Representing Carlitz formula with q-shift operator

TL;DR

This paper introduces a new q-shift operator formula that offers a novel proof and extension of the Carlitz formula, with applications to q-congruences on cyclotomic polynomials.

Contribution

It presents a new q-shift operator formula that generalizes the Carlitz formula and applies it to prove new q-congruences on cyclotomic polynomials.

Findings

New q-shift operator formula for Carlitz formula

Extended Carlitz formula with new proofs

Proved two new q-congruences on cyclotomic polynomials

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.