Laguerre polynomials

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11: 18.9 Recurrence Relations and Derivatives

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

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$p_n(x)$	$a_n$	$b_n$	$c_n$
…
$L_n^{(\alpha )}\left(x\right)$	$-n-1$	$2n+\alpha +1$	$-n-\alpha$
…

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

ⓘ

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), §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

… ►

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 of $f$ with respect to $x$ , $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.17(i), §18.39(ii), §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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12: 8.7 Series Expansions

§8.7 Series Expansions

►For the functions $e_n(z)$, ${𝗂}_n^{(1)}\left(z\right)$, and $L_n^{(\alpha )}\left(x\right)$ see (8.4.11), §§10.47(ii), and 18.3, respectively. … ►

8.7.6 $$\mathrm{\Gamma }(a,x)=x^a{\mathrm{e}}^{-x}\sum _{n=0}^{\mathrm{\infty }}\frac{L_n^{(a)}\left(x\right)}{n+1},$$ $x>0$, .

ⓘ

Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\mathrm{\Re }$: real part, $x$: real variable, $z$: complex variable, $a$: parameter and $n$: nonnegative integer

Proof sketch:

Integrate (18.12.13) from $z=0$ to $z=1$ and for the integral on the left-hand side use the substitution $z=\frac{t}{1+t}$. The resulting integral is (8.6.5). The constraint follows from (18.15.14).

Referenced by:

Erratum (V1.1.8) for Equation (8.7.6)

Permalink:

http://dlmf.nist.gov/8.7.E6

Encodings:

pMML, png, TeX

Addition (effective with 1.1.8):

The constraint was updated to include .

Suggested 2022-10-14 by Walter Gautschi

See also:

Annotations for §8.7 and Ch.8

…

13: 18.5 Explicit Representations

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Table 18.5.1: Classical OP’s: Rodrigues formulas (18.5.5).

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$p_n(x)$	$w(x)$	$F(x)$	${\kappa }_n$
…
$L_n^{(\alpha )}\left(x\right)$	${\mathrm{e}}^{-x}{x}^{\alpha }$	$x$	$n!$
…

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… ►

$L_0\left(x\right)=1,$

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$L_1\left(x\right)=-x+1,$

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$L_0^{(\alpha )}\left(x\right)=1,$

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$L_1^{(\alpha )}\left(x\right)=-x+\alpha +1,$

…

14: 18.7 Interrelations and Limit Relations

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18.7.19 $$H_{2n}\left(x\right)={(-1)}^n{2}^{2n}n!L_n^{(-\frac{1}{2})}\left(x^2\right),$$

ⓘ

Symbols:

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

A&S Ref:

22.5.40

Referenced by:

§18.16(v), §18.18(iii), §18.18(iv), §18.18(viii), §18.7(ii)

Permalink:

http://dlmf.nist.gov/18.7.E19

Encodings:

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See also:

Annotations for §18.7(ii), §18.7 and Ch.18

►

18.7.20 $$H_{2n+1}\left(x\right)={(-1)}^n{2}^{2n+1}n!x{L}_n^{(\frac{1}{2})}\left(x^2\right).$$

ⓘ

Symbols:

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

A&S Ref:

22.5.41

Referenced by:

§18.16(v), §18.18(iii), §18.18(iv), §18.7(ii), §18.7(ii)

Permalink:

http://dlmf.nist.gov/18.7.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §18.7(ii), §18.7 and Ch.18

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18.7.21 $$\underset{\beta \to \mathrm{\infty }}{lim}{P}_n^{(\alpha ,\beta )}\left(1-\left(2x/\beta \right)\right)=L_n^{(\alpha )}\left(x\right).$$

ⓘ

Symbols:

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

A&S Ref:

22.15.5

Referenced by:

§18.10(ii), §18.7(iii)

Permalink:

http://dlmf.nist.gov/18.7.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §18.7(iii), §18.7(iii), §18.7 and Ch.18

►

18.7.22 $$\underset{\alpha \to \mathrm{\infty }}{lim}{P}_n^{(\alpha ,\beta )}\left((2x/\alpha )-1\right)={(-1)}^n{L}_n^{(\beta )}\left(x\right).$$

ⓘ

Symbols:

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

Referenced by:

§18.7(iii)

Permalink:

http://dlmf.nist.gov/18.7.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §18.7(iii), §18.7(iii), §18.7 and Ch.18

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18.7.26 $$\underset{\alpha \to \mathrm{\infty }}{lim}{\left(\frac{2}{\alpha }\right)}^{\frac{1}{2}n}{L}_n^{(\alpha )}\left({(2\alpha )}^{\frac{1}{2}}x+\alpha \right)=\frac{{(-1)}^n}{n!}{H}_n\left(x\right).$$

