  We introduce a method for calculating the divided differences of the
exponential function by means of addition and removal of items from the input
list to the function. Our technique exploits a new identity related to divided
differences recently derived by F. Zivcovich [Dolomites Research Notes on
Approximation 12, 28-42 (2019)]. We show that upon adding an item to or
removing an item from the input list of an already evaluated exponential, the
re-evaluation of the divided differences can be done with only $O(s n)$
floating point operations and $O(s n)$ bytes of memory, where $[z_0,\dots,z_n]$
are the inputs and $s \propto \max_{i,j} |z_i - z_j|$. We demonstrate our
algorithm's ability to deal with input lists that are orders-of-magnitude
longer than the maximal capacities of the current state-of-the-art. We discuss
in detail one practical application of our method: the efficient calculation of
weights in the off-diagonal series expansion quantum Monte Carlo algorithm.
