Associated Laguerre polynomial

The associated Laguerre polynomials (also called generalized Laguerre polynomials) $𝐿 𝑝 𝑞$ are polynomials of degree $𝑞$ defined (by one convention¹) for $𝑝 \geq 0$ as #m/def/fun

𝐿 𝑝 𝑞 (𝑥) = (- 1) 𝑝 (𝑑 𝑑 𝑥) 𝑝 𝐿 𝑝 + 𝑞 (𝑥)

where $𝐿 𝑝 + 𝑞$ is a Laguerre polynomial, or equivalently,

𝐿 𝑝 𝑞 (𝑥) = 𝑥 - 𝑝 𝑒 𝑥 𝑞! (𝑑 𝑑 𝑥) 𝑞 (𝑒 - 𝑥 𝑥 𝑝 + 𝑞)

Mathematica

The associated Laguerre polynomial $𝐿 𝑝 𝑞 (𝑥)$ may be generated in Wolfram Mathematica with LaguerreL[p,q,x].

Properties

Clearly $𝐿 0 𝑞 = 𝐿 𝑞$ .

#state/develop | #lang/en | #SemBr

2018. Introduction to quantum mechanics, §4.2, pp. 149–150 ↩

Associated Laguerre polynomial

Properties

Footnotes