Lucasnomial Fuss-Catalan Numbers and Related Divisibility Questions

For all integers $n\ge1$ we define the generalized Lucasnomial Fuss-Catalan numbers

$\begin{displaymath}C_{U,a,r}(n):=\frac{U_r}{U_{(a-1)n+r}}\binom{an+r-1}{n}_U,\end{displaymath}$

and prove their integrality. Here U is a fundamental Lucas sequence, $a\ge2$ and $r\ge1$ are integers, and $\binom{*}{*}_U$ denotes a Lucasnomial coefficient. If U = I, where I_n = n, then the C_I,a,r(n) are the usual generalized Fuss-Catalan numbers. With the assumption that U is regular, we show that U_(a-1)n+k divides $\binom{an}{n}_U$ for all $n\ge1$ but a set of asymptotic density 0 if $k\ge1$ , but only for a small set if $k\le0$ . This small set is finite when $U\not=I$ and at most of upper asymptotic density $1-\log 2$ when U = I. We also determine all triples (U,a,k), where $k\ge2$ , for which the exceptional set of density 0 is actually finite, and in fact empty.

Lucasnomial Fuss-