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 19 Main Result. For any n > 1 and fl,Ä

£R

(a/0) if G

n(a, x + b, x) = 0, then (13) \x\ <max(|fl| + |6|,2), i.e. the absolute values of all zeros of all polynomial terms of polynomial sequence G

n(a

, x + 6, x) (n = 1, 2,3,.. .) have a common upper bound, and by (13) this bound depends only on a and b in explicit way. We mention that Theorem B can be obtained as a special case (a = b = 1) of our Main Result. Proofs Proof of Theorem 1. It is known that the characteristic polynomial f

n(x) of matrix A

n can be obtained by the determinant of matrix x\

n - A

n, where l

n is the n X n unit matrix. So fx + b ai 0 ••• 0 0 0 \ i x i ••• 0 0 0 (14) f

n(x) = det (xl

n - A

n) = det 0 i x • • • 0 0 0 0 0 0 ••• i x i \ 0 0 0 • • • 0 i x ) We prove the theorem by induction on n. It can be seen directly that fi(x) = x + b = Gi (a, x -f 6, x) and /2(2) = x

2 -f bx + a = G

2(a, x -f b. x). Let us suppose that /

n_

2(z) = G

n2(a

, x + 6, x) and f

n-i(x) = G

n_i(a, x + b, x) hold for an integer n > 3. Then developing (14) with respect to the last column and the resulting determinant with respect to the last row, we get f

n{x) = xf

n-i(x) - ilf

n2{x

) = xf

n-i(x) + f

n2(x), i.e. by our induction hipothesis fn(x) = xG

n-i(a,x + b,x) + Gn2(a.x -f b,x) and so by (1) fn(x) = G

n(a

, x + 6, x) holds for every integer n > 1. Proof of the Theorem 2. From the matrix A

n we determine the socalled Gershgorin-circles. By the definition of A

n and (6) now there are only