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

The asymptotic behavior of the real roots of Fibonacci-like polynomials 61 and XQ — a 2 (17 — == and z

0 > 0 a XQ + v^o + 4 Since (16) and (17) are identical to (8) and (7), respectively, therefore all of the statements of our theorem follows from the Theorem 2. Thus the theorem is proved. Concluding Remarks Using our Theorem 2 for a = 1 we get that g

n - g'

n ~ § + (-l)

n ff4~

n, which matches perfectly with the result of H. Prodinger. On the other hand it is quite likely that similar results can be obtained for seed-polynomials GQ(X) — X±a and GQ(X) = a or for other polynomials. This could be the subject of further research work. References [1] D. GREENE and D. KNUTH, Mathematics for the Analysis of Algorithms, Birkhäuser, 1981. '[2] V. E. HOGGAT, JR. and M. BICKNELL, Roots of Fibonacci Polynomials, The Fibonacci Quarterly 11.3 (1973), 271-274. [3] HONGQUAN YU, YI WANG and MINGFENG HE, On the Limit of Generalized Golden Numbers, The Fibonacci Quarterly 34.4 (1996), 320-322. [4] F. MÁTYÁS, Real Roots of Fibonacci-like Polynomials, Proceedings of Number Theory Conference, Eger (1996) (to appear) [5] G. A. MOORE, The Limit of the Golden Numbers is 3/2, The Fibonacci Quarterly 32.3 (1994), 211-217. [6] H. PRODINGER, The Asymptotic Behavior of the Golden Numbers, The Fibonacci Quarterly 35.3 (1996), 224-225. FERENC MÁTYÁS ESZTERHÁZY KÁROLY TEACHERS' TRAINING COLLEGE DEPARTMENT OF MATHEMATICS LEÁNYKA U. 4. 3301 EGER, PF. 43. HUNGARY E-mail: matyas@gemini.ektf.hu