Meaning of $\alpha$ in Laguerre polynomials

Question

I found that generalized Laguerre polynomials are:

$$ L_n^{\alpha} (x) = \sum_{i=0}^n (-1)^i {n+\alpha \choose n-i} \frac{x^i}{i!}.$$

However, I wonder what is the meaning of $\alpha$ in this expression. How does influence in the polynomial?

I know that it may shift it because if we evaluate it int $x=0$, we get

$$L_n^{\alpha}(0)= {n+\alpha\choose n} \approx \frac{n^\alpha}{\Gamma(\alpha+1)}.$$

But, do these polynomials still converge using the Least Square approximant if $\alpha \neq 0$??

EDIT: this last question is referring to use Laguerre series to approximate a function $f(x)$ by

$$f_m = \displaystyle\sum_{i=0}^m c_k L_k^{\alpha}(x),$$

where $c_k$ are the fourier coefficients of Laguerre:

$$c_k = \langle f,L_k^{\alpha}\rangle = \displaystyle\int_0^{\infty} f(x) L_k^{\alpha}\,w(x) dx,$$

and $w(x)$ is the corresponding weight function, namely

$$w(x)=x^{\alpha}e^{-x}.$$

I know that this $f_m$ converges to $f$ when $\alpha=0$, my question is if it does for other cases.

Answer 1

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 $L_n^{(\alpha)}$.