Binomial coefficients can be defined by

Edouard Lucas defined the Fibonomial coefficients analogously by

where *F*_{j} is the *j*th Fibonacci number.

Binomial coefficients satisfy

and Fibonomial coefficients satisfy an analogous identity

Incidentally, this identity can be used to show that Fibonomial coefficients are integers, which isn’t obvious from the definition.

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Has this notion proven useful for anything?

Not that I know of, but I just ran across it tonight, so I’m no authority.

I imagine it has applications in combinatorics, but I’m guessing. Several people have written papers on it.

Um… to prove that Fibonacci are integers, isn’t it easier to use the fact that they satisfy the recursion

F_k = F_{k-1} + F_{k-2}

My comment wasn’t about the Fibonacci numbers but the Fibonomial coefficients. It’s obvious that Fibonacci numbers are integers, but it’s not obvious that the ratio of Fibonacci products defining the Fibonomial coefficients is always an integer.