I need to take a $d$-dimensional convolution of the last element of a matrix exponential, but am having trouble. Some specifics:
For a positive integer $g$, I have the $g \times g$ matrix
$$ Q = \begin{bmatrix} 0 & g & & & & & \\ 1 & 0 & (g-1) & & & & \\ & 2 & 0 & (g-2) & & & \\ & & 3 & \ddots & \ddots & & \\ & & & \ddots & & 3 & \\ & & & & (g-2) & 0 & 2 \\ & & & & & (g-1) & 0 \\ \end{bmatrix} $$
Let $0<r<1$, $\hat{x}_g = [0,0,\cdots ,0,1]^T$ be the column unit vector in the $g^{th}$ dimension, and let $H(r)$ be the Heaviside step function, zero for $r<0$. Then define
$$ \tau (r) = H(r)\, \hat{x}_g^T\, e^{Q r}\, \hat{x}_g $$
$$ \sigma (r) = H(r)\, \hat{x}_g^T\, e^{Q r}\, 1 $$
I.e., the last element of the matrix exponential for $\tau$, or sum of the last row for $\sigma$.
For all positive integers $d$, I need to find the following three quantities:
$$ \left( \otimes_{i=1}^d \tau \right)(r) $$
$$ \left( \left( \otimes_{i=1}^{d-1} \tau \right) \otimes \sigma \right)(r) $$ $$ \left( \left( \otimes_{i=1}^{d-2} \tau \right) \otimes \sigma \otimes \sigma \right)(r) $$
where $\otimes$ denotes convolution. Ideally this would be closed form to use in an infinite sum, but otherwise numerically for a truncated sum.
I've been trying to use the Laplace transform convolution theorem, but I'm having a hard time making it work for arbitrary $g$ and $d$. $g$ only needs to go up to 20 or so, but $d$ needs to go to at least around 100. Also, if anyone knows any good approximations, that could also be super helpful. Thanks!