Generalized LCM matrices

TL;DR

This paper introduces a generalized class of LCM matrices where entries depend on functions of the least common multiple and other parameters, expanding the scope of classical LCM matrix theory.

Contribution

It extends the traditional LCM matrix concept by defining new matrices with entries as functions of multiple variables, broadening the theoretical framework.

Findings

Defined new classes of generalized LCM matrices.

Explored properties and potential applications of these matrices.

Provided initial theoretical results on their structure.

Abstract

Let $f$ be an arithmetical function. The matrix $[f[i,j]]_{n\times n}$ given by the value of $f$ in least common multiple of $[i,j]$, $f\big([i,j]\big)$ as its $i,\; j$ entry is called the least common multiple (LCM) matrix. We consider the generalization of this matrix where the elements are in the form $f\big(n,[i,j]\big)$ and $f\big(n,i,j,[i,j]\big)$.