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On some identities for the Fibonomial coefficients. (English) Zbl 1108.11019
Summary: The Fibonomial coefficients $$\Bigl [\begin{smallmatrix} n\\k\end{smallmatrix} \Bigr ]$$ are defined for positive integers $$n\geq k$$ as follows $\begin{bmatrix} n \\ k \end{bmatrix} =\frac {F_n F_{n-1}\cdots F_{n-k+1}}{F_1 F_2 \cdots F_k}\;,$ with $$\Bigl [\begin{smallmatrix} n\\0 \end{smallmatrix} \Bigr ]=1$$, where the Fibonacci numbers are given by the recurrence relation $$F_{n+2}=F_{n+1}+F_{n}$$, $$F_0=0$$, $$F_1=1$$. In this paper new identities for the Fibonomial coefficients are derived. These identities are related to the generating function of the $$k$$th powers of the Fibonacci numbers. Their proofs are based on a reasonable manipulation with these generating functions.

##### MSC:
 11B39 Fibonacci and Lucas numbers and polynomials and generalizations 05A15 Exact enumeration problems, generating functions 05A19 Combinatorial identities, bijective combinatorics
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##### References:
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