Prove a relation of Laguerre polynomials Prove this relation for Laguerre polynomials $L_{n}^{(\alpha)}(x)$:
$$L_{n}^{(\alpha)}(cx)=(\alpha+1)_n\sum_{k=0}^{n}\frac{c^k(1-c)^{n-k}}{(n-k)!(\alpha+1)_k}L_{k}^{(\alpha)}(x).$$
I tried to prove using the generating function of Laguerre polynomials and equating the coefficients of $x^n$ on both sides, but I don't get something that brings this relation.
Have someone any proof or idea?
 A: Using both the closed form representation of the Laguerre polynomials
\begin{equation}
L_n^{(\alpha)}(x)=\sum_{k=0}^n(-1)^k\binom{n+\alpha}{n-k}\frac{x^k}{k!}
\end{equation} 
and the decomposition
\begin{equation}
\frac{x^k}{k!}=\sum_{p=0}^k(-1)^k\binom{k+\alpha}{k-p}L_p^{(\alpha)}(x)
\end{equation} 
we have
\begin{align}
L_n^{(\alpha)}(cx)&=\sum_{k=0}^n(-1)^k\binom{n+\alpha}{n-k}c^k\sum_{p=0}^k(-1)^p\binom{k+\alpha}{k-p}L_p^{(\alpha)}(x)\\
&=\sum_{k=0}^n\sum_{p=0}^k(-1)^{p+k}\binom{n+\alpha}{n-k}\binom{k+\alpha}{k-p}c^kL_p^{(\alpha)}(x)
\end{align}
But
\begin{equation}
\binom{n+\alpha}{n-k}\binom{k+\alpha}{k-p}=\frac{\Gamma(n+\alpha+1)}{(n-k)!(k-p)!\Gamma(\alpha+p+1)}
\end{equation} 
and thus
\begin{align}
L_n^{(\alpha)}(cx)&=\Gamma(n+\alpha+1)\sum_{k=0}^n\sum_{p=0}^k\frac{(-1)^{p+k}}{(n-k)!(k-p)!\Gamma(\alpha+p+1)}c^kL_p^{(\alpha)}(x)\\
&=\Gamma(n+\alpha+1)\sum_{p=0}^n\frac{(-1)^p}{\Gamma(\alpha+p+1)}L_p^{(\alpha)}(x)\sum_{k=p}^n\frac{(-1)^{k}}{(n-k)!(k-p)!}c^k\\
&=\Gamma(n+\alpha+1)\sum_{p=0}^n\frac{(-1)^p}{\Gamma(\alpha+p+1)}L_p^{(\alpha)}(x)\sum_{k'=0}^{n-p}\frac{(-1)^{k'+p}}{(n-p-k')!(k')!}c^{k'+p}\\
&=\Gamma(n+\alpha+1)\sum_{p=0}^n\frac{c^p(1-c)^{n-p}}{\Gamma(\alpha+p+1)(n-p)!}L_p^{(\alpha)}(x)
\end{align}
which, after introduction of the Pochhammer symbols, gives the desired result:
\begin{equation}
L_{n}^{(\alpha)}(cx)=(\alpha+1)_n\sum_{p=0}^{n}\frac{c^p(1-c)^{n-p}}{(n-p)!(\alpha+1)_p}L_{p}^{(\alpha)}(x)
\end{equation} 
