limit of Laguerre polynomials

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1: 18.6 Symmetry, Special Values, and Limits to Monomials

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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

2: 18.11 Relations to Other Functions

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Laguerre

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

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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

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3: 37.7 Parabolic Biangular Region with Weight Function ${\left(1-x\right)}^{\alpha }{\left(x-y^2\right)}^{\beta }$

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37.7.20 $$2^{k}k!\underset{\beta \to \mathrm{\infty }}{lim}{\beta }^{-\frac{1}{2}k}{P}_{k,n}^{\alpha ,\beta }(1-{\beta }^{-1}x,{\beta }^{-\frac{1}{2}}y)=L_{n-k}^{(\alpha )}\left(x\right)H_k\left(y\right),$$ $0\le k\le n$,

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Symbols:

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

Proof sketch:

Use (37.7.3), (18.7.21) and (18.7.23).

Referenced by:

§37.7(v), §37.7(v)

Permalink:

http://dlmf.nist.gov/37.7.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §37.7(v), §37.7, Part 37.II and Ch.37

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37.7.21 $${(-1)}^{n-k}\underset{\beta \to \mathrm{\infty }}{lim}{\beta }^{\frac{1}{2}k}{R}_{k,n}^{\alpha ,\beta }(1-{\beta }^{-1}x,{\beta }^{-\frac{1}{2}}y)=L_{n-k}^{(\alpha +k+\frac{1}{2})}\left(x+y^2\right){(x+y^2)}^{\frac{1}{2}k}{P}_k^{(\alpha ,\alpha )}\left(\frac{y}{\sqrt{x+y^2}}\right),$$ $0\le k\le n$.

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Symbols:

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

Proof sketch:

Use (37.7.16) and (18.7.22).

Referenced by:

§37.7(v), §37.7(v)

Permalink:

http://dlmf.nist.gov/37.7.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §37.7(v), §37.7, Part 37.II and Ch.37

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37.7.24 $${(-1)}^{n-k}\underset{\alpha \to \mathrm{\infty }}{lim}{\alpha }^{\frac{1}{2}k}{P}_{k,n}^{\alpha ,\beta }({\alpha }^{-1}x,{\alpha }^{-\frac{1}{2}}y)=L_{n-k}^{(\beta +k+\frac{1}{2})}\left(x\right)x^{\frac{1}{2}k}{P}_k^{(\beta ,\beta )}\left(x^{-\frac{1}{2}}y\right),$$ $0\le k\le n$,

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Symbols:

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

Proof sketch:

Use (37.7.3) and (18.7.22).

Referenced by:

§37.7(vi), §37.7(vi)

Permalink:

http://dlmf.nist.gov/37.7.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §37.7(vi), §37.7, Part 37.II and Ch.37

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37.7.25 $$2^{k}k!\underset{\alpha \to \mathrm{\infty }}{lim}{\alpha }^{-\frac{1}{2}k}{R}_{k,n}^{\alpha ,\beta }({\alpha }^{-1}x,{\alpha }^{-\frac{1}{2}}y)=L_{n-k}^{(\beta )}\left(x-y^2\right)H_k\left(y\right),$$ $0\le k\le n$.

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Symbols:

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

Proof sketch:

Use (37.7.16), (18.7.21) and (18.7.23).

Referenced by:

§37.7(vi), §37.7(vi)

Permalink:

http://dlmf.nist.gov/37.7.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §37.7(vi), §37.7, Part 37.II and Ch.37

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4: 18.7 Interrelations and Limit Relations

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

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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

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

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

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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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5: Bibliography C

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Asymptotic behaviour of the zeros of the (generalized) Laguerre polynomial $L_n^{\alpha }(x)$ as the index $\alpha \to \mathrm{\infty }$ and limiting formula relating Laguerre polynomials of large index and large argument to Hermite polynomials. Lett. Nuovo Cimento (2) 23 (3), pp. 101–102.

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External Links:

ISSN 0024-1318, Document, MathReview (R. A. Askey)

Referenced by:

§18.7(iii).

