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 59 This will be much nicer when we substitute (9) g

n = u - -. u Without loss of generality we can assume that u > 0 and we get the equality (10) (r

+tt+

13

(t

t"

V=-(-"

a)

wSince XQ = u — ^ holds for u = a + 1 and u = — ^y therefore it is plain to see that, for large n, (9) can only hold if u is either close to a + 1 or — ^ry

. In both cases this would mean that g

n is close to x

0 • Let us assume that u is close to a + 1 and so a > 0 because of u > 0. It is clear from (10) that the cases when n is even or odd have to be distinguished. We start with n = 2m and rewrite (10) as (11) a + 1 - U = + . „-4m. v ; u + l We get the asymptotic behavior by a process known as "bootstrapping" which is explained in [1]. First we insert u = a + 1 + Si into the left-hand side of (11) and u = a + 1 into the right-hand side of (11). So we get an approximation for Si

. Then we insert w = a-fl-f-<$i+^2 into the left-hand side of (11) and u = a + 1 -f into the rihgt-hand side of (11) and get an approximation for Si- This procedure can be repeated to get better and better estiamations for u. Now we determine only the number Si

. Prom (11) we have a(a

2 +2a + 2) 1 ~ —— (a + 1) a + Z and so a(a

2 + 2a + 2) . , 4tt

7 U = a + 1 SI ~a-fl + —

1 1 (a + l)

_4

m. a + 2 v 1 Substituting u into (9) we get that 1 a(a + 2) (a(a

2 -f 2a + 2)

2 . 12 g

2m=a+l+Si —— \ a+1 a + 1-l-Ö! a+1 (a + lj^a + z) If n — 2m -f 1 then (10) can be rewrite as (au + u -f l)(u - 1) _

4

m a + l-u=- — Lu 4

m. w

2(w + 1) — 4m