An approximation of matrix exponential by a truncated Laguerre series

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.