Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1997. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 24)

MÁTYÁS, F., The asymptotic behavior of the real roots of Fibonacci-like polynomials

The asymptotic behavior of the real roots of Fibonacci-like polynomials FERENC MÁTYÁS* Abstract. The Fibouacci-like polynomials G

n(x) are defined by the recursive formula G

T

l(:r)=;rG

n_

1(:E)+G

n_2(:c) for n>

2, where G

0(a;) and Gi(;c) are given seedpolynomials. In this paper the non-zero accumulation points of the set of the real roots of Fibonacci-like polynomials are determined if either both of the seed-polynomials are constants or G

0(a;) = -a and G

x(x)=x±a (a6R\{0}). The theorems generalize the results of G. A. Moore and H. Prodinger who investigated this problem if G

0(a;) = —

1 and Gi(x) = a:-1, furthermore we extend a result of Hongquan Yu, Yi Wang and Mingfeng He. Introduction The Fibonacci-like polynomials G

n(x

) are defined by the following manner. For n > 2 (1) G

n(x) = xG

n-i(x) + G

n-2(x), where GQ(X) and are fixed polynomials (so-called seed-polynomials) with real coefficients. Ifit is necessary to denote the seed-polynomials, then we will use the notation G

N(X

) = G

n (GQ(X), G\ (a;), x), too. The polynomials (?„((), 1, a;) are the original Fibonacci polynomials and the numbers G

n(0,l,l) are the well-known Fibonacci numbers. Recently, G. A. Moore [5] investigated the maximal real roots g'

n of the polynomials G

n( — l,x — 1

,2:) and proved that g'

n exists for every n > 1 and lim g' = 3/2. (These numbers g' are called as "golden numbers". ) H. n—

+ oo Prodinger [6] gave the asymptotic formula g'

n ~ | + ( —l)

n ||4~" Hongquan Yu, Yi Wang and Mingfeng He [3] investigated the limit of the maximal real roots g'

n of polynomials G

n(—a, x - a, x) if a E R+. For brevity let us introduce the following notations. B denotes the set of the real roots of polynomials G

n(x) (n — 0,1, 2,...) and A denotes the set of the the accumulation points of set B. In [4] we investigated these sets. Research supported by the Hungarian National Research Science Foundation, Operating Grant Number OTKA T 020295.