A normal time discrete convolution operation could be written $(f*g)[n]$. In mathematical language this is the same as multiplying with a Toeplitz matrix (if zero padded convolution) or circulant matrix (if circular convolution).
Integrals if represented discretely by matrices can be written with triangular matrices filled with ones. But we also know we can calculate the same thing by iterating +1 and storing the intermediate result. If we have summed 16 values to get the sum up to 17 we can just add the 17th element to the previous sum instead of restarting the whole sum from first to 17th.
Mathematically we can write this ($\prod$ being matrix multiplication):
$${\bf S} = \prod_{\forall k} \bf S_k$$
Where $S_k$ is the slightly modified identity-matrix. Only row $k$ having 2 non-zero elements:
$${\bf S_1 S_2} \cdots {\bf S_n} =\\\left[\begin{array}{cccccc}1& & & & & \\ &1& & & & \\ & &1& & & \\ & & &1& & \\ & & & &1& \\ & & & &1&1\end{array}\right]\cdot \left[\begin{array}{cccccc}1& & & & & \\ &1& & & & \\ & &1& & & \\ & & &1& & \\ & & &1&1& \\ & & & & &1\end{array}\right]\cdots \left[\begin{array}{cccccc}1& & & & & \\1&1& & & & \\ & &1& & & \\ & & &1& & \\ & & & &1& \\ & & & & &1\end{array}\right]=$$
$$\left[\begin{array}{cccccc}1& & & & & \\1&1& & & & \\1&1&1& & & \\1&1&1&1& & \\1&1&1&1&1& \\1&1&1&1&1&1\end{array}\right] = {\bf S}$$
The left hand side multiplied on column vectors to the right will perform the summing-up. While this is not a convolution ( none of the matrices are Toeplitz or circulant ). The operation for each row is relatively the same: the only active operation, (the [1,1]) pair moves along the diagonal one step for each new matrix multiplication in a way that resembles a Toeplitz matrix. Like a convolution but each intermediate step is stored ( computer scientists sometimes call this in-place calculation which is nice as we don't have to allocate and juggle with the memory as much which usually slows down calculations )
So the question is.. do we have some mathematical name or notation for this? We could call it a sparse factorization, but it does not quite capture all the properties of the factors.