Difference between revisions of "Q-Fibonacci polynomials"
From specialfunctionswiki
(Created page with "The $q$-Fibonacci polynomials are defined by $$F_n(x,s,q) = \left\{ \begin{array}{ll} 0 &; n=0 \\ 1 &; n=1 \\ xF_{n-1}(x,s,q)+t(q^{n-2}s)F_{n-2}(x,s,q), \end{array} \right.$$...") |
|||
| Line 19: | Line 19: | ||
=References= | =References= | ||
[http://www.fq.math.ca/Scanned/41-1/cigler.pdf $q$-Fibonacci polynomials] | [http://www.fq.math.ca/Scanned/41-1/cigler.pdf $q$-Fibonacci polynomials] | ||
| + | |||
| + | [[Category:SpecialFunction]] | ||
Revision as of 18:55, 24 May 2016
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: █
