Recurrence relations of the associated Laguerre Polynomials

The recurrence relations of the associated Laguerre polynomials describe how each polynomial in the sequence can be obtained from its predecessors.

Some useful recurrence relations of the associated Laguerre polynomials include

$(n+1)L_{n+1}^m=(2n+m+1-x)L_n^m-(n+m)L_{n-1}^m\;\;\;\;\;\;\;\;450$

$x(L_{n}^m)'=nL_n^m-(n+m)L_{n-1}^m\;\;\;\;\;\;\;\;451$

Eq450 can be proven by differentiating the generating function for the associated Laguerre polynomials (see eq448) with respect to $t$ to give

$\frac{(m+1)(1-t)-x}{(1-t)^{m+3}}e^{-\frac{xt}{1-t}}=\sum_{n=0}^{\infty}nL_n^mt^{n-1}$

Substituting eq448 yields

$[(m+1)(1-t)-x]\sum_{n=0}^{\infty}L_n^mt^{n}=(1-t)^2\sum_{n=0}^{\infty}nL_n^mt^{n-1}$

Equating the coefficients of and rearranging them results in eq450. Eq451 can be proven by replacing $n$ with $n+m$ in the recurrence relations of the Laguerre polynomials (see eq436) to give

$xL_{n+m}'=(n+m)L_{n+m}-(n+m)L_{n+m-1}$

Differentiating this equation $m$ times with respect to $x$ using Leibniz’ theorem and multiplying through by $(-1)^m$ yields

$(-1)^m\sum_{k=0}^m\begin{pmatrix}m\\k\end{pmatrix}\frac{d^{m-k}x}{dx^{m-k}}\frac{d^{k+1}L_{n+m}}{dx^{k+1}}=(n+m)(-1)^m\frac{d^mL_{n+m}}{dx^m}-(n+m)(-1)^m\frac{d^mL_{n+m-1}}{dx^m}$