Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1998. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 25)
MÁTYÁS , F., Bounds for the zeros of Fibonacci-like polynomials
Bounds for the zeros of Fibonacci-like polynomials FERENC MÁTYÁS Abstract. The Fibonacci-like polynomials G„(x) are defined by the recursive formula G 7 l(x) = xG n_ 1(a:)+G I l_ 2(x) for n> 2, where G 0(x) and G^x) are given seedpolynomials. The notation G n(x) = G„(G 0(x),Gi(x),x) is also used. In this paper we determine the location of the zeros of polynomials G n(a,x+6,x) and give some bounds for the absolute values of complex roots of these polynomials if a,b£R and a^O. Our result generalizes the result of P. E. RICCI who investigated this problem in the case a=6=l. Introduction Let GQ(X ) and G\{x) be polynomials with real coefficients. For any nEN\{0,l} the polynomial G n(x) is defined by the recurrence relation (1) G n(x) - xG n-i(x) + G n2(x) and these polynomials are called Fibonacci-like polynomials. If it is necessary then the initial or seed polynomials Co (a:) and G\(x) can also be detected and in this case we use the form G n(x) — G n{G{){x)^ G\ (x), z). Note that G n(0,1,1) = F N where F N is the n t h Fibonacci number. In some earlier papers the Fibonacci-like polynomials and other polynomials, defined by similar recursions, were studied. G. A. MOORE [5] and H. PRODINGER [6] investigated the maximal real roots (zeros) of the polynomials G n(-l,x - l.Z) (N > 1). HONGQUAN Yu, Yl WANG and MING FENG HE [2] studied the Hmit of maximal real roots of the polynomials G„(Ű, x — a,x) if a £ R + as n tends to infinity. Under some restrictions in [3] we proved a necessary and sufficient condition for seed-polynomials when the set of the real roots of polynomials G n{Go{x), G\ (z), x) (n = 0,1, 2,. . .) has nonzero accumulation points. These accumulation points can be effectively determined. In [4], using this result, we proved the following Theorem A. Ifa<0 or 2<a then, apart from 0, the single accumulation point of the set of real roots of polynomials G n(a, x ± a, x) (n = Research supported by the Hungarian OTKA Foundation, No. T 020295.
