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.

翻译：拉盖尔函数 $l_{n,\tau}^\alpha$，$n=0,1,\dots$，由广义拉盖尔多项式构造而成。这些函数依赖于两个参数：尺度参数 $\tau>0$ 和广义阶参数 $\alpha>-1$，并构成 $L_2[0,\infty)$ 空间中的正交基。设方阵 $A$ 的谱位于左半开平面内，则矩阵指数 $H_A(t)=e^{At}$，$t>0$，属于 $L_2[0,\infty)$。因此，矩阵指数 $H_A$ 可展开为级数 $H_A=\sum_{n=0}^\infty S_{n,\tau,\alpha,A}\,l_{n,\tau}^\alpha$。本文提出了范数 $\Bigl\lVert H_A-\sum_{n=0}^N S_{n,\tau,\alpha,A}\,l_{n,\tau}^\alpha\Bigr\rVert_{L_2[0,\infty)}$ 的估计。讨论了通过优化 $\tau$ 和 $\alpha$ 使该估计值最小化的方法。数值算例表明，最优 $\alpha$ 往往接近于0，这极大地简化了问题。