Laguerre polynomials

SearchAdvancedHelp

(0.008 seconds)

21—30 of 50 matching pages

21: 18.16 Zeros

… ►

§18.16(iv) Laguerre

►The zeros of $L_n^{(\alpha )}\left(x\right)$ are denoted by $x_{n,m}$, $m=1,2,\mathrm{\dots },n$, with … ►

Asymptotic Behavior

… ►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 $H_n\left(x\right)$. … ►

18.16.20 $$\mathrm{Disc}\left(L_n^{(\alpha )}\right)=\prod _{j=1}^n{j}^{j-2n+2}{(j+\alpha )}^{j-1}.$$

ⓘ

Symbols:

$\mathrm{Disc}\left(x\right)$: discriminant function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial and $n$: nonnegative integer

Proved:

Ismail (2009, (3.4.15))(proved)

Permalink:

http://dlmf.nist.gov/18.16.E20

Encodings:

pMML, png, TeX

See also:

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

…

22: 18.12 Generating Functions

… ►

Laguerre

►

18.12.13 $${(1-z)}^{-\alpha -1}\exp \left(\frac{xz}{z-1}\right)=\sum _{n=0}^{\mathrm{\infty }}{L}_n^{(\alpha )}\left(x\right)z^n,$$ .

ⓘ

Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\exp z$: exponential function, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.9.15

Referenced by:

§18.12, §18.18(ii), §18.18(iv), §18.18(viii), §18.2(xii), (8.7.6)

Permalink:

http://dlmf.nist.gov/18.12.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §18.12, §18.12 and Ch.18

►

18.12.14 $$\mathrm{\Gamma }\left(\alpha +1\right){(xz)}^{-\frac{1}{2}\alpha }{\mathrm{e}}^z{J}_{\alpha }\left(2\sqrt{xz}\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{L_n^{(\alpha )}\left(x\right)}{{\left(\alpha +1\right)}_n}{z}^n.$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.9.16

Referenced by:

§18.12

Permalink:

http://dlmf.nist.gov/18.12.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §18.12, §18.12 and Ch.18

…

23: 17.17 Physical Applications

… ►See Kassel (1995). … ►It involves $q$-generalizations of exponentials and Laguerre polynomials, and has been applied to the problems of the harmonic oscillator and Coulomb potentials. …

24: 18.15 Asymptotic Approximations

… ►

§18.15(iv) Laguerre

… ►

18.15.14 $$L_n^{(\alpha )}\left(x\right)=\frac{n^{\frac{1}{2}\alpha -\frac{1}{4}}{\mathrm{e}}^{\frac{1}{2}x}}{{\pi }^{\frac{1}{2}}{x}^{\frac{1}{2}\alpha +\frac{1}{4}}}\left(\cos {\theta }_n^{(\alpha )}(x)\left(\sum _{m=0}^{M-1}\frac{a_m(x)}{n^{\frac{1}{2}m}}+O\left(\frac{1}{n^{\frac{1}{2}M}}\right)\right)+\sin {\theta }_n^{(\alpha )}(x)\left(\sum _{m=1}^{M-1}\frac{b_m(x)}{n^{\frac{1}{2}m}}+O\left(\frac{1}{n^{\frac{1}{2}M}}\right)\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\sin z$: sine function, $m$: nonnegative integer, $n$: nonnegative integer, ${\theta }_n^{(\alpha )}(x)$ , $a_m(x)$: coefficients, $b_m(x)$: coefficients and $x$: real variable

Referenced by:

(8.7.6)

Permalink:

http://dlmf.nist.gov/18.15.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §18.15(iv), §18.15(iv), §18.15 and Ch.18

