On the Generalized Binomial Coefficients Defined by Strong Divisibility Sequences

Shiro Ando and Daihachiro Sato

A chapter in Applications of Fibonacci Numbers, 1999, pp 1-10 from Springer

Abstract: Abstract Given a positive integer sequence {a n}, where n = 1,2,3,…, we replace n! in the defining form $$\binom{n}{r}=\frac{n!}{r!s!}, \ \ r+s=n$$ of the binomial coefficients with $$\prod {{{a}_{n}}} = {{a}_{n}}{{a}_{{n - 1}}} \cdots {{a}_{2}}{{a}_{1}} $$ , where we define $$\prod a_{0}=1$$ . When the resulting numbers $$\begin{bmatrix}n \\ r \end{bmatrix}=\frac{\prod a_{n}}{\prod a_{r}\prod a_{s}}, \ \ r+s=n$$ are all integers, we call them the generalized binomial coefficients defined by {a n}, and the sequence {a n} a Raney sequence. In particular, the Fibonacci sequence {F n} is a Raney sequence, and the generalized binomial coefficients defined by it are called Fibonomial coefficients.

Keywords: 11A05; 11B65; 11B83 (search for similar items in EconPapers)
Date: 1999
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DOI: 10.1007/978-94-011-4271-7_1

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