New generalizations of Fibonacci and Lucas sequences

Abstract

We consider the sequences {fn}∞n=0 and {ln}∞n=0 which are generated bythe recurrence relations fn=2afn-1+(b2-a)fn-2 and ln=2aln-1+(b2-a)ln-2 with the initial conditions f0=0, f1=1 and l0=2, l1=2a where a and b are any non - zero real numbers. We obtain generating functions, Binet formulas for these two sequences and give generalizations of some well - known identities.