… ►

18.15.19 $$L_n^{(\alpha )}\left(\nu x\right)=\frac{{\mathrm{e}}^{\frac{1}{2}\nu x}}{2^{\alpha }{x}^{\frac{1}{2}\alpha +\frac{1}{4}}{(1-x)}^{\frac{1}{4}}}\left({\xi }^{\frac{1}{2}}{J}_{\alpha }\left(\nu \xi \right)\sum _{m=0}^{M-1}\frac{A_m(\xi )}{{\nu }^{2m}}+{\xi }^{-\frac{1}{2}}{J}_{\alpha +1}\left(\nu \xi \right)\sum _{m=0}^{M-1}\frac{B_m(\xi )}{{\nu }^{2m+1}}+{\xi }^{\frac{1}{2}}\mathrm{env}{J}_{\alpha }\left(\nu \xi \right)O\left(\frac{1}{{\nu }^{2M-1}}\right)\right),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $O\left(x\right)$: order not exceeding, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\mathrm{env}{J}_{\nu }\left(x\right)$: envelope of Bessel function $J_{\nu }\left(x\right)$ , $\mathrm{e}$: base of natural logarithm, $m$: nonnegative integer, $n$: nonnegative integer, $\nu$ , $\xi$ , $A_m(\xi )$: coefficient, $B_m(\xi )$: coefficient and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.15.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §18.15(iv), §18.15(iv), §18.15 and Ch.18

… ►

18.15.22 $$L_n^{(\alpha )}\left(\nu x\right)={(-1)}^n\frac{{\mathrm{e}}^{\frac{1}{2}\nu x}}{2^{\alpha -\frac{1}{2}}{x}^{\frac{1}{2}\alpha +\frac{1}{4}}}{\left(\frac{\zeta }{x-1}\right)}^{\frac{1}{4}}\left(\frac{\mathrm{Ai}\left({\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{1}{3}}}\sum _{m=0}^{M-1}\frac{E_m(\zeta )}{{\nu }^{2m}}+\frac{{\mathrm{Ai}}^{\prime }\left({\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{5}{3}}}\sum _{m=0}^{M-1}\frac{F_m(\zeta )}{{\nu }^{2m}}+\mathrm{envAi}\left({\nu }^{\frac{2}{3}}\zeta \right)O\left(\frac{1}{{\nu }^{2M-\frac{2}{3}}}\right)\right),$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $O\left(x\right)$: order not exceeding, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\sim$: Poincaré asymptotic expansion, $\mathrm{envAi}\left(x\right)$: envelope of Airy function $\mathrm{Ai}\left(x\right)$ , $\mathrm{e}$: base of natural logarithm, $m$: nonnegative integer, $n$: nonnegative integer, $\nu$ , $\zeta$ , $E_m$: coefficient, $F_m$: coefficient and $x$: real variable

Referenced by:

Erratum (V1.0.11) for Equation (18.15.22)

Permalink:

http://dlmf.nist.gov/18.15.E22

Encodings:

pMML, png, TeX

Errata (effective with 1.0.11):

Originally this equation was expressed in terms of the asymptotic symbol $\sim$. As a consequence of the use of the $O$ order symbol on the right-hand side, $\sim$ was replaced by $=$.

See also:

Annotations for §18.15(iv), §18.15(iv), §18.15 and Ch.18

… ►For asymptotic approximations of Jacobi, ultraspherical, and Laguerre polynomials in terms of Hermite polynomials, see López and Temme (1999a). …

25: 13.18 Relations to Other Functions

… ►

Laguerre Polynomials

►

13.18.17 $$W_{\frac{1}{2}\alpha +\frac{1}{2}+n,\frac{1}{2}\alpha }\left(z\right)={(-1)}^n{\left(\alpha +1\right)}_n{M}_{\frac{1}{2}\alpha +\frac{1}{2}+n,\frac{1}{2}\alpha }\left(z\right)={(-1)}^{n}n!{\mathrm{e}}^{-\frac{1}{2}z}{z}^{\frac{1}{2}\alpha +\frac{1}{2}}{L}_n^{(\alpha )}\left(z\right).$$

ⓘ

Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: nonnegative integer and $z$: complex variable

Referenced by:

(33.14.14)

Permalink:

http://dlmf.nist.gov/13.18.E17

Encodings:

pMML, png, TeX

See also:

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

…

26: 18.30 Associated OP’s

… ►

§18.30(iii) Associated Laguerre Polynomials

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

$L_{-1}^{\lambda }(x;c)=0,$

►

$L_0^{\lambda }(x;c)=1,$

… ►

18.30.9 $$(n+c+1)L_{n+1}^{\lambda }(x;c)=(2n+2c+\lambda +1-x)L_n^{\lambda }(x;c)-(n+c+\lambda )L_{n-1}^{\lambda }(x;c),$$ $n=0,1,\mathrm{\dots }$.

