On the location of roots of the characteristic polynomial of \((p, q)\)-distance Fibonacci sequences (Q2091185)

Let \(k,\ell,p\) and \(q\) and be positive integers. The \((k,\ell,p,q,)\)-Fibonacci sequence \((F_{k,\ell,p,q})_{n\ge 0}\) is the four-parameter sequence defined by the recurrence relation \[ F_{k,\ell,p,q}(n)=kF_{k,\ell,p,q}(n-p)+\ell F_{k,\ell,p,q}(n-q), \] with suitable initial conditions. In the paper under review the authors examine the geometric, algebraic, and analytic properties of the roots of the characteristic polynomial of the above recurrence relation. Also, the authors discuss the connection between this four-parameter Fibonacci sequence and some well-known sequences including Fibonacci, Lucas, Pell, Jacobsthal, Padovan, and Narayana sequences, as well as the sequence of Perrin numbers. In support of their arguments the authors of the paper under review utilize Baker's method (transcendental method), Descartes' sign rule, Eneström-Kakeya theorem, Rouché's theorem, Möbius transformation, Bernoulli inequality, mean value theorem, and intermediate value theorem.