Generalized Fibonacci sequences via orthogonal polynomials

Using tools from the theory of orthogonal polynomials we obtain and extend some recent results on generalized Fibonacci sequences, by stating a Binet’s-type formula for a sequence of real or complex numbers {Qn}n=0∞ defined byQ0=0,Q1=1,Qm=ajQm-1+bjQm-2,m≡j(modk),where k⩾3k⩾3 is a fix integer number, and a0,a1,…,ak-1a0,a1,…,ak-1 and b0,b1,…,bk-1b0,b1,…,bk-1 are 2k2k given real or complex numbers, with bj≠0 for 0⩽j⩽k-10⩽j⩽k-1.