Q-Fibonacci polynomials

The $q$-Fibonacci polynomials are defined by $$F_n(x,s,q) = \left\{ \begin{array}{ll} 0 &; n=0 \\ 1 &; n=1 \\ xF_{n-1}(x,s,q)+t(q^{n-2}s)F_{n-2}(x,s,q), \end{array} \right.$$ where $t(\xi) \neq 0$ is a function of a (real) variable $s$ and $q \neq 0$ is a real number.

Properties

Theorem: ($q$-Euler-Cassini formula) The $q$-Fibonacci polynomials satisfy the polynomial identity $$F_{n-1}(x,qs,q)F_{n+k}(x,s,q)-F_n(x,s,q)F_{n+k-1}(x,qs,q) = (-1)^nt(qs) \ldots t(q^{n-1}s)F_k(x,q^ns,q).$$

Proof: █

References

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