Hermite

SearchAdvancedHelp

(0.001 seconds)

21—30 of 60 matching pages

21: 18.21 Hahn Class: Interrelations

… ►

Charlier $\to$ Hermite

►

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

ⓘ

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

… ►

Meixner–Pollaczek $\to$ Hermite

… ►

►►

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

22: 18.9 Recurrence Relations and Derivatives

… ►

Table 18.9.1: Classical OP’s: recurrence relations (18.9.1).

► ►►

$p_n(x)$	$A_n$	$B_n$	$C_n$
…

►

… ►

Table 18.9.2: Classical OP’s: recurrence relations (18.9.2_1).

► ►►

$p_n(x)$	$a_n$	$b_n$	$c_n$
…

►

… ►

Hermite

►

18.9.25 $$\frac{d}{dx}{H}_n\left(x\right)=2n{H}_{n-1}\left(x\right),$$

ⓘ

Symbols:

$H_n\left(x\right)$: Hermite polynomial, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.8.7

Referenced by:

§18.17(i), §18.9(iii)

Permalink:

http://dlmf.nist.gov/18.9.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §18.9(iii), §18.9(iii), §18.9 and Ch.18

… ►

18.9.27 $$\frac{d}{dx}{\mathrm{𝐻𝑒}}_n\left(x\right)=n{\mathrm{𝐻𝑒}}_{n-1}\left(x\right),$$

ⓘ

Symbols:

${\mathrm{𝐻𝑒}}_n\left(x\right)$: Hermite polynomial, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.8.7

Permalink:

http://dlmf.nist.gov/18.9.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §18.9(iii), §18.9(iii), §18.9 and Ch.18

…

23: 13.18 Relations to Other Functions

… ►

Hermite Polynomials

►

13.18.14 $$M_{\frac{1}{4}+n,-\frac{1}{4}}\left(z^2\right)={(-1)}^n\frac{n!}{(2n)!}{\mathrm{e}}^{-\frac{1}{2}{z}^2}\sqrt{z}{H}_{2n}\left(z\right),$$

ⓘ

Symbols:

$H_n\left(x\right)$: Hermite polynomial, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.18.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §13.18(v), §13.18(v), §13.18 and Ch.13

►

13.18.15 $$M_{\frac{3}{4}+n,\frac{1}{4}}\left(z^2\right)={(-1)}^n\frac{n!}{(2n+1)!}\frac{{\mathrm{e}}^{-\frac{1}{2}{z}^2}\sqrt{z}}{2}{H}_{2n+1}\left(z\right),$$

ⓘ

Symbols:

$H_n\left(x\right)$: Hermite polynomial, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.18.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §13.18(v), §13.18(v), §13.18 and Ch.13

►

13.18.16 $$W_{\frac{1}{4}+\frac{1}{2}n,\frac{1}{4}}\left(z^2\right)=2^{-n}{\mathrm{e}}^{-\frac{1}{2}{z}^2}\sqrt{z}{H}_n\left(z\right).$$

ⓘ

Symbols:

$H_n\left(x\right)$: Hermite polynomial, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.18.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §13.18(v), §13.18(v), §13.18 and Ch.13

…

24: 18.29 Asymptotic Approximations for $q$-Hahn and Askey–Wilson Classes

… ►For a uniform asymptotic expansion of the Stieltjes–Wigert polynomials, see Wang and Wong (2006). ►For asymptotic approximations to the largest zeros of the $q$-Laguerre and continuous $q^{-1}$-Hermite polynomials see Chen and Ismail (1998).

25: 18.14 Inequalities

… ►

Hermite

… ►

Hermite

►

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

ⓘ

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

… ►

Hermite

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

26: 18.30 Associated OP’s

… ►

§18.30(iv) Associated Hermite Polynomials

►The recursion relation for the associated Hermite polynomials, see (18.30.2), and (18.30.3), is ►

$H_{-1}(x;c)=0,$

►

$H_0(x;c)=1,$

… ►

18.30.13 $$H_{n+1}(x;c)=2x{H}_n(x;c)-2(n+c)H_{n-1}(x;c),$$ $n=0,1,\mathrm{\dots }$.

ⓘ

Symbols:

$H_n(x;c)$: associated Hermite polynomial, $n$: nonnegative integer and $x$: real variable

Referenced by:

(18.30.20)

Permalink:

http://dlmf.nist.gov/18.30.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §18.30(iv), §18.30 and Ch.18

…

27: 13.6 Relations to Other Functions

… ►

Hermite Polynomials

►

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

ⓘ

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

►

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

ⓘ

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

►

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

ⓘ

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

…

28: 18.12 Generating Functions

… ►The $z$-radii of convergence will depend on $x$, and in first instance we will assume $x\in [-1,1]$ for Jacobi, ultraspherical, Chebyshev and Legendre, $x\in [0,\mathrm{\infty })$ for Laguerre, and $x\in ℝ$ for Hermite. … ►

Hermite

►

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

ⓘ

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

►

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

ⓘ

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

►

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

ⓘ

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

…

29: 18.28 Askey–Wilson Class

… ►

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

… ►

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

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

From Continuous $q$-Ultraspherical to Continuous $q$-Hermite

… ►

From Continuous $q$-Hermite to Hermite

…

30: 18.16 Zeros

… ►

§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 $H_n\left(x\right)$. … ►

Hermite

…