Representing Carlitz formula with q-shift operator

Abstract

This paper presents a new formula for the q-shift operator, building on the techniques by Liu and Sears. This formula provides fresh proof of the Carlitz formula and extends it naturally. As applications, we derive an equivalent form of the generalized Carlitz formula to prove two $q$-congruences on cyclotomic polynomials, which expand upon the results of Guo et al.