On integral representations of $q$-difference operators and their applications

Integral representations of two $q$-difference operators are provided in terms of special functions arising in the theory of asymptotic solutions to $q$-difference equations in the complex domain. Both representations are unified through the so-called $(p,q)$-differential operator, for which a kernel-like function is provided, generating the sequence of $(p,q)$-factorials.

💡 Research Summary

The paper investigates integral representations of two fundamental q‑difference operators and unifies them through the (p,q)‑differential operator. Starting from the classical setting where q>1, the authors recall the basic q‑numbers, q‑factorials, and two q‑exponential functions: the standard q‑exponential exp_q(z)=∑_{n≥0}z^n/