ⓘ

Symbols:

$L_n^{\lambda }(x;c)$: associated Laguerre polynomial, $n$: nonnegative integer and $x$: real variable

Referenced by:

(18.30.19)

Permalink:

http://dlmf.nist.gov/18.30.E9

Encodings:

pMML, png, TeX

See also:

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

…

27: 18.13 Continued Fractions

… ►

Laguerre

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

28: 13.6 Relations to Other Functions

… ►

Laguerre Polynomials

►

13.6.19 $$U(-n,\alpha +1,z)={(-1)}^n{\left(\alpha +1\right)}_{n}M(-n,\alpha +1,z)={(-1)}^{n}n!L_n^{(\alpha )}\left(z\right).$$

ⓘ

Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $U(a,b,z)$: Kummer confluent hypergeometric function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $n$: nonnegative integer and $z$: complex variable

A&S Ref:

13.6.9 and 13.6.27

Referenced by:

§13.6(v), §18.17(iv), §18.9(iii)

Permalink:

http://dlmf.nist.gov/13.6.E19

Encodings:

pMML, png, TeX

See also:

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

…

29: 33.14 Definitions and Basic Properties

… ►

33.14.14 $${\varphi }_{n,\mathrm{\ell }}(r)={(-1)}^{\mathrm{\ell }+1+n}{(2/n^3)}^{1/2}s(-1/n^2,\mathrm{\ell };r)=\frac{{(-1)}^{\mathrm{\ell }+1+n}}{n^{\mathrm{\ell }+2}}{\left(\frac{(n-\mathrm{\ell }-1)!}{(n+\mathrm{\ell })!}\right)}^{1/2}{(2r)}^{\mathrm{\ell }+1}{\mathrm{e}}^{-r/n}{L}_{n-\mathrm{\ell }-1}^{(2\mathrm{\ell }+1)}\left(2r/n\right)$$

ⓘ

Defines:

${\varphi }_{n,\mathrm{\ell }}(r)$: function (locally)

Symbols:

$s(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{\ell }$: nonnegative integer and $r$: real variable

Referenced by:

(33.14.14), (33.14.15), §33.14(iv), §33.22(v), Erratum (V1.0.24) for Equation (33.14.15), Erratum (V1.0.24) for Subsection 33.14(iv)

Permalink:

http://dlmf.nist.gov/33.14.E14

Encodings:

pMML, png, TeX

Addition (effective with 1.0.24):

For the second equation in (33.14.14) first, via (33.14.9) and (33.14.4), express ${\varphi }_{n,\mathrm{\ell }}(r)$ in terms of Whittaker functions and then use (13.18.17).

See also:

Annotations for §33.14(iv), §33.14 and Ch.33

…

30: 1.17 Integral and Series Representations of the Dirac Delta

… ►

Laguerre Polynomials (§18.3)

►

1.17.23 $$\delta \left(x-a\right)={\mathrm{e}}^{-(x+a)/2}\sum _{k=0}^{\mathrm{\infty }}{L}_k\left(x\right)L_k\left(a\right).$$

ⓘ

Symbols:

$\delta \left(x-a\right)$: Dirac delta (or Dirac delta function), $\mathrm{e}$: base of natural logarithm, $L_n\left(x\right)=L_n^{(0)}\left(x\right)$: Laguerre polynomial and $k$: integer

Permalink:

http://dlmf.nist.gov/1.17.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §1.17(iii), §1.17(iii), §1.17 and Ch.1

…