18.13 Continued Fractions

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Hermite

► $H_n\left(x\right)$ is the denominator of the $n$th approximant to: …

24: 18.38 Mathematical Applications

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Integrable Systems

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

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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. …

25: 18.12 Generating Functions

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Hermite

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18.12.15 $${\mathrm{e}}^{2xz-z^2}=\sum _{n=0}^{\mathrm{\infty }}\frac{H_n\left(x\right)}{n!}{z}^n,$$

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $z$: complex variable, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.9.17

Referenced by:

§18.12, §18.17(v), §18.18(ii), §18.2(xii)

Permalink:

http://dlmf.nist.gov/18.12.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §18.12, §18.12 and Ch.18

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18.12.16 $${\mathrm{e}}^{xz-\frac{1}{2}{z}^2}=\sum _{n=0}^{\mathrm{\infty }}\frac{{\mathrm{𝐻𝑒}}_n\left(x\right)}{n!}{z}^n,$$

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Symbols:

${\mathrm{𝐻𝑒}}_n\left(x\right)$: Hermite polynomial, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.17(v)

Permalink:

http://dlmf.nist.gov/18.12.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §18.12, §18.12 and Ch.18

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18.12.17 $$\frac{1+2xz+4z^2}{{\left(1+4z^2\right)}^{\frac{3}{2}}}\exp \left(\frac{4x^2{z}^2}{1+4z^2}\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{H_n\left(x\right)}{\lfloor n/2\rfloor !}{z}^n,$$ .

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, $\exp z$: exponential function, $!$: factorial (as in $n!$), $\lfloor x\rfloor$: floor of $x$ , $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Source:

Koekoek et al. (2010, (9.15.14))

Referenced by:

§18.12, Erratum (V1.2.0) Section 18.12

Permalink:

http://dlmf.nist.gov/18.12.E17

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.12, §18.12 and Ch.18

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26: 32.10 Special Function Solutions

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§32.10(iv) Fourth Painlevé Equation

… ►When $a+\frac{1}{2}$ is zero or a negative integer the $U$ parabolic cylinder functions reduce to Hermite polynomials (§18.3) times an exponential function; thus ►

32.10.20 $$w(z;-m,-2{(m-1)}^2)=-\frac{H_{m-1}^{\prime }\left(z\right)}{H_{m-1}\left(z\right)},$$ $m=1,2,3,\mathrm{\dots }$,

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, $m$: integer and $z$: real

Permalink:

http://dlmf.nist.gov/32.10.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §32.10(iv), §32.10 and Ch.32

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32.10.21 $$w(z;-m,-2{(m+1)}^2)=-2z+\frac{H_m^{\prime }\left(z\right)}{H_m\left(z\right)},$$ $m=0,1,2,\mathrm{\dots }$.

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, $m$: integer and $z$: real

Permalink:

http://dlmf.nist.gov/32.10.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §32.10(iv), §32.10 and Ch.32

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27: 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).

28: 18.28 Askey–Wilson Class

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§18.28(vi) Continuous $q$-Hermite Polynomials

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§18.28(vii) Continuous $q^{-1}$-Hermite Polynomials

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18.28.18 $$h_n\left(\sinh t|q\right)=\sum _{\mathrm{\ell }=0}^n{q}^{\frac{1}{2}\mathrm{\ell }(\mathrm{\ell }+1)}\frac{{(q^{-n};q)}_{\mathrm{\ell }}}{{(q;q)}_{\mathrm{\ell }}}{\mathrm{e}}^{(n-2\mathrm{\ell })t}={\mathrm{e}}^{nt}{}_1{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{q^{-n}}{0};q,-q{\mathrm{e}}^{-2t})={\mathrm{i}}^{-n}{H}_n\left(\mathrm{i}\sinh t|q^{-1}\right).$$

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Defines:

$h_n\left(x|q\right)$: continuous $q^{-1}$-Hermite polynomial

Symbols:

$H_n\left(x|q\right)$: continuous $q$-Hermite polynomial, $\mathrm{e}$: base of natural logarithm, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, $t$: real variable, $q$: real variable, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.28.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(vii), §18.28 and Ch.18

►For continuous $q^{-1}$-Hermite polynomials the orthogonality measure is not unique. … ►

18.28.33 $$\underset{q\to 1}{lim}\frac{H_n\left(x\sqrt{(1-q)/2}|q\right)}{{((1-q)/2)}^{n/2}}=H_n\left(x\right).$$

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Symbols:

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

Source:

Koekoek et al. (2010, (14.26.14))

Referenced by:

§18.28(x)

Permalink:

http://dlmf.nist.gov/18.28.E33

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(x), §18.28(x), §18.28 and Ch.18

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29: 18.14 Inequalities

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Hermite

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Hermite

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18.14.13 $${(H_n\left(x\right))}^2\ge H_{n-1}\left(x\right)H_{n+1}\left(x\right),$$ .

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Symbols:

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

Referenced by:

§18.14(ii)

Permalink:

http://dlmf.nist.gov/18.14.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §18.14(ii), §18.14(ii), §18.14 and Ch.18

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Hermite

►The successive maxima of $|H_n\left(x\right)|$ form a decreasing sequence for $x\le 0$, and an increasing sequence for $x\ge 0$. …

30: 18.16 Zeros

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§18.16(v) Hermite

►All zeros of $H_n\left(x\right)$ lie in the open interval $(-\sqrt{2n+1},\sqrt{2n+1})$. … ►For an asymptotic expansion of $x_{n,m}$ as $n\to \mathrm{\infty }$ that applies uniformly for $m=1,2,\mathrm{\dots },\lfloor \frac{1}{2}n\rfloor$, see Olver (1959, §14(i)). … ►Lastly, in view of (18.7.19) and (18.7.20), results for the zeros of $L_n^{(\pm \frac{1}{2})}\left(x\right)$ lead immediately to results for the zeros of $H_n\left(x\right)$. … ►

18.16.21 $$\mathrm{Disc}\left(H_n\right)=2^{\frac{3}{2}n(n-1)}\prod _{j=1}^n{j}^j.$$

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Symbols:

$\mathrm{Disc}\left(x\right)$: discriminant function, $H_n\left(x\right)$: Hermite polynomial and $n$: nonnegative integer

Proved:

Ismail (2009, (3.4.14))(proved)

Referenced by:

Erratum (V1.2.0) Section 18.16

Permalink:

http://dlmf.nist.gov/18.16.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §18.16(vii), §18.16(vii), §18.16 and Ch.18