First Order Linear Proportional Difference Equation with Integration Factor the $(s,t)$-Pantograph Function

In this paper, we find solutions to first-order linear proportional difference equations via the $(s,t)$-integration factor method. The $(s,t)$-integration factor involves the $(s,t)$-Pantograph function, which is a generalization of the partial Theta function. Other equations are solved including the $(s,t)$-analog of the Bernoulli equation.