Laguerre polynomials

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1: 18.3 Definitions

§18.3 Definitions

►Table 18.3.1 provides the definitions of Jacobi, Laguerre, and Hermite polynomials via orthogonality and normalization (§§18.2(i) and 18.2(iii)). … ►

Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, normalizations, 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 exact values of the coefficients of the Jacobi polynomials $P_n^{(\alpha ,\beta )}\left(x\right)$, the ultraspherical polynomials $C_n^{(\lambda )}\left(x\right)$, the Chebyshev polynomials $T_n\left(x\right)$ and $U_n\left(x\right)$, the Legendre polynomials $P_n\left(x\right)$, the Laguerre polynomials $L_n\left(x\right)$, and the Hermite polynomials $H_n\left(x\right)$, 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: 18.18 Sums

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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:

§18.18(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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Laguerre

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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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4: 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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5: 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). …

6: 18.9 Recurrence Relations and Derivatives

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18.9.13 $$L_n^{(\alpha )}\left(x\right)=L_n^{(\alpha +1)}\left(x\right)-L_{n-1}^{(\alpha +1)}\left(x\right),$$

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Symbols:

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

A&S Ref:

22.7.30

Referenced by:

§18.17(i), §18.9(ii)

Permalink:

http://dlmf.nist.gov/18.9.E13

Encodings:

pMML, png, TeX

See also:

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

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18.9.14 $$x{L}_n^{(\alpha +1)}\left(x\right)=-(n+1)L_{n+1}^{(\alpha )}\left(x\right)+(n+\alpha +1)L_n^{(\alpha )}\left(x\right).$$

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Symbols:

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

A&S Ref:

22.7.31

Referenced by:

§18.9(ii)

Permalink:

http://dlmf.nist.gov/18.9.E14

Encodings:

pMML, png, TeX

See also:

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

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Laguerre

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18.9.23 $$\frac{d}{dx}{L}_n^{(\alpha )}\left(x\right)=-L_{n-1}^{(\alpha +1)}\left(x\right),$$

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Symbols:

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

Referenced by:

§18.17(i), §18.9(iii)

Permalink:

http://dlmf.nist.gov/18.9.E23

Encodings:

pMML, png, TeX

See also:

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

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18.9.24 $$\frac{d}{dx}\left({\mathrm{e}}^{-x}{x}^{\alpha }{L}_n^{(\alpha )}\left(x\right)\right)=(n+1){\mathrm{e}}^{-x}{x}^{\alpha -1}{L}_{n+1}^{(\alpha -1)}\left(x\right).$$

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\frac{df}{dx}$: derivative, $\mathrm{e}$: base of natural logarithm, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.9(iii)

Permalink:

http://dlmf.nist.gov/18.9.E24

Encodings:

pMML, png, TeX

See also:

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

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7: 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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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: 18.1 Notation

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Classical OP’s

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Laguerre: $L_n^{(\alpha )}\left(x\right)$ and $L_n\left(x\right)=L_n^{(0)}\left(x\right)$. ($L_n^{(\alpha )}\left(x\right)$ with $\alpha \ne 0$ is also called Generalized Laguerre.)

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$q$-Laguerre: $L_n^{(\alpha )}(x;q)$.

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10: 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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$L_n^{(\alpha )}\left(x\right)$	$x$	$\alpha +1-x$	$0$	$n$
${\mathrm{e}}^{-\frac{1}{2}{x}^2}{x}^{\alpha +\frac{1}{2}}{L}_n^{(\alpha )}\left(x^2\right)$	$1$	$0$	$-x^2+(\frac{1}{4}-{\alpha }^2)x^{-2}$	$4n+2\alpha +2$
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