Zeros of a Class of Fibonacci-type Polynomials

with the initial values G0(x) = a and G1(x) = x + b. The polynomials Gn−1(1, 0;x) and Gn(2, 0;x)(n ≥ 1) are just the usual Fibonacci polynomials Fn(x) and the Lucas polynomials Ln(x), respectively. Our concern in this paper is to study some of the properties of the zeros of the Fibonacci-type polynomials Gn(a, b;x). The zeros of Fn(x) and Ln(x) have been given explicitly by V. E. Hoggatt, Jr. and M. Bicknell[3] and N. Georgieva[2](see MR52#5634 for corrections by Reviewer). However there are no general formulae for the zeros of the Fibonacci-type polynomials. There have been quite a few papers concerned the properties of the zeros of the Fibonacci-type polynomials in recent years. For example, G. A. Moore[8] and H. Prodinger[9] investigated the asymptotic behavior of the maximal real zeros of Gn(−1,−1;x) respectively. The authors etc.[11] and F. Mátyás[5] investigated the same problem for Gn(a, a;x)(a < 0) and Gn(a,±a;x)(a 6= 0) respectively. In [6], F. Mátyás shown that the absolute values of complex zeros of polynomials Gn(a, b;x) do not exceed max{2, |a|+ |b|}, which generalizes the result of P. E. Ricci[10] who investigated the problem in the case a = b = 1.