Laguerre polynomials

The Laguerre polynomials are are a sequence of polynomials indexed by a nonnegative integer nn given by

Ln(x)=exn!dndxn(xne−x)L_{n}(x)=\frac{e^{x}}{n!} \frac{d^{n}}{dx^{n}}(x^{n}e^{-x})

The n=0n=0 case is trivial and simply resolves to L0=1L_{0}=1. These polynomials arise as the nontrivial solutions to Laguerre's differential equation for nonnegative nn. More generally, the Laguerre functions are the nontrivial solutions for any real nn, not just integers. An alternate definition uses the generating function

U(t,x)=∑n=0∞tnLn(x)=11−te−tx/(1−t)U(t,x)=\sum_{n=0}^{\infty} t^{n}L_{n}(x)=\frac{1}{1-t}e^{-tx/(1-t)}

The associated Laguerre polynomials are the nontrivial solution to the generalized Laguerre differential equation and are given by

Ln(α)=x−αexn!dndxn(xn+αe−x)L_{n}^{(\alpha)}=\frac{x^{-\alpha}e^{ x }}{n!} \frac{d^{n}}{dx^{n}}(x^{n+\alpha}e^{-x})

or alternatively the generating function

U(t,x;α)=∑n=0∞tnLn(α)=1(1−t)1+αe−tx/(1−t)U(t,x;\alpha)=\sum_{n=0}^{\infty} t^{n}L_{n}^{(\alpha)}=\frac{1}{(1-t)^{1+\alpha}}e^{-tx/(1-t)}

or even as derivatives of Ln(x)L_{n}(x) if α∈N\alpha \in \mathbb{N}

Ln(α)(x)=dαdxαLn(x)L_{n}^{(\alpha)}(x)=\frac{d^{\alpha}}{dx^{\alpha}}L_{n}(x)

In physics it is common to use an alternative definition of the Laguerre polynomials (both regular and associated) that eschews the n!n! factor:

Ln(x)=exdndxn(xne−x),Ln(α)=x−αexdndxn(xn+αe−x)L_{n}(x)=e^{ x } \frac{d^{n}}{dx^{n}}(x^{n}e^{-x}),\quad L_{n}^{(\alpha)}=x^{-\alpha}e^{ x } \frac{d^{n}}{dx^{n}}(x^{n+\alpha}e^{-x})

They are related to the Hermite polynomials Hn(x)H_{n}(x) by

H2n(x)=(−1)n2nn! Ln(−1/2)(x2)H2n+1(x)=(−1)n2n+1n! xLn(1/2)(x2)\begin{align} H_{2n}(x)&=(-1)^{n}2^{n}n!\ L_{n}^{(-1/2)}(x^{2}) \\ H_{2n+1}(x)&=(-1)^{n}2^{n+1}n!\ xL_{n}^{(1/2)}(x^{2}) \end{align}