LaguerreL

Identifier LaguerreL is used in Mathematica to denote the Laguerre polynomial and the associated Laguerre functions.

Definitions
Laguerre polynomial can be defined with

\(\displaystyle \mathrm{LaguerreL}[n,x]= L_n(x)=\sum_{m=0}^n \frac{(-1)^m}{m!} \mathrm{Binomial}(n,m) \, x^m =\sum_{m=0}^n \frac{(-1)^m}{m!} \, \frac{n!}{m! \, (n\!-\!m)!} \, x^m \)

Associated Laguerre function appears as

\(\displaystyle \mathrm{LaguerreL}[n,k,x]= L_n^k(x)=(-1)^k \, \partial_x^k L_{n+k}(x) =\sum_{m=0}^n (-1)^m \frac{ (n\!+\!k)!}{(n\!-\!m)!\, (k\!+\!m)!\, m!} \, x^m \)

For non-negative integer values of the second argument, it is also polynomial. In particular the 0th associated Laguerre is just Laguerre polynomial.

First polynomials
TeXForm[TableForm[Table[Table[LaguerreL[n, m, z], {n, 0, 5}], {m, 0, 8}]]]

gives the following table:

\(\begin{array}{cccccc} 1 & 1-z & \frac{1}{2} \left(z^2-4 z+2\right) &  \frac{1}{6} \left(-z^3+9 z^2-18 z+6\right) &   \frac{1}{24} \left(z^4-16 z^3+72 z^2-96 z+24\right) &   \frac{1}{120} \left(-z^5+25 z^4-200 z^3+600 z^2-600 z+120\right) \\ 1 & 2-z & \frac{1}{2} \left(z^2-6 z+6\right) &  \frac{1}{6} \left(-z^3+12 z^2-36 z+24\right) &   \frac{1}{24} \left(z^4-20 z^3+120 z^2-240 z+120\right)   & \frac{1}{120} \left(-z^5+30 z^4-300 z^3+1200 z^2-1800 z+720\right) \\ 1 & 3-z & \frac{1}{2} \left(z^2-8 z+12\right) &  \frac{1}{6} \left(-z^3+15 z^2-60 z+60\right) &   \frac{1}{24} \left(z^4-24 z^3+180 z^2-480 z+360\right)   & \frac{1}{120} \left(-z^5+35 z^4-420 z^3+2100 z^2-4200 z+2520\right) \\ 1 & 4-z & \frac{1}{2} \left(z^2-10 z+20\right) &  \frac{1}{6} \left(-z^3+18 z^2-90 z+120\right) &   \frac{1}{24} \left(z^4-28 z^3+252 z^2-840 z+840\right)   & \frac{1}{120} \left(-z^5+40 z^4-560 z^3+3360 z^2-8400 z+6720\right) \\ 1 & 5-z & \frac{1}{2} \left(z^2-12 z+30\right) &  \frac{1}{6} \left(-z^3+21 z^2-126 z+210\right) &   \frac{1}{24} \left(z^4-32 z^3+336 z^2-1344 z+1680\right) & \frac{1}{120} \left(-z^5+45 z^4-720 z^3+5040 z^2-15120 z+15120\right) \\ 1 & 6-z & \frac{1}{2} \left(z^2-14 z+42\right) &  \frac{1}{6} \left(-z^3+24 z^2-168 z+336\right) &   \frac{1}{24} \left(z^4-36 z^3+432 z^2-2016 z+3024\right) & \frac{1}{120} \left(-z^5+50 z^4-900 z^3+7200 z^2-25200 z+30240\right) \\ 1 & 7-z & \frac{1}{2} \left(z^2-16 z+56\right) &  \frac{1}{6} \left(-z^3+27 z^2-216 z+504\right) &   \frac{1}{24} \left(z^4-40 z^3+540 z^2-2880 z+5040\right) & \frac{1}{120} \left(-z^5+55 z^4-1100 z^3+9900 z^2-39600 z+55440\right) \\ 1 & 8-z & \frac{1}{2} \left(z^2-18 z+72\right) &  \frac{1}{6} \left(-z^3+30 z^2-270 z+720\right) &   \frac{1}{24} \left(z^4-44 z^3+660 z^2-3960 z+7920\right) & \frac{1}{120} \left(-z^5+60 z^4-1320 z^3+13200 z^2-59400 z+95040\right) \\ 1 & 9-z & \frac{1}{2} \left(z^2-20 z+90\right) &  \frac{1}{6} \left(-z^3+33 z^2-330 z+990\right) &   \frac{1}{24} \left(z^4-48 z^3+792 z^2-5280 z+11880\right) & \frac{1}{120} \left(-z^5+65 z^4-1560 z^3+17160 z^2-85800 z+154440\right) \\ \end{array} \)

Applications
The Laguerre polynomials appear in many applications. In particular, in the following:

The Gauss-Laguerre quadrature formula for the numerical integration of a smooth function (especially if the main trend of the integrand is decaying exponential)

The Hydrogen radial wave function, solution of the stationary Schroedinger for the single particle in the Coulomb potential :

\(\displaystyle \psi_{n,\ell}( r) = \sqrt{\frac{(n\!-\!\ell \!-\!1)! }{(n\!+\!\ell)! } }\exp(-r/n) \left(\frac{2 r}{n}\right)^\ell \frac{2}{n^2} \mathrm{LaguerreL}(n\!-\ell\!-\!1, 2 \ell\! +\!1, 2r/n)\)

P[n_, l_, r_] := Sqrt[ (n-l-1)!/(n+l)!] E^(-(r/n)) ((2 r)/n)^l 2/ n^2 LaguerreL[n-l-1, 2l+1, (2r)/n]

With the definition above the test for orthogonality can be written as follows:

Integrate[P[1, 0, r] P[2, 0, r] r^2, {r, 0, Infinity}]

Orthogonality
For each integer non-negative \(k\), the \(k\)th Laguerre functions also form the orthogonal basis at the integration with the exponential weight:

\(\displaystyle \int_0^\infty \mathrm e^{-x}\, L_{n,k}(x)\, L_{m,k}(x) \, x^k \, \mathrm d x=\frac{(n\!+\!k)!}{n!} \delta_{n,m}\)

where \(\delta_{m,n}\) is the Kronecker delta.

Zeros
For the applications the zeros of LaguerreL are important. Luigi Gatteschi suggests the asymptotics for them; however, no simple estimate useful for the numerical implementation is supplied.

Keywords
Atomic physics, Hydrogen wave function, Laplacian, LegendreP, Quantum mechanics,