ⓘ

Symbols:

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

Referenced by:

§18.7(iii)

Permalink:

http://dlmf.nist.gov/18.7.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §18.7(iii), §18.7(iii), §18.7 and Ch.18

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15: 18.21 Hahn Class: Interrelations

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Meixner $\to$ Laguerre

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18.21.8 $$\underset{c\to 1}{lim}{M}_n({(1-c)}^{-1}x;\alpha +1,c)=\frac{L_n^{(\alpha )}\left(x\right)}{L_n^{(\alpha )}\left(0\right)}.$$

ⓘ

Symbols:

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

Referenced by:

§18.21(ii)

Permalink:

http://dlmf.nist.gov/18.21.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §18.21(ii), §18.21(ii), §18.21 and Ch.18

… ►

Meixner–Pollaczek $\to$ Laguerre

►

18.21.12 $$\underset{\varphi \to 0}{lim}{P}_n^{(\frac{1}{2}\alpha +\frac{1}{2})}(-{(2\varphi )}^{-1}x;\varphi )=L_n^{(\alpha )}\left(x\right).$$

ⓘ

Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $P_n^{(\lambda )}(x;\varphi )$: Meixner–Pollaczek polynomial, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.21(ii), §18.24, §18.30(v)

Permalink:

http://dlmf.nist.gov/18.21.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §18.21(ii), §18.21(ii), §18.21 and Ch.18

… ►

►►

Figure 18.21.1: Askey scheme. … Magnify

16: 18.10 Integral Representations

… ►

Laguerre

►

18.10.6 $$L_n^{(\alpha )}\left(x^2\right)=\frac{2{(-1)}^n}{{\pi }^{\frac{1}{2}}\mathrm{\Gamma }\left(\alpha +\frac{1}{2}\right)n!}{\int }_0^{\mathrm{\infty }}{\int }_0^{\pi }{(x^2-r^2+2ixr\cos \varphi )}^n{\mathrm{e}}^{-r^2}{r}^{2\alpha +1}{(\sin \varphi )}^{2\alpha }d\varphi dr,$$ $\alpha >-\frac{1}{2}$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $\sin z$: sine function, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.10(ii)

Permalink:

http://dlmf.nist.gov/18.10.E6

Encodings:

pMML, png, TeX

See also:

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

… ►

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
…
$L_n^{(\alpha )}\left(x\right)$	${\mathrm{e}}^x{x}^{-\alpha }$	$z{(z-x)}^{-1}$	$z^{\alpha }{\mathrm{e}}^{-z}$	$x$	$0$ outside $C$.
…

►

… ►

Laguerre

►

18.10.9 $$L_n^{(\alpha )}\left(x\right)=\frac{{\mathrm{e}}^x{x}^{-\frac{1}{2}\alpha }}{n!}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{n+\frac{1}{2}\alpha }{J}_{\alpha }\left(2\sqrt{xt}\right)dt,$$ $\alpha >-1$.

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $t$: real variable, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.10.14

Referenced by:

§18.10(iv)

Permalink:

http://dlmf.nist.gov/18.10.E9

Encodings:

pMML, png, TeX

See also:

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

…

17: 18.11 Relations to Other Functions

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Laguerre

►

18.11.2 $$L_n^{(\alpha )}\left(x\right)=\frac{{\left(\alpha +1\right)}_n}{n!}M(-n,\alpha +1,x)=\frac{{(-1)}^n}{n!}U(-n,\alpha +1,x)=\frac{{\left(\alpha +1\right)}_n}{n!}{x}^{-\frac{1}{2}(\alpha +1)}{\mathrm{e}}^{\frac{1}{2}x}{M}_{n+\frac{1}{2}(\alpha +1),\frac{1}{2}\alpha }\left(x\right)=\frac{{(-1)}^n}{n!}{x}^{-\frac{1}{2}(\alpha +1)}{\mathrm{e}}^{\frac{1}{2}x}{W}_{n+\frac{1}{2}(\alpha +1),\frac{1}{2}\alpha }\left(x\right).$$

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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), $M_{\kappa ,\mu }\left(z\right)$: 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 $x$: real variable

Referenced by:

§13.6(v), §18.11(i)

Permalink:

http://dlmf.nist.gov/18.11.E2

Encodings:

pMML, png, TeX

See also:

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

… ►

Laguerre

►

18.11.6 $$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n^{\alpha }}{L}_n^{(\alpha )}\left(\frac{z}{n}\right)=\frac{1}{z^{\frac{1}{2}\alpha }}{J}_{\alpha }\left(2z^{\frac{1}{2}}\right).$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

