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

60 Ferenc Mátyás Using the "bootstrapping" method for u = a -f 1 + 6[ we get the estimation a(a

2 + 2a + 2

) 4

m (G + 2)(a + l)

2 ' which imphes the following form: #2m + l = a + 1 + a + 1 + S[ (13) V ' a(fl + 2) a(a

2 -f 2a + 2)

2 4

m ~ — — — [ a + 1) a + 1 (a + 2)(a + l)

4 ; Comparing (12) and (13) the desired approximation yields since a > 0. One can verify in the same manner that the estimation for g

n also holds when a < — 2. This completes the proof. Remark. From our proof one can see that for large n g

n = g'

n if a > 0 while g

n is the minimal real root if a < —2. A similar result can be proved for the polynomials G

N{ — a, x + a, x). Theorem 3. Let GQ{X

) = —a and GI(x) — x + a where a £ R \ {0}. If etiher a > 0 or a < -2 then A \ {0} = {-^p^}, while A \ {0} = 0 if — 2 < a < 0. Furthermore for large n a(a + 2

) q(q

2 + 2a + 2)

2 2

n ~ -^rr

+ (o+i)»(a+2)

(a+x

) ' where G

n(g

n) = 0 and lim ^ 0. n—*oo Proof. For a real number Xo, by our Lemma 2, Xo £ A \ {0} if and only if (14) = ?== and xo > 0 a x

0+y/x

2

0 + 4 or (15) ° — — and XQ < 0 a x

0-y/xl + 4 holds. Substituting — XQ for XQ into (14) and (15) we get that . . X

Q — a 2 (16) — . and xo < 0 a x

0- ^Jx\ + 4