Determinants of matrices associated with arithmetic functions on finitely many quasi-coprime divisor chains

Let S={x1,…,xn}S={x1,…,xn} be a set of n distinct positive integers and f   be an arithmetic function. We use (f(S))=(f(xi,xj))f(S)=f(xi,xj) (resp. (f[S])=(f[xi,xj])f[S]=f[xi,xj]) to denote the n×nn×n matrix having f   evaluated at the greatest common divisor (resp. the least common multiple) of xixi and xjxj as its i,ji,j-entry. The set S   is called a divisor chain if there is a permutation σσ of {1,…,n}{1,…,n} such that xσ(1)|…|xσ(n)xσ(1)|…|xσ(n). If S   can be partitioned as S=⋃i=1kSi with all Si(1⩽i⩽k) being divisor chains and (max(Si)(Si), max(Sj)(Sj)) = gcd(S  ) for 1⩽i≠j⩽k, then we say that S   consists of finitely many quasi-coprime divisor chains. In this paper, we introduce a new method to give the formulas for the determinants of the matrices (f(S))(f(S)) and (f[S])(f[S]) on finitely many quasi-coprime divisor chains S  . We show also that det(f(S))|det(f[S])det(f(S))|det(f[S]) holds under some natural conditions. These extend the results obtained by Tan and Lin (2010) and Tan et al. (2013), respectively.