22.15.2

Referenced by:

§18.11(ii)

Permalink:

http://dlmf.nist.gov/18.11.E6

Encodings:

pMML, png, TeX

See also:

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

…

18: 18.34 Bessel Polynomials

… ►For the confluent hypergeometric function ${}_1{}^{}F_1^{}$ and the generalized hypergeometric function ${}_2{}^{}F_0^{}$, the Laguerre polynomial $L_n^{(\alpha )}$ and the Whittaker function $W_{\kappa ,\mu }$ see §16.2(ii), §16.2(iv), (18.5.12), and (13.14.3), respectively. ►

18.34.1 $$y_n(x;a)={}_2{}^{}F_0^{}(\genfrac{}{}{0pt}{}{-n,n+a-1}{-};-\frac{x}{2})={\left(n+a-1\right)}_n{\left(\frac{x}{2}\right)}^n{}_1{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-n}{-2n-a+2};\frac{2}{x})=n!{\left(-\frac{1}{2}x\right)}^n{L}_n^{(1-a-2n)}\left(2x^{-1}\right)={\left(\frac{1}{2}x\right)}^{1-\frac{1}{2}a}{\mathrm{e}}^{1/x}{W}_{1-\frac{1}{2}a,\frac{1}{2}(a-1)+n}\left(2x^{-1}\right).$$

ⓘ

Defines:

$y_n(x;a)$: Bessel polynomial

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, ${}_1{}^{}F_1^{}(a;b;z)$: $=M(a,b,z)$ notation for the 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), $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: nonnegative integer and $x$: real variable

Proved:

Grosswald (1978, p.38, Theorem 1); Combine (18.5.12) and items (i), (v), (vii) of the mentioned theorem for $b=2$(proved)

Referenced by:

(18.34.2), (18.34.7_1), §18.34(i), §18.34(i), Erratum (V1.2.0) for Equation (18.34.1)

Permalink:

http://dlmf.nist.gov/18.34.E1

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was updated to include the definition of Bessel polynomials in terms of Laguerrepolynomials and the Whittaker confluent hypergeometric function.

See also:

Annotations for §18.34(i), §18.34 and Ch.18

… ►

18.34.7_1 $${\varphi }_n(x;\lambda )={\mathrm{e}}^{-\lambda {\mathrm{e}}^{-x}}{\left(2\lambda {\mathrm{e}}^{-x}\right)}^{\lambda -\frac{1}{2}}{y}_n({\lambda }^{-1}{\mathrm{e}}^x;2-2\lambda )/n!={(-1)}^n{\mathrm{e}}^{-\lambda {\mathrm{e}}^{-x}}{\left(2\lambda {\mathrm{e}}^{-x}\right)}^{\lambda -n-\frac{1}{2}}{L}_n^{(2\lambda -2n-1)}\left(2\lambda {\mathrm{e}}^{-x}\right)={(2\lambda )}^{-\frac{1}{2}}{\mathrm{e}}^{x/2}{W}_{\lambda ,n+\frac{1}{2}-\lambda }\left(2\lambda {\mathrm{e}}^{-x}\right)/n!,$$ $n=0,1,\mathrm{\dots },N=\lceil \lambda -\frac{3}{2}\rceil$, $\lambda >\frac{1}{2}$,

ⓘ

Symbols:

$y_n(x;a)$: Bessel polynomial, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\lceil x\rceil$: ceiling of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $N$: positive integer, $n$: nonnegative integer and $x$: real variable

Proof sketch:

The second and third equality follow from (18.34.1).

Referenced by:

(18.34.7_2), (18.34.7_3), §18.34(iii), (18.39.17), (18.39.18), §18.39(i), Erratum (V1.2.0) Section 18.34

Permalink:

http://dlmf.nist.gov/18.34.E7_1

Encodings:

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Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.34(iii), §18.34 and Ch.18

►expressed in terms of Romanovski–Bessel polynomials, Laguerre polynomials or Whittaker functions, we have …

19: 18.27 $q$-Hahn Class

… ►

Little $q$-Laguerre polynomials

… ►

18.27.14_6 $$\underset{q\to 1}{lim}{p}_n((1-q)x;q^{\alpha },0;q)=\frac{n!}{{\left(\alpha +1\right)}_n}{L}_n^{(\alpha )}\left(x\right).$$

ⓘ

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 $n!$), $p_n(x;a,b;q)$: little $q$-Jacobi polynomial, $q$: real variable, $n$: nonnegative integer and $x$: real variable

Source:

Koekoek et al. (2010, (14.12.12), (14.20.12))

Referenced by:

