5,863 research outputs found
On Binomial Identities in Arbitrary Bases
We extend the digital binomial identity as given by Nguyen el al. to an
identity in an arbitrary base , by introducing the ary binomial
coefficients. We then study the properties of these coefficients such as
orthogonality, a link to Lucas' theorem and the corresponding ary Pascal
triangles
A matrix generalization of a theorem of Fine
In 1947 Nathan Fine gave a beautiful product for the number of binomial
coefficients , for in the range , that are
not divisible by . We give a matrix product that generalizes Fine's formula,
simultaneously counting binomial coefficients with -adic valuation
for each . For each this information is naturally encoded in
a polynomial generating function, and the sequence of these polynomials is
-regular in the sense of Allouche and Shallit. We also give a further
generalization to multinomial coefficients.Comment: 9 pages; publication versio
- …