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

TL;DR

This paper introduces a novel method using the $(s,t)$-integration factor involving the $(s,t)$-Pantograph function to solve first-order linear proportional difference equations, generalizing classical functions and equations.

Contribution

It develops the $(s,t)$-integration factor method and applies it to solve first-order linear proportional difference equations and their $(s,t)$-analogues, extending existing solution techniques.

Findings

Solutions to first-order linear proportional difference equations are obtained.

The $(s,t)$-Pantograph function generalizes the partial Theta function.

The method is applied to the $(s,t)$-analog of Bernoulli's equation.

Abstract

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.