  The Laguerre functions $l_{n,\tau}^\alpha$, $n=0,1,\dots$, are constructed
from generalized Laguerre polynomials. The functions $l_{n,\tau}^\alpha$ depend
on two parameters: scale $\tau>0$ and order of generalization $\alpha>-1$, and
form an orthogonal basis in $L_2[0,\infty)$. Let the spectrum of a square
matrix $A$ lie in the open left half-plane. Then the matrix exponential
$H_A(t)=e^{At}$, $t>0$, belongs to $L_2[0,\infty)$. Hence the matrix
exponential $H_A$ can be expanded in a series $H_A=\sum_{n=0}^\infty
S_{n,\tau,\alpha,A}\,l_{n,\tau}^\alpha$. An estimate of the norm $\Bigl\lVert
H_A-\sum_{n=0}^N
S_{n,\tau,\alpha,A}\,l_{n,\tau}^\alpha\Bigr\rVert_{L_2[0,\infty)}$ is proposed.
Finding the minimum of this estimate over $\tau$ and $\alpha$ is discussed.
Numerical examples show that the optimal $\alpha$ is often almost 0, which
essentially simplifies the problem.
