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)
It is well known (cf. ) that the Fibonacci polynomials 1 2 1 2 0 1 (,) n j n j n j n j F x s s x j and the Lucas polynomials 2 2 0 (,) n j n j n j n j n L x s s x j n j with 0 (,) 2 L x s satisy (,). n n n L x y xy x y Consider the Kyle map 2 (1) (1) j j n j j j j j K q q which (as Richard Stanley  observed)… (More)