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# Mathematica Slovaca

Editor-in-Chief: Pulmannová, Sylvia

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Volume 64, Issue 4

# On some new identities for the Fibonomial coefficients

Diego Marques
/ Pavel Trojovský
• Department of Mathematics Faculty of Science, University of Hradec Králové, Rokitanského 62, Hradec Králové, Czech Republic
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Published Online: 2014-09-07 | DOI: https://doi.org/10.2478/s12175-014-0241-7

## Abstract

Let F n be the nth Fibonacci number. The Fibonomial coefficients $$\left[ {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right]_F$$ are defined for n ≥ k > 0 as follows $$\left[ {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right]_F = \frac{{F_n F_{n - 1} \cdots F_{n - k + 1} }} {{F_1 F_2 \cdots F_k }},$$ with $$\left[ {\begin{array}{*{20}c} n \\ 0 \\ \end{array} } \right]_F = 1$$ and $$\left[ {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right]_F = 0$$. In this paper, we shall provide several identities among Fibonomial coefficients. In particular, we prove that $$\sum\limits_{j = 0}^{4l + 1} {\operatorname{sgn} (2l - j)\left[ {\begin{array}{*{20}c} {4l + 1} \\ j \\ \end{array} } \right]_F F_{n - j} = \frac{{F_{2l - 1} }} {{F_{4l + 1} }}\left[ {\begin{array}{*{20}c} {4l + 1} \\ {2l} \\ \end{array} } \right]_F F_{n - 4l - 1} ,}$$ holds for all non-negative integers n and l.

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## About the article

Published Online: 2014-09-07

Published in Print: 2014-08-01

Citation Information: Mathematica Slovaca, Volume 64, Issue 4, Pages 809–818, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918,

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