Symmetry property for Laguerre polynomials

Question

The classic type of orthogonal polynomials $H_n(x)$, $P_n(x)$, $T_n(x)$ and $U_n(x)$ have symmetry property. Does the Laguerre polynomials $L_n(x)$ also has symmetry property?

$L_n(-x)=L_n(x)$ and $L_n(-x)=-L_n(x)$

Pls help me. Thanks

Answer 1

From glancing at the first few, the answer is clearly no. In particular, $L_1(x) = -x+1$.

Answer 2

No, as you can see by writing even the first two or three Laguerre polynomials. But we shouldn't expect them to be since they form an orthogonal family on $[0,\infty)$ (with respect to the weight function $e^{-x}$), rather than a symmetric interval about the origin.