On (q, h)-differentiation: divided differences, quotient rules, and applications

This paper investigates a class of time scales for which the forward jump function is given by \(\sigma (t)=qt+h\), where q, and h are constants. This framework allows us to treat the standard, h-, q-, and (q, h)-derivatives simultaneously as special cases of the delta derivative. We establish a key connection between the nth delta derivative and specific nth divided difference, which serves as the foundation for generalizing several classical results from q-calculus to the broader context of (q, h)-calculus. In the second part of the paper, we present explicit formulas for the nth delta derivative of a quotient of two functions, extending familiar results from classical calculus. As an application, we use the obtained results to study the (q, h)-analogs of the power and exponential functions, yielding explicit expressions for the nth derivatives of their reciprocals and leading to a novel q-binomial identity.