# Common Convention for Denoting "stateful" functions?

In computer programming it's common to define "functions" (well not really functions) that have a state that persists between calls. For example, in Python and I want to define a temporal difference and running sum operations:

class TemporalDifference:
def __init__(self, initial=0):
self.last_x = initial
def __call__(self, x):
delta = x-self.last_x
self.last_x = x
return delta

class RunningSum:
def __init__(self, initial=0):
self.sum = initial
def __call__(self, x):
self.sum = self.sum + x
return self.sum

# Which can be used like:
td = TemporalDifference()  # A stateful function
dx = [td(xt) for xt in np.sin(np.linspace(0, 10, 100))]


Is there a common convention for expressing this kind of thing mathematically? I was thinking something like

$$\Delta(x; x_{last}) := x-x_{last} : x \rightarrow x_{last} \\ \Sigma(x; s) := s+x : s+x \rightarrow s$$

I'd like to use this notation to conveniently express identities like:

$$(\Sigma \circ \Delta) (x_t) = x_t \forall t$$