I am interested in sufficient conditions on non-negative sequences of coefficients $\{c_{2n}\}_{n\ge 0}$ guaranteeing that $$%\begin{equation}\label{cond} \sum_{n=0}^\infty c_{2n} L_{2n}^{(1)}(x)\ge 0,\quad \forall x\ge 0, %%\end{equation} $$ where $L_{2n}^{(1)}$ is the first associated Laguerre polynomial of degree $2n$.

More specifically, I would like to know for which rapidly decaying sequences $\{c_{2n}\}$ the above inequality holds.

For example, if $c_{2n}=\theta^{2n}$, where $0<\theta<1$, using the generating function $$ \sum_{n=0}^\infty L_{n}^{(1)}(x) z^n = \frac 1{(1-z)^2} \exp\Bigl( -\frac {zx}{1-z}\Bigr), $$ one has that $$ \sum_{n=0}^\infty L_{2n}^{(1)}\theta^{2n}= \frac 1{2(1-\theta^2)^2} \exp\Bigl( -\frac {\theta^2x}{1-\theta^2}\Bigr) + \frac 1{2(1+\theta^2)^2} \exp\Bigl( \frac {\theta^2x}{1+\theta^2}\Bigr)\ge 0. $$

Will the inequality $%\begin{equation}\label{cond} \sum_{n=0}^\infty c_{2n} L_{2n}^{(1)}(x)\ge 0,\quad \forall x\ge 0, %%\end{equation} $ remain true if $$ 0<c_{2n}\le \theta^{2n} $$ or $$ 0<c_{2n+2}\le c_{2n} \theta^2 $$ for all $n$ and $\theta^2$ sufficiently small (e.g, $\theta^2<0.5$)?