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We introduce a new q-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.

Morse code sequences are very useful to give combinatorial interpretations of various properties of Fibonacci numbers. In this note we study some algebraic and combinatorial aspects of Morse code sequences and obtain several q-analogues of Fibonacci numbers and Fibonacci polynomials and their generalizations. 1 Morse code polynomials Morse code sequences… (More)

- Johann Cigler
- 2000

0. INTRODUCTION Let MC be the monoid of all Morse code sequences of dots a(:=®) and dashes b(: =-) with respect to concatenation. MC consists of all words in a and b. Let P be the algebra of all polyno-mials H veMC K v w ^h r e a l coefficients. We are interested in: a) polynomials in P which we call abstract Fibonacci polynomials. They are defined by the… (More)

- Johann Cigler
- 2014

In this note we give a survey about polynomials whose moments are multiples of super Catalan numbers and explore two different kinds of q analogues. The moments of Lucas polynomials and of Chebyshev polynomials of the first kind are (multiples of) central binomial coefficients and the moments of Fibonacci polynomials and of Chebyshev polynomials of the… (More)

- Johann Cigler
- 2013

We give a survey of some interesting q analogues of Fibonacci, Lucas, Chebyshev and related polynomials and their moments. 0. Introduction The moments of the Fibonacci polynomials or equivalently of the Chebyshev polynomials of the second kind are (multiples of) the Catalan numbers 2 1 1 n n C n n and the moments of the Lucas polynomials… (More)

- Johann Cigler
- 2009

We give simple proofs for the Hankel determinants of q − exponential polynomials. Let (,) S n k be the Stirling numbers of the second kind. Christian Radoux ([6]) has shown that the Hankel determinants of the exponential polynomials 0 () (,) n k n k B x S n k x = = ∑ are given by () 1 1 2 , 0 0 det () !. In [2] I have proved some q − analogues of this… (More)

- JOHANN CIGLER, JIANG ZENG
- 2009

Two well-known q-Hermite polynomials are the continuous and discrete q-Hermite polynomials. In this paper we consider two new q-Hermite polynomials and prove several curious properties about these polynomials. One striking property is the connection with q-Fibonacci polynomials and the recent works on the combinatorics of the Matrix Ansatz of the PASEP.

- Johann Cigler
- 2008

In [1] and in my unpublished survey paper [2] I have calculated Hankel determinants for the coefficients of two classes of generating functions which are associated with certain q − exponential functions. It turns out that the methods used in [3] give simpler proofs of these facts. Since the proofs are simple extensions of the proofs in [3] I shall only… (More)