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Báo cáo toán học: "A new class of q-Fibonacci polynomials"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: A new class of q-Fibonacci polynomials | A new class of q-Fibonacci polynomials Johann Cigler Institut fur Mathematik Universitat Wien A - 1090 Wien Osterreich Johann.Cigler@univie.ac.at Submitted Mar 24 2003 Accepted May 2 2003 Published May 7 2003 MR Subject Classifications primary 05A30 05A15 secondary 15A15 Abstract We introduce a new ợ-analogue of the Fibonacci polynomials and derive some of its properties. Extra attention is paid to a special case which has some interesting connections with Euler s pentagonal number theorem. 1 Introduction The Fibonacci polynomials fn x s are defined by the recursion fn x s xfn-1 x s sfn-2 x s with initial values f0 x s 0 f1 x s 1. They are given by the explicit yf1 formula fn x s p n k xn 1 2ksk. L. Carlitz 3 has defined a q-analogue which k 0 has been extensively studied cf. e.g. 6 2 8 . 2-ij In 7 I found that Fn x s P n k 1 q 2 xn1 - 2ksk is another natural q-k 0 analogue which satisfies the simple but rather unusual recursion 2.8 . This recursion does not lend itself to the computation of special values. Therefore I was surprised as I Lyf J zfcA learned that it has been shown in 9 and 13 that Fn 1 1 p 1 kq 2 n k has Q k 0 the simple evaluation 3.2 . This fact led me to a thorough study of this q-analogue via a combinatorial approach based on Morse code sequences. We show that these q-Fibonacci polynomials satisfy some other recurrences too generalize some well-known facts for ordinary Fibonacci polynomials to this case derive their generating function and study the special values Fn 1 q and Fn 1 1 which turn out to be intimately connected with Euler s pentagonal number series. Finally we show that the Hankel determinants det .Fi j k 1 1 b can be explicitly evaluated. q i j 0 THE ELECTRONIC JOURNAL OF COMBINATORICS 10 2003 R19 1 I want to thank H. Prodinger for pointing out to me identity 4.7 in 1 and the paper 4 S.O. Warnaar for some helpful remarks and drawing my attention to 9 and R. Chapman and C. Krattenthaler for providing another simple proof of 3.2 . .