Laguerre 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: 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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Laguerre	$L_n^{(\alpha )}\left(x\right)$	$(0,\mathrm{\infty })$	${\mathrm{e}}^{-x}{x}^{\alpha }$	$\mathrm{\Gamma }\left(n+\alpha +1\right)/n!$	${(-1)}^n/n!$	$-n(n+\alpha )$	$\alpha >-1$
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… ►For explicit power series coefficients up to $n=12$ for these polynomials and for coefficients up to $n=6$ for Jacobi and ultraspherical polynomials see Abramowitz and Stegun (1964, pp. 793–801). …

2: 18.4 Graphics

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Figure 18.4.5: Laguerre polynomials $L_n\left(x\right)$, $n=1,2,3,4,5$. Magnify

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Figure 18.4.6: Laguerre polynomials $L_3^{(\alpha )}\left(x\right)$, $\alpha =0,1,2,3,4$. Magnify

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Figure 18.4.8: Laguerre polynomials $L_3^{(\alpha )}\left(x\right)$, $0\le \alpha \le 3$, $0\le x\le 10$. Magnify 3D Help

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Figure 18.4.9: Laguerre polynomials $L_4^{(\alpha )}\left(x\right)$, $0\le \alpha \le 3$, $0\le x\le 10$. Magnify 3D Help

3: 18.36 Miscellaneous Polynomials

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§18.36(v) Non-Classical Laguerre Polynomials $L_n^{(-k)}\left(x\right)$, $k=1,2\mathrm{\dots }$

… ►For the Laguerre polynomials $L_n^{(\alpha )}\left(x\right)$ this requires, omitting all strictly positive factors, … ►The resulting EOP’s, ${\widehat{L}}_n^{(k)}\left(x\right)$, $n=1,2,\mathrm{\dots }$ satisfy … ►

18.36.5 $${\widehat{L}}_n^{(k)}\left(x\right)=-(x+k+1)L_{n-1}^{(k)}\left(x\right)+L_{n-2}^{(k)}\left(x\right),$$ $n\ge 1$,

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, ${\widehat{L}}_n^{(k)}\left(x\right)$: exceptional Laguerre polynomial, $k$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.36.E5

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

… ►The $y(x)={\widehat{L}}_n^{(k)}\left(x\right)$ satisfy a second order Sturm–Liouville eigenvalue problem of the type illustrated in Table 18.8.1, as satisfied by classical OP’s, but now with rational, rather than polynomial coefficients: …

4: 37.5 Quarter Plane with Weight Function $x^{\alpha }{y}^{\beta }{\mathrm{e}}^{-x-y}$

… ►For the Laguerre polynomial $L_n^{(\alpha )}\left(x\right)$ and the Jacobi polynomial $P_n^{(\alpha ,\beta )}\left(x\right)$ in one variable see Table 18.3.1. … ►

37.5.4 $$L_k^{(\alpha )}\left(x\right)L_{n-k}^{(\beta )}\left(y\right),$$ $0\le k\le n$.

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $y$: real variable

Referenced by:

§37.5(ii), §37.5(iii)

Permalink:

http://dlmf.nist.gov/37.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §37.5(i), §37.5, Part 37.II and Ch.37

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37.5.9 $$[x{D}_{xx}+(1+\alpha -x)D_x]L_k^{(\alpha )}\left(x\right)L_{n-k}^{(\beta )}\left(y\right)=-k{L}_k^{(\alpha )}\left(x\right)L_{n-k}^{(\beta )}\left(y\right),$$

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $y$: real variable

Proof sketch:

Use the differential equation for $L_k^{(\alpha )}\left(x\right)$ in Table 18.8.1.

Permalink:

http://dlmf.nist.gov/37.5.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §37.5(ii), §37.5, Part 37.II and Ch.37

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37.5.12 $$\underset{\gamma \to \mathrm{\infty }}{lim}{P}_{k,n}^{\alpha ,\beta ,\gamma }({\gamma }^{-1}x,{\gamma }^{-1}y)={(-1)}^n{L}_{n-k}^{(\alpha )}\left(x\right)L_k^{(\beta )}\left(y\right),$$

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/37.5.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §37.5(iii), §37.5, Part 37.II and Ch.37

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37.5.13 $$\underset{\gamma \to \mathrm{\infty }}{lim}{Q}_{k,n}^{\alpha ,\beta ,\gamma }({\gamma }^{-1}x,{\gamma }^{-1}y)={(-1)}^{n-k}{L}_k^{(\alpha )}\left(x\right)L_{n-k}^{(\beta )}\left(y\right),$$

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/37.5.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §37.5(iii), §37.5, Part 37.II and Ch.37

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5: 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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8	$L_n^{(\alpha )}\left(x\right)$	$x$	$\alpha +1-x$	$0$	$n$
9	${\mathrm{e}}^{-\frac{1}{2}{x}^2}{x}^{\alpha +\frac{1}{2}}{L}_n^{(\alpha )}\left(x^2\right)$	$1$	$0$	$-x^2+(\frac{1}{4}-{\alpha }^2)x^{-2}$	$4n+2\alpha +2$
10	${\mathrm{e}}^{-\frac{1}{2}x}{x}^{\frac{1}{2}\alpha }{L}_n^{(\alpha )}\left(x\right)$	$x$	$1$	$-\frac{1}{4}x-\frac{1}{4}{\alpha }^2{x}^{-1}$	$n+\frac{1}{2}(\alpha +1)$
11	${\mathrm{e}}^{-n^{-1}x}{x}^{\mathrm{\ell }+1}{L}_{n-\mathrm{\ell }-1}^{(2\mathrm{\ell }+1)}\left(2n^{-1}x\right)$	$1$	$0$	$\frac{2}{x}-\frac{\mathrm{\ell }(\mathrm{\ell }+1)}{x^2}$	$-\frac{1}{n^2}$
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6: 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 $H_n\left(x\right)$. … ►For $P_n\left(x\right)$, $L_n\left(x\right)$, and $H_n\left(x\right)$ see §3.5(v). …

7: 18.14 Inequalities

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Laguerre

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Laguerre

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18.14.12 $${(L_n^{(\alpha )}\left(x\right))}^2\ge L_{n-1}^{(\alpha )}\left(x\right)L_{n+1}^{(\alpha )}\left(x\right),$$ , $\alpha \ge 0$.

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Symbols:

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

Referenced by:

§18.14(ii)

Permalink:

http://dlmf.nist.gov/18.14.E12

Encodings:

pMML, png, TeX

See also:

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

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Laguerre

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18.14.24

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Symbols:

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

Referenced by:

§18.14(iii)

Permalink:

http://dlmf.nist.gov/18.14.E24

Encodings:

pMML, png, TeX

See also:

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

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8: 18.6 Symmetry, Special Values, and Limits to Monomials

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Laguerre

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18.6.1 $$L_n^{(\alpha )}\left(0\right)=\frac{{\left(\alpha +1\right)}_n}{n!}.$$

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$) and $n$: nonnegative integer

A&S Ref:

22.4.7 (see column for $f_n(0)$)

Referenced by:

§18.17(i), §18.6(i), §18.6(ii)

Permalink:

http://dlmf.nist.gov/18.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.6(i), §18.6(i), §18.6 and Ch.18

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§18.6(ii) Limits to Monomials

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18.6.5 $$\underset{\alpha \to \mathrm{\infty }}{lim}\frac{L_n^{(\alpha )}\left(\alpha x\right)}{L_n^{(\alpha )}\left(0\right)}={(1-x)}^n.$$

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Symbols:

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

Referenced by:

§18.6(ii)

Permalink:

http://dlmf.nist.gov/18.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §18.6(ii), §18.6 and Ch.18

9: 18.18 Sums

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Laguerre

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18.18.10 $$L_n^{({\alpha }_1+\mathrm{\cdots }+{\alpha }_r+r-1)}\left(x_1+\mathrm{\cdots }+x_r\right)=\sum _{m_1+\mathrm{\cdots }+m_r=n}{L}_{m_1}^{({\alpha }_1)}\left(x_1\right)\mathrm{\cdots }{L}_{m_r}^{({\alpha }_r)}\left(x_r\right).$$

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Symbols:

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

Referenced by:

§18.18(ii), §18.18(viii)

Permalink:

http://dlmf.nist.gov/18.18.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §18.18(ii), §18.18(ii), §18.18 and Ch.18

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18.18.12 $$\frac{L_n^{(\alpha )}\left(\lambda x\right)}{L_n^{(\alpha )}\left(0\right)}=\sum _{\mathrm{\ell }=0}^n\left(\genfrac{}{}{0.0pt}{}{n}{\mathrm{\ell }}\right){\lambda }^{\mathrm{\ell }}{(1-\lambda )}^{n-\mathrm{\ell }}\frac{L_{\mathrm{\ell }}^{(\alpha )}\left(x\right)}{L_{\mathrm{\ell }}^{(\alpha )}\left(0\right)}.$$

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

(18.17.34_5), §18.18(iii), §18.18(iv)

Permalink:

http://dlmf.nist.gov/18.18.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §18.18(iii), §18.18(iii), §18.18 and Ch.18

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18.18.37 $$\sum _{\mathrm{\ell }=0}^n{L}_{\mathrm{\ell }}^{(\alpha )}\left(x\right)=L_n^{(\alpha +1)}\left(x\right),$$

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.18(viii)

Permalink:

http://dlmf.nist.gov/18.18.E37

Encodings:

pMML, png, TeX

See also:

Annotations for §18.18(viii), §18.18(viii), §18.18 and Ch.18

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18.18.38 $$\sum _{\mathrm{\ell }=0}^n{L}_{\mathrm{\ell }}^{(\alpha )}\left(x\right)L_{n-\mathrm{\ell }}^{(\beta )}\left(y\right)=L_n^{(\alpha +\beta +1)}\left(x+y\right).$$

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $y$: real variable, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.12.6

Referenced by:

§18.18(viii)

Permalink:

http://dlmf.nist.gov/18.18.E38

Encodings:

pMML, png, TeX

See also:

Annotations for §18.18(viii), §18.18(viii), §18.18 and Ch.18

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10: 18.17 Integrals

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Laguerre

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Laguerre

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Laguerre

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Laguerre

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Laguerre

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