§18.27(iv), §18.27(iv), Erratum (V1.2.0) Section 18.27

Permalink:

http://dlmf.nist.gov/18.27.E14_6

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.27(iv), §18.27(iv), §18.27 and Ch.18

… ►

18.27.16 $${\int }_0^{\mathrm{\infty }}{L}_n^{(\alpha )}(x;q)L_m^{(\alpha )}(x;q)\frac{x^{\alpha }}{{(-x;q)}_{\mathrm{\infty }}}dx=\frac{{(q^{\alpha +1};q)}_n}{{(q;q)}_n{q}^n}{h}_0^{(1)}{\delta }_{n,m},$$ $\alpha >-1$,

ⓘ

Symbols:

${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $\int$: integral, $L_n^{(\alpha )}(x;q)$: $q$-Laguerre polynomial, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: real variable, $m$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $h_n$

Permalink:

http://dlmf.nist.gov/18.27.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §18.27(v), §18.27 and Ch.18

… ►

18.27.17 $$\sum _{y=-\mathrm{\infty }}^{\mathrm{\infty }}{L}_n^{(\alpha )}(c{q}^y;q)L_m^{(\alpha )}(c{q}^y;q)\frac{q^{y(\alpha +1)}}{{(-c{q}^y;q)}_{\mathrm{\infty }}}=\frac{{(q^{\alpha +1};q)}_n}{{(q;q)}_n{q}^n}{h}_0^{(2)}{\delta }_{n,m},$$ $\alpha >-1$, $c>0$,

ⓘ

Symbols:

${\delta }_{j,k}$: Kronecker delta, $L_n^{(\alpha )}(x;q)$: $q$-Laguerre polynomial, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $y$: real variable, $q$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $h_n$

Permalink:

http://dlmf.nist.gov/18.27.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §18.27(v), §18.27 and Ch.18

… ►

18.27.17_3 $$\underset{q\to 1}{lim}{L}_n^{(\alpha )}((1-q)x;q)=L_n^{(\alpha )}\left(x\right).$$

ⓘ

Symbols:

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

Source:

Koekoek et al. (2010, (14.21.18))

Referenced by:

§18.27(v), Erratum (V1.2.0) Section 18.27

Permalink:

http://dlmf.nist.gov/18.27.E17_3

Encodings:

pMML, png, TeX

Modification (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.27(v), §18.27(v), §18.27 and Ch.18

…

20: 18.39 Applications in the Physical Sciences

… ► $p$ here being the order of the Laguerre polynomial, $L_p^{(2l+1)}$ of Table 18.8.1, line 11, and $l$ the angular momentum quantum number, and where … ►

18.39.34 $${\psi }_{n,l}(r)=\frac{1}{n}\sqrt{\frac{Z(n-l-1)!}{(n+l)!}}{\mathrm{e}}^{-{\rho }_n/2}{\rho }_n^{l+1}{L}_{n-l-1}^{(2l+1)}\left({\rho }_n\right),$$ $n=1,2,\mathrm{\dots },l=0,1,\mathrm{\dots }n-1$,

ⓘ

Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $l$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/18.39.E34

Encodings:

pMML, png, TeX

See also:

Annotations for §18.39(ii), §18.39(ii), §18.39 and Ch.18

… ►

18.39.40 $${𝐋}_{p+m}^m(\rho )={(-1)}^m(p+m)!L_p^{(m)}\left(\rho \right),$$

ⓘ

Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $!$: factorial (as in $n!$) and $m$: nonnegative integer

Permalink:

http://dlmf.nist.gov/18.39.E40

Encodings:

pMML, png, TeX

See also:

Annotations for §18.39(ii), §18.39(ii), §18.39 and Ch.18

… ►

18.39.41 $$L_{n-l-1}^{(2l+1)}\left({\rho }_n\right)=-{𝐋}_{n+l}^{2l+1}({\rho }_n)/(n+l)!,$$

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $!$: factorial (as in $n!$), $l$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/18.39.E41

Encodings:

pMML, png, TeX

See also:

Annotations for §18.39(ii), §18.39(ii), §18.39 and Ch.18

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18.39.44 $${\varphi }_{n,l}(sr)={(sr)}^{l+1}{\mathrm{e}}^{-sr/2}{L}_n^{(2l+1)}\left(sr\right),$$ $n=0,1,2,\mathrm{\dots }$,

ⓘ

Symbols:

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

Referenced by:

§18.39(iv), §18.39(iv), §18.39(iv)

Permalink:

http://dlmf.nist.gov/18.39.E44

Encodings:

pMML, png, TeX

See also:

Annotations for §18.39(iv), §18.39 and Ch.18

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