LCM of sequences of polynomials
Let \({\mathbf P}= \{P_ k(T)\mid k\geq n_ 0\}\) be a sequence of monic polynomials with complex coefficients. Write \(\Pi(n,T,{\mathbf P})\) and \(M(n,T,{\mathbf P})\) for the product and least common multiple of \(\{P_ k(T)\mid n_ 0\leq k\leq n\}\), and let \(r(n,{\mathbf P})\) be the ratio of the degree of \(M(n,T,{\mathbf P})\) to the degree of \(\Pi(n,T,{\mathbf P})\). This paper is concerned with evaluating \(r({\mathbf P})= \lim_{n\to\infty} r(n,{\mathbf P})\) for certain sequences of polynomials, namely \(P_ k(T)= T^{kn}-w\), \(T^{km+q}-w\), and \((T^{k(nm+1)}- w)/(T^ k- w)\), where \(w\) is a primitive \(m\)-th root of unity. This has applications to second order linear recurrences. For example, for \(P_ k(T)=T^ k-w\), one has \[ r({\mathbf P})= {{6m\Omega_ 1(m) \Omega_ 2(m)} \over {\varphi(m)\pi^ 2}}, \] where \(\Omega_ 1(n)= \prod_{p\mid n}({p\over p+1})\), and \(\Omega_ 2(n)= \sum\{{1\over k^ 2}\mid 1\leq k\leq n\), \((k,n)=1\}\). (The English version of the abstract contains a number of misprints: one which is potentially misleading is the omission of ``the degree of'' before ``the LCM'' and ``the product of'').
monic polynomials with complex coefficients
scientific article; zbMATH DE number 202534