Definition:Fibonomial Coefficient

Definition

Let $n \in \Z_{\ge 0}$ and $k \in \Z$.

Then the Fibonomial coefficient $\dbinom n k$ is defined as:

$\dbinom n k_\FF = \begin{cases} 0 & : n < 0, n > k \\

1 & : n \ge 0, k = 0 \\ \dfrac {F_n F_{n - 1} \cdots F_{n - k + 1} } {F_k F_{k - 1} \cdots F_1} = \ds \prod_{j \mathop = 1}^k \dfrac {F_{n - k + j} } {F_j} & : \text{otherwise} \end{cases}$

where $F_n$ denotes the $n$th Fibonacci number.

Also known as

Some sources use the more verbose but also more descriptive Fibonacci-binomial coefficient.

Also see

• Results about Fibonomial coefficients can be found here.

Source of Name

This entry was named for Leonardo Fibonacci.

Historical Note

The Fibonomial coefficients were initially defined by François Édouard Anatole Lucas.

Sources

• 1878: Édouard Lucas: Théorie des Fonctions Numériques Simplement Périodiques (American Journal of Mathematics Vol. 1, no. 3: pp. 197 – 240)  www.jstor.org/stable/2369311
• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.8$: Fibonacci Numbers: Exercise $29$