A study on differential equations associated with ( q , h ) -Frobenius-Euler polynomials

‎In recent years‎, ‎utilizing the generalized quantum exponential function (also known as the \((q,h)\)-exponential function) that extends and unifies the \(q\)‎- ‎and \(h\)-exponential functions into a single and convenient form‎, ‎\((q,h)\)-generalizations of the diverse polynomials and numbers‎, ‎such as Euler and tangent polynomials and numbers‎, ‎have been introduced and studied‎. ‎Inspired by these studies‎, ‎in this work‎, ‎we focus on defining and analyzing extensions of Frobenius-Euler polynomials and numbers using the \((q,h)\)-exponential function‎. ‎Also‎, ‎we show that the mentioned polynomials are solutions to some higher-order differential equations‎. ‎Furthermore‎, ‎we examine that \((q,h)\)-Frobenius-Euler polynomials are solutions to higher-order differential equations combined with the \(q\)-Bernoulli‎, ‎\(q\)-Euler‎, ‎and \(q\)-Genocchi numbers and polynomials‎, ‎respectively‎. ‎Finally‎, ‎we use a computer program to visualize the approximate roots of the mentioned polynomials‎.