Hermite

SearchAdvancedHelp

(0.001 seconds)

11—20 of 60 matching pages

11: 18.7 Interrelations and Limit Relations

§18.7 Interrelations and Limit Relations

… ►

Hermite

… ►

Jacobi $\to$ Hermite

… ►

Ultraspherical $\to$ Hermite

… ►

Laguerre $\to$ Hermite

…

12: 18.17 Integrals

… ►

Hermite

… ►

Hermite

… ►

Hermite

… ►

Hermite

… ►

Hermite

…

13: 18.11 Relations to Other Functions

… ►

Hermite

►

18.11.3 $$H_n\left(x\right)=2^{n}U(-\frac{1}{2}n,\frac{1}{2},x^2)=2^{n}xU(-\frac{1}{2}n+\frac{1}{2},\frac{3}{2},x^2)=2^{\frac{1}{2}n}{\mathrm{e}}^{\frac{1}{2}{x}^2}U(-n-\frac{1}{2},2^{\frac{1}{2}}x),$$

ⓘ

Symbols:

$H_n\left(x\right)$: Hermite polynomial, $U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.55, 22.5.58 ((factor $x$ missing on RHS of 22.5.55))

Referenced by:

§18.11(i), §18.15(v)

Permalink:

http://dlmf.nist.gov/18.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §18.11(i), §18.11(i), §18.11 and Ch.18

►

18.11.4 $${\mathrm{𝐻𝑒}}_n\left(x\right)=2^{\frac{1}{2}n}U(-\frac{1}{2}n,\frac{1}{2},\frac{1}{2}{x}^2)=2^{\frac{1}{2}(n-1)}xU(-\frac{1}{2}n+\frac{1}{2},\frac{3}{2},\frac{1}{2}{x}^2)={\mathrm{e}}^{\frac{1}{4}{x}^2}U(-n-\frac{1}{2},x).$$

ⓘ

Symbols:

${\mathrm{𝐻𝑒}}_n\left(x\right)$: Hermite polynomial, $U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.59

Permalink:

http://dlmf.nist.gov/18.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.11(i), §18.11(i), §18.11 and Ch.18

… ►

Hermite

►

18.11.7 $$\underset{n\to \mathrm{\infty }}{lim}\frac{{(-1)}^n{n}^{\frac{1}{2}}}{2^{2n}n!}{H}_{2n}\left(\frac{z}{2n^{\frac{1}{2}}}\right)=\frac{1}{{\pi }^{\frac{1}{2}}}\cos z,$$

ⓘ

Symbols:

$H_n\left(x\right)$: Hermite polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $!$: factorial (as in $n!$), $z$: complex variable and $n$: nonnegative integer

A&S Ref:

22.15.3

Referenced by:

§18.11(ii)

Permalink:

http://dlmf.nist.gov/18.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §18.11(ii), §18.11(ii), §18.11 and Ch.18

…

14: 18.18 Sums

… ►

Hermite

… ►

Hermite

… ►

Hermite

… ►

Hermite

… ►

Hermite and Laguerre

…

15: 18.10 Integral Representations

… ►

Hermite

►

18.10.7 $$H_n\left(x\right)=\frac{2^n}{{\pi }^{\frac{1}{2}}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{(x+it)}^n{\mathrm{e}}^{-t^2}dt.$$

ⓘ

Symbols:

$H_n\left(x\right)$: Hermite polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $t$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.10(ii)

Permalink:

http://dlmf.nist.gov/18.10.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §18.10(ii), §18.10(ii), §18.10 and Ch.18

… ►for the Jacobi, Laguerre, and Hermite polynomials. … ►

Table 18.10.1: Classical OP’s: contour integral representations (18.10.8).

► ►►

$p_n(x)$	$g_0(x)$	$g_1(z,x)$	$g_2(z,x)$	$c$	Conditions
…

►

… ►

Hermite

…

16: 18.13 Continued Fractions

… ►

Hermite

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

17: 18.5 Explicit Representations

§18.5 Explicit Representations

… ►

Hermite

… ►For corresponding formulas for Chebyshev, Legendre, and the Hermite ${\mathrm{𝐻𝑒}}_n$ polynomials apply (18.7.3)–(18.7.6), (18.7.9), and (18.7.11). … ►

Hermite

… ►

18: 18.1 Notation

… ►

Hermite: $H_n\left(x\right)$, ${\mathrm{𝐻𝑒}}_n\left(x\right)$.

… ►

Discrete $q$-Hermite I: $h_n(x;q)$.

►

Discrete $q$-Hermite II: ${\tilde{h}}_n(x;q)$.

… ►

Continuous $q$-Hermite: $H_n\left(x|q\right)$.

►

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

…

19: 18.27 $q$-Hahn Class

… ►

From Stieltjes–Wigert to Hermite

… ►

§18.27(vii) Discrete $q$-Hermite I and II Polynomials

►

Discrete $q$-Hermite I

… ►

Discrete $q$-Hermite II

… ►For discrete $q$-Hermite II polynomials the measure is not uniquely determined. …

20: 18.38 Mathematical Applications

… ►

Integrable Systems

… ►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. … ►Hermite EOP’s appear in solutions of a rationally modified Schrödinger equation in §18.39. …