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

16 Ferenc Mátyás 1,2,

.. .) is while in the case 0 < a < 2 the above set has no nonzero accumulation point. According to Theorem A, apart from finitely many real roots, all of the real roots of polynomials G

n(a, x ± a, x) (a E R\{0}, n — 1,2,...) can be found in the open intervals a) , a(2 - a) , a(2 ± - £, ±+ £ or (-£,e), a — 1 a — 1 where £ is an arbitrary positive real number. Investigating the complex zeros of Fibonacci-like polynomials V. E. HOGATT, JR. and M. BLCKNELL [1] proved that the roots of the equation G

n(

0,1, x) = 0 are x

k = 2i cos (A; = 1, 2, 1), i.e. apart from 0 if n is even, all of the roots are purely imaginary and their absolute values are less than 2. P. E. Ricci [7] among others studied the location of zeros of polynomials G

n(l,x -f l,x) and proved the following result. Theorem B. All of the complex zeros of polynomials G

n(l,x + 1, x) (n = 1, 2,

...) are in or on the circle with midpoint (0, 0) and radius 2 in the Gaussian plane. The purpose of this paper is to generalize the result of P. E. Ricci for the polynomials G

n(a,x + b,x) where a, b 6 R and a / 0, i.e. to give bounds for the absolute values of zeros. To prove our results we are going to use linear algebraic methods as it was applied by P. E. RlCCl [7], too. At the end of this part we list some terms of the polynomial sequence G

n(x) = G

n(a, x + 6, x) (n = 2, 3,.. .). We have (^(x) = x

2 -f bx + a, G

3(X) = x

3 + bx

2 + (a + l)x + b, G

4(x) = X

4 + bx

3 + (a + 2)x

2 + 2 bx + a, G

5(X) = x

5 + bx

4 + (a + 3)x

3 + 3 bx

2 + (2a + l)x + b

, Ge{x) = x

6 + bx

5 + (a + 4)z

4 + 4 bx

3 + (3a + 3)x

2 + 3 bx + a. Known facts from linear algebra To estimate the absolute values of zeros of polynomials G

n(a,x + b,x) (n > 1) we need the following notations and theorem. Let A = (a^) be an n X n matrix with complex entries, A; (I = 1, 2,..., n) and f(x) denote the eigenvalues and the characteristic polynomial of A, respectively. It is known that (2) /(A,) = 0