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
56 Ferenc Mátyás Although, the main result of [4] is formulated for seed-polynomials with integer coefficients but it is true for seed-polynomials with real coefficients, too. Since we are going to apply it, therefore we cite it as a lemma. Lemma 1. Let GQ{X) and Gi(x) be two fixed polynomials with real coefficients , Go(0) • Gi(0) ^ 0 and x 0 E R %o G A if and only if one of the following conditions holds: (i) = 0; (») -Sd = wfcy «"» < o; (in) x 0 = 0, where , s x + Vx 2 -f 4 x - yjx 2 + 4 (2) a(x) = and = . The purpose of this paper is to investigate the asymptotic behavior of the elements of the set B in the cases of simple seed-polynomials. In our discussion we are going to use the following explicit formulae for the polynomial G n{x) = G n (Co(^), Ci(x), x). It is known that (3) G n(x)=p(x)a n(x)-q(x)ß n(x) for n > 0, where a(a:) and ß(x) are defined in (2), while _ G^x) - ß(x)G 0(x ) _ Gtfr)-«,(»)<?„(») P( X>- a(x)-ß{x) ^ q{X )~ a{x)-ß{x) ' These formulae can be obtained by standard methods or see in [2]. Since we want to investigate the roots of the polynomials G n(x), therefore it is worth rephasing the expression G n(x) = 0 as p(x) ( ß{x) q(x) that is m Gi{x) - ß(x)G 0(x ) = / n U G l(x) - a(x)Go(x) {x + V^T4j ' Let us consider the polynomial G n (Go(x),Gi(x), x). It is obvious that G n(0,0,;r) is identical to the zero polynomial for every n > 0. Using (3)
