orthogonal-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?
Very roughly speaking: Laguerre polynomials look like the family of trigonometric functions ($\sin nx$ or $\cos nx$) in the region where the weight is concentrated. That is, they have moderate size and slowly increasing oscillation with $n$. Outside of this region, they look pretty much like any random collection of polynomials.
When $\alpha=0$, the weight $e^{-x}$ is concentrated near $0$. On the graph below the weight is shown in black, and the first five Laguerre polynomials $L_n$ are in various colors. You can see that they look like $1-\sin nx$ near $0$, and then wildly diverge when the weight becomes small.
For comparison, I did the same with $\alpha=2$. The polynomials have civilized appearance where the weight is not too small, roughly in the interval $[1,4]$. They wildly diverge both to the left and to the right.
do these polynomials still converge using the Least Square approximant
When $\alpha=0$, the Laguerre polynomial form on orthonormal basis of $L^2([0,\infty),e^{-x})$ (reference here). Therefore, for any function $f\in L^2([0,\infty),e^{-x})$ we have $\sum c_n L_n\to f$ in the norm, where $c_n=\langle f,L_n\rangle$.
When $\alpha>0$, the Laguerre polynomial form on orthogonal (but not normalized) basis* of $L^2([0,\infty),x^\alpha e^{-x})$. Therefore, for any function $f\in L^2([0,\infty),x^\alpha e^{-x})$ we have $\sum c_n L_n^{(\alpha)}\to f$ in the norm, where $c_n=\langle f,L_n^{(\alpha)}\rangle / \|L_n^{(\alpha)}\|$. (The value of $\|L_n^{(\alpha)}\|$ is given on Wikipedia).
(*) Disclaimer: I don't have a reference for the the completeness of $\{L_n^{(\alpha)}\}$, but I just put a bounty on unanswered question On the completeness of the generalized Laguerre polynomials in the hope someone does. Or you can consult the books G. Polya Orthogonal polynomials and G. Sansone Orthogonal functions, in case they treat $L_n^{(\alpha)}$.
A straightforward idea is to put the series for $J_\alpha$ directly into the integral (and exchange integration with summation, admissible because of absolute convergence). Recognizing an $n$-th derivative middleways, we reduce everything to the Rodrigues formula for $L_n^{(\alpha)}(x)$: \begin{align}\mathrm{RHS}&=x^{-\alpha}e^x\int_0^\infty y^n e^{-y}\sum_{k=0}^\infty\frac{(-1)^k(xy)^{k+\alpha}}{k!\Gamma(k+\alpha+1)}~dy\\&=x^{-\alpha}e^x\frac{\partial^n}{\partial x^n}\int_0^\infty e^{-y}\sum_{k=0}^\infty\frac{(-1)^k(xy)^{k+n+\alpha}}{k!\Gamma(k+n+\alpha+1)}~dy\\&=x^{-\alpha}e^x\frac{\partial^n}{\partial x^n}\sum_{k=0}^\infty\frac{(-1)^k x^{k+n+\alpha}}{k!\Gamma(k+n+\alpha+1)}\int_0^\infty y^{k+n+\alpha}e^{-y}~dy\\&=x^{-\alpha}e^x\frac{\partial^n}{\partial x^n}\left(x^{n+\alpha}\sum_{k=0}^\infty\frac{(-x)^k}{k!}\right)=\mathrm{LHS}.\end{align}
Best Answer
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}