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 57 the identities G„(0, x) = Gi{x) • G n(0, 1, x) and G n (G 0(x), 0,ar) = Go(x) • G n(l,0, x) yield. But it is known from [2] and can be obtained easily from (4) that neither the Fibonacci polynomials G n(0,l,a;) nor the polynomials G n(l,0,x) have real root x' except x' — 0 if n is even or odd, respectively. Therefore investigating the asymptotic behavior of the roots of polynomials G n(Go(x) JGi(x),x) we can assume that the seedpolynomials differ from the zero polynomial and at least one of them is a monic polynomial (since one can simplify the left-hand side of (4) with the leading coefficient of the polynomial G\(x) or Theorems and Proofs First of all we need the following lemma, which deals with the properties of the functions a(x) and ß(x) defined in (2). Lemma 2. (a), On the interval [0, oo) the function -y-^-y is continuous and strictly monotonically decreasing, its graph is convex and 1 > —t—- > 0. (b) On the interval (—oo,0] the function -^y is continuous and strictly monotonically decreasing, its graph is concave and 0 > j^j > — 1. Proof. By (2) it is obvious that the functions ^y and -^y are continuous on the above mentioned intervals. The rest of the statement can be proved easily using the methods of differential calculus. Further on we deal with the set A if Co(a;) = 1 and G\{x) = a. In this case, using Lemma 1, the set A can be determined in a very simple manner. Theorem 1. Let a G R \ {0} and G n{\,a,x ) be Fibonacci-like polinomials. If 0 < |a| < 1 then A \ {0} = {^V 1}; while in the case |a| > 1 A\{O} = 0. Proof. According to Lemma 1 to get the elements of the set A \ {0} we have to solve the equations 2 (5) -a - ——____ for x > 0 x + Vx 2 + 4 and 2 (6) -a = for x < 0. X — Y/X 2 + 4
