Associated Laguerre L

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Let $\lambda \in \mathbb{R}$. The associated Laguerre polynomials, $L_n^{(\lambda)}(x)$ are solutions of the differential equation $$x\dfrac{d^2y}{dx^2} + (\lambda+1-x)\dfrac{dy}{dx} + ny=0.$$

The first few Laguerre polynomials are given by $$\begin{array}{ll} L_0^{(\lambda)}(x) &= 1 \\ L_1^{(\lambda)}(x) &= -x+\lambda+1 \\ L_2^{(\lambda)}(x) &= \dfrac{x^2}{2} -(\lambda+2)x+\dfrac{(\lambda+2)(\lambda+1)}{2} \\ L_3^{(\lambda)}(x) &= -\dfrac{x^3}{6} + \dfrac{(\lambda+3)x^2}{2} - \dfrac{(\lambda+2)(\lambda+3)x}{2} + \dfrac{(\lambda+1)(\lambda+2)(\lambda+3)}{6} \\ \vdots \end{array}$$

Properties[edit]

Orthogonal polynomials