On a (p,q)-Hahn Difference Operator: Algebraic Properties, Integral Formulation and Applications

In this paper, we introduce a new difference operator, called the (p,q)-Hahn difference operator, which extends both the classical Hahn operator and the (p,q)-difference operator by incorporating an additional shift parameter, ω. This extension allows the operator to reflect translation effects together with scaling behavior within a common framework. We investigate the basic algebraic properties of the operator, including linearity, product and quotient rules, and explicit formulas for power functions. A generalized Leibniz rule is established using (p,q)-binomial coefficients. In addition, a corresponding (p,q)-Hahn integral is defined, and a fundamental relation between the operator and the integral is obtained under suitable assumptions. Furthermore, several special and limiting cases are analyzed in order to clarify the connection between the proposed operator and existing difference operators. In particular, it is shown that the operator reduces to the classical derivative, the q-difference operator, and the standard (p,q)-difference operator under appropriate parameter choices. Finally, applications to (p,q)-Hahn Sturm–Liouville-type problems and hypergeometric-type difference equations are discussed. These results suggest that the proposed operator provides a consistent extension of existing difference-calculus structures.