I'm working on a problem that has brought up for me the need to address infinite series of the following form, $ \sum_{i=k}^\infty \frac{1}{i!}A^{i-k+1} $ where $A$ is an $n\times n$ matrix. If $k = 0$, then this is clearly the power series of a matrix exponential. If $A$ were a real number, $x$, then it's easy to see that $ \sum_{i=k}^\infty \frac{1}{i!}x^{i-k+1} = \frac{1}{x^{k-1}}\left(e^x-\sum_{i=0}^k\frac{1}{i!}x^i \right), $ by simply equating to the power series of the exponential function (at least I think I got those indices right).
I assume that the same thing would hold if $A$ were invertible; unfortunately, in my case $A$ is not invertible. However I have verified that my sums do converge to something, so I figured I'd ask if anyone had any idea of a way to simplify my infinite series.