LaguerreL

The Laguerre polynomial $L_n(x)$, and its generalization $L_n^{(a)}(x)$ with a parameter $a$, orthogonal on $(0, \infty)$ with weight $x^a e^{-x}$. Provided by @enumeratio/analytic.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaLaguerre polynomials↗MathWorldLaguerrePolynomial↗DLMF18.3↗18.1.I1.ix7.p1↗Wolfram LanguageLaguerreL✓↗

LaguerreL(n, x)the Laguerre polynomial $L_n(x)$ at $x$, for an integer $n \ge 0$.

LaguerreL(n, a, x)the generalized Laguerre polynomial $L_n^{(a)}(x)$ at $x$, for an integer $n \ge 0$ and any $a$.

Typed signatures

Type	Package	Applies
(number, number, number?) -> number	analytic	always

Domain: Special functions

Details

• $L_0^{(a)} = 1$, $L_1^{(a)}(x) = 1 + a - x$, and $(k+1)\,L_{k+1}^{(a)}(x) = (2k+1+a-x)\,L_k^{(a)}(x) - (k+a)\,L_{k-1}^{(a)}(x)$. The two-argument $L_n(x)$ is $L_n^{(0)}(x)$.
• $L_n^{(a)}(x) = \sum_{k=0}^{n} (-1)^k \binom{n+a}{n-k} \dfrac{x^k}{k!}$, a polynomial in $x$ of degree $n$ whose leading coefficient is $(-1)^n / n!$.
• $\dfrac{d}{dx} L_n^{(a)}(x) = -L_{n-1}^{(a+1)}(x)$, so D(LaguerreL(n, x), x) is $-L_{n-1}^{(1)}(x)$ at a symbolic order too.
• $L_n(0) = 1$, and $L_n^{(a)}(0) = \binom{n+a}{n}$.
• $\int_0^\infty e^{-x} L_m(x)\,L_n(x)\,dx = \delta_{mn}$.
• At an integer $n$ and a symbolic or exact $x$ the result is the explicit sum, so a symbolic $a$ stays symbolic. At a floating-point or complex $x$ and a numeric $a$ the recurrence runs directly on the number, and the head stays unevaluated if it overflows the double range.
• A symbolic or negative order stays unevaluated, and so does a non-integer one. Wolfram continues $L_\nu^{(a)}$ to those orders through the confluent hypergeometric $_1F_1$, which is not implemented here.

Examples

Basic #

$L_1(x) = 1 - x$

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In$\mathrm{LaguerreL}(1, x)$Out$1-x$

$L_2(x) = \tfrac12 x^2 - 2x + 1$

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In$\mathrm{LaguerreL}(2, x)$Out$\frac{x^2}{2}-2x+1$

$L_3(x) = -\tfrac16 x^3 + \tfrac32 x^2 - 3x + 1$

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In$\mathrm{LaguerreL}(3, x)$Out$-\frac{x^3}{6}+\frac{3x^2}{2}-3x+1$

At an integer the value is exact: $L_3(2) = -\tfrac13$

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In$\mathrm{LaguerreL}(3, 2)$Out$-\frac{1}{3}$

At a floating-point $x$

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In$\mathrm{LaguerreL}(4, 0.7)$Out$-0.548\,662\,5$

$L_1^{(a)}(x) = 1 + a - x$, with a symbolic parameter $a$

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In$\mathrm{LaguerreL}(1, a, x)$Out$a-x+1$

$L_2^{(a)}(x) = \tfrac12(a+1)(a+2) - (a+2)x + \tfrac12 x^2$

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In$\mathrm{LaguerreL}(2, a, x)$Out$\frac{x^2}{2}+\frac{1}{2}((a+1)(a+2))-x×(a+2)$

At a floating-point $a$ and $x$

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In$\mathrm{LaguerreL}(3, 0.5, 1.3)$Out$-0.908\,666\,666\,666\,666\,7$

Scope #

At a Gaussian integer the value is exact: $L_2(2+3i) = -\tfrac{11}{2}$

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In$\mathrm{LaguerreL}(2, 2+3i)$Out$-\frac{11}{2}$

At a complex parameter $a = \tfrac12 + i$

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In$\mathrm{LaguerreL}(2, 0.5+i, 0.7)$Out$-0.13+1.3i$

A symbolic order stays unevaluated

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In$\mathrm{LaguerreL}(n, x)$Out$\mathrm{LaguerreL}(n, x)$

Properties #

$L_n'(x) = -L_{n-1}^{(1)}(x)$

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In$\frac{\mathrm{d}\mathrm{LaguerreL}(n, x)}{\mathrm{d}x}$Out$-\mathrm{LaguerreL}(n-1, 1, x)$

$\partial_x L_n^{(a)}(x) = -L_{n-1}^{(a+1)}(x)$

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In$\frac{\mathrm{d}\mathrm{LaguerreL}(n, a, x)}{\mathrm{d}x}$Out$-\mathrm{LaguerreL}(n-1, a+1, x)$

$L_n(0) = 1$

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In$\mathrm{LaguerreL}(5, 0)$Out$1$

$L_3^{(a)}(0) = \binom{a+3}{3}$

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In$\mathrm{LaguerreL}(3, a, 0)$Out$\frac{1}{6}((a+1)(a+2)(a+3))$

$3\,L_3(x) = (5-x)\,L_2(x) - 2\,L_1(x)$, here at $x = 3$

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In$3\mathrm{LaguerreL}(3, 3)-((5-3)\mathrm{LaguerreL}(2, 3)-2\mathrm{LaguerreL}(1, 3))$Out$0$

test #

A higher order at a floating-point $x$

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In$N(\mathrm{LaguerreL}(10, 1.3))$Out$0.545\,955\,765\,661\,49$

About

Examples	17
Checked against	Wolfram Language 17 agree
Crosswalk	WikipediaMathWorldDLMF ×2Wolfram Language

Implementation

referenceenginenotatio

The explicit sum over $k$ up to $n$, with $a = 0$ for the two-argument form. The kernel runs the three-term recurrence at a number instead, and the two are checked against each other.

nativeenginetypescriptpackages/symbols/analysis/analytic/src/laguerre.ts

mappedexternalwolfram / mpmath / sympypackages/oracle/src/mappings.ts

LaguerreL[n, x] and LaguerreL[n, a, x]; mpmath.laguerre(n, a, x); sympy.assoc_laguerre(n, a, x).

See also: HermiteH, ChebyshevT, LegendrePolynomial, LaplaceTransform