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

18 Ferenc Mátyás Theorems and the Main Result Let us consider the n X n matrix 0 0 0 \ 0 0 0 0 0 0 , -i 0 -i 0-20/ where 6 E R and a £ R \ {0}. Further on we prove the following Theorem 1. Let n > 1 and a, 6 G R (a / 0). The characteristic polynomial of matrix A

n is the polynomial G

n(a, x + b, x). Let n > 2 and a, 6 £ R (a ^ 0). If A

n

i

, A

n2,. .., A

n

n denote the zeros of the polynomial G

n(a

, x + 6, x) then, using the norms defined by (4) and (5) for the matrix A

n, one can get the following estimations by (2),(3) and Theorem 1. /-b -ai 0 -i 0 -i 0 -i 0 0 0 0 V 0 0 0 (10) max |A

m j < n max(|a|

, |6|

, 1) 1 < i < n and (11) max |A

mj < y/a

2 + b

2 + 2n - 3. 1 < i < n From (10) and (11) it can be seen that these bounds depend on a, 6 and n but using the Gershgorin-circles we can get a more precise bound for IA m-| and this bound depends only on a and b. We shall prove Theorem 2. Let n > 2 and a, b E R (a / 0) and let us denote by K\ the set K\ = {z : \z + 6| < |a|} and by K

2 the set Ii

2 - {z : \z\ < 2} in the Gaussian plane. Then (12) A n 1 1 ^n2 •> • • • 1 ^nn 6 K1 U K

2

. Now we are able to formulate our main result.