harmonic analysis – Laguerre Polynomials formula

Please. I would like to get rid of the denominator in the following quotient:
$$frac{L_{k}^{(alpha)}(y)}{ L_{n}^{(alpha)}(y)}$$
For $k=0, dots, n-1$. Is there any induction formula for this type of quotient? If no can we compute it for the cases $y=1,2$. We already have the Rodrigues formula:
$$L_{n}^{(alpha)}(y)=frac{y^{-alpha} e^{x}}{n!} frac{d^{n}}{d y^{n}}(e^{-x}y^{n+ alpha})$$
and also the following sum
$$L_{n}^{(alpha)}(y)= sum_{k=0}^{n} begin{pmatrix} n+ alpha \ n-k end{pmatrix} (-1)^{k} frac{x^{k}}{k!}$$
Please can someone help. I am so confused.

Thank you very much and respectful regards for everyone.