Johann Cigler

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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(More)
1 Morse code polynomials Morse code sequences are finite sequences of dots (•) and dashes (−). If a dot has length 1 and a dash has length 2 then the number of all such sequences of total length n−1 is the Fibonacci number Fn, which is defined as the sequence of numbers satisfying the recursion Fn = Fn−1 + Fn−2 with initial conditions F0 = 0 and F1 = 1. If(More)
of n words of length r obtained by splitting all permutations of {O, 1, ... , rn I} into n pieces of equal length r. It suffices therefore to give a bijection between the set of all n-fold products formed from (r l)n + 1 different elements XQ, Xl' ... , x(r-l)n and the set A~. To simplify the exposition we shall write i instead of Xi and consider the usual(More)
Two well-known q-Hermite polynomials are the continuous and discrete q-Hermite polynomials. In this paper we consider a new family of q-Hermite polynomials and prove several curious properties about these polynomials. One striking property is the connection with q-Fibonacci and q-Lucas polynomials. The latter relation yields a generalization of the(More)
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 polynomials HveMCK ^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 recursion Fn(a, b) =(More)
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.
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