I am trying to prove that the Laguerre polynomial $$L=L_n^{\alpha}(x)=\frac{x^{-\alpha}e^{-x}}{n!}\frac{d^n}{dx^n}(e^{-x}x^{n+\alpha})= \frac{(\alpha+1)_n}{n!}\sum_{k=0}^{n}\frac{(-n)_kx^k}{(\alpha+1)_kk!}$$ is a solution to the differential equation $$xf''(x)+(\alpha+1-x)f'(x)+nf(x)=0$$ , where $ (a)_n=a(a+1)\cdots (a+n-1)$ is the Pochhammer symbol.
I did not learned any theory of DE before, so my only strategy is to directly put the polynomial into the DE and hope it indeed equals zero, but the calculation gets extremely messy and yet I still don't get what I want. I have tried both the differential definition and the series definition but both ways are not promising, can anyone please gives me a solution or online reference for the proof?
I think there are some identities which could simplify the calculation, namely $$L_n^{\alpha}(x)'=x^{-1}(nL_n^{\alpha}(x-(n+k)L_{n-1}^{\alpha}(x)).$$ and $$L_n^{\alpha}(x)=L_n^{\alpha+1}(x)-L_{n-1}^{\alpha+1}(x).$$
Thank you so much!