Associated Laguerre L

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

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

Properties

Theorem: The following formula holds: $$L_n^{(\alpha)}(x) = \displaystyle\sum_{k=0}^n (-1)^k {n+\alpha \choose n-k} \dfrac{x^k}{k!}.$$

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Theorem: The following formula holds: $$L_n^{(\alpha+\beta+1)}(x+y) = \displaystyle\sum_{k=0}^n L_k^{(\alpha)}(x)L_{n-k}^{(\beta)}(x).$$

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Theorem: The following formula holds: $$L_n^{(\alpha)}(x) = L_n^{(\alpha+1)}(x)-L_{n-1}^{(\alpha+1)}(x).$$

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Theorem: The following formula holds: $$nL_n^{(\alpha+1)}(x) = (n-x)L_{n-1}^{(\alpha)}(x)-(n+\alpha)L_{n-1}^{(\alpha)}(x).$$

Proof: █

Theorem: The following formula holds: $$xL_n^{(\alpha+1)}(x)= (n+\alpha)L_{n-1}^{(\alpha)}(x)-(n-x)L_n^{(\alpha)}(x).$$

Proof: █