A note on rational numbers and certain operators

TL;DR

This paper investigates conditions involving a generalized shift operator that determine when rational numbers can be represented using extended forms of q-ary expansions, expanding understanding of number representations.

Contribution

It introduces new conditions based on a generalized shift operator for representing rational numbers with positive q-ary-like expansions.

Findings

Derived criteria for rational number representations

Extended the concept of q-ary expansions

Provided theoretical conditions for generalized shift operators

Abstract

This paper is devoted to conditions defined in terms of the generalized shift operator for a rational number to be representable by certain positive generalizations of $q$-ary expansions.