Generalized GCD matrices

Abstract

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