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
58 Ferenc Mátyás By Lemma 2 the functions —j-v = tUf= and -J^r = J , , , axe continuous, 1 > ^y > 0 for any x > 0 and 0 > ^y > -1 for any x < 0, therefore 0 < \a\ < 1 is a necessary and sufficient condition for the solvability of (5) and (6). Solving (5) and (6) we get that the single real root x 0 is = ^-f 1, where £ O>0if-l<a<0 and x 0<OifO<a<l. This completes the proof. In the following theorems we prove asymptotic formulae for those real roots g n of the polynomials G n(—a,x ± a,x) which do not tend to 0 if n tends to infinity. Theorem 2. Let GQ(X) = —A and G\(x) = x — a, where a 6 K \ {0}. If either a > 0 or a < -2 then A \ {0} = Í }, while in the case -2 < a < 0 we have A \ {0} = 0. Furthermore for large n q(q + 2) a(a 2 + 2a + 2) 2 2 n 9n + 1 7—rTm —rrrr' ' a + 1 (a + 1) (a + 2) Proof. According to Lemma 1, XQ 6 A \ {0} if and only if XQ — a 2 7 — = ^ and x 0 > 0 a XQ + y/x 2 0 + 4 or Xq — a 2 8) — = 7== and < 0 a x 0- V xo + 4 holds. Using the statements of Lemma 2 one can verify that (7) has a solution for XQ if and only if a > 0, while (8) has a solution for XQ if and only if a < —2. Solving (7) and (8) we get that a(a + 2) xo = ~r~a -f 1 To determine the asymptotic behavior of g n we apply (4), which in our case has the following form 2 ( g n - a) + q (g n - yjg* + 4 ) ^ / g n - y/gl+ l\ " 2(g n -a) + a (g n -f y/g£+Í) \9n + y/ffl+4 J