Encodings:

BibTeX

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6: 18.27 $q$-Hahn Class

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From Little $q$-Laguerre to Laguerre

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

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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 $n!$), $p_n(x;a,b;q)$: 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) §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

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From $q$-Laguerre to Laguerre

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

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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) §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

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7: null

error generating summary

8: 37.18 Orthogonal Polynomials on Quadratic Domains

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37.18.14 $${𝐑}_n((𝐱,t),(𝐲,s))=\frac{2^{\mu +\frac{d-5}{2}}\mathrm{\Gamma }(\mu +\frac{1}{2})\mathrm{\Gamma }(\mu +\frac{d}{2})}{\pi \mathrm{\Gamma }(\mu )}{\int }_{-1}^1{\int }_0^{\pi }{L}_n^{(2\mu +d-1)}\left(t+s+2\rho \cos \theta \right){\mathrm{e}}^{-\rho \cos \theta }{\left(\rho \sin \theta \right)}^{-\frac{1}{2}(\mu +d-3)}{J}_{\frac{\mu +d-3}{2}}(\rho \sin \theta ){(\sin \theta )}^{2\mu +d-2}{\left(1-u^2\right)}^{\mu -1}d\theta du,$$ $\rho =\sqrt{\frac{1}{2}(ts+⟨𝐱,𝐲⟩+\sqrt{t^2-{\Vert 𝐱\Vert }^2}\sqrt{t^2-{\Vert 𝐱\Vert }^2}u)}$,

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $⟨𝐮,𝐯⟩$: inner product over vectors, $\int$: integral, $(a,b)$: open interval, $\sin z$: sine function, $\Vert 𝐯\Vert$: vector norm ($l$²), $j$: nonnegative integer, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space, $z$: complex number and ${𝐑}_n$: reproducing kernel

Proved:

Xu (2021b, Theorem 5.4); Take there in (5.5) with ${\mathrm{\Phi }}_{\kappa }$ given by (2.15) the limit case for ${\kappa }_1=\mathrm{\cdots }={\kappa }_d=0$ and use the formula for $j_{\alpha }(z)$ on p.4 in that paper.(proved)

Referenced by:

§37.18(iii)

Permalink:

http://dlmf.nist.gov/37.18.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §37.18(iii), §37.18(iii), §37.18, Part 37.III and Ch.37

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9: 37.12 Orthogonal Polynomials on Quadratic Surfaces

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37.12.13 $${𝐑}_n(w_{-1};(𝐱,t),(𝐲,s))=\frac{1}{\sqrt{\pi }}{2}^{\frac{d-4}{2}}\mathrm{\Gamma }(\frac{d-1}{2}){\int }_0^{\pi }{L}_n^{(d-2)}\left(t+s+2\rho \cos \theta \right){\mathrm{e}}^{-\rho \cos \theta }{(\rho \sin \theta )}^{-\frac{1}{2}(d-4)}{J}_{\frac{d-4}{2}}(\rho \sin \theta ){(\sin \theta )}^{d-3}d\theta ,$$ $\rho =\sqrt{\frac{ts+⟨𝐱,𝐲⟩}{2}}$, $d>2$,

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $⟨𝐮,𝐯⟩$: inner product over vectors, $\int$: integral, $(a,b)$: open interval, $\sin z$: sine function, $j$: nonnegative integer, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space, $z$: complex number and ${𝐑}_n$: reproducing kernel

Proved:

Xu (2021b, Theorem 3.7); Take in (3.11) there the limit case for ${\kappa }_1=\mathrm{\cdots }={\kappa }_d=0$ and use (2.15) and the formula on p.4 for $j_{\alpha }(z)$ in that paper.(proved)

Permalink:

http://dlmf.nist.gov/37.12.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §37.12(iii), §37.12(iii), §37.12, Part 37.III and Ch.37

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10: 18.30 Associated OP’s

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18.30.19 $$L_n^{\lambda }(x;c)=\underset{\varphi \to 0}{lim}{𝒫}_n^{(\lambda +1)/2}(\frac{-x}{2\sin \varphi };\varphi ,c),$$

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Symbols:

$L_n^{\lambda }(x;c)$: associated Laguerre polynomial, ${𝒫}_n^{\lambda }(x;\varphi ,c)$: associated Meixner–Pollaczek polynomial, $\sin z$: sine function, $n$: nonnegative integer and $x$: real variable

Proof sketch:

Using the limit will convert (18.30.16) into (18.30.9). See also Askey and Wimp (1984, §2).

Permalink:

http://dlmf.nist.gov/18.30.E19

Encodings:

pMML, png, TeX

See also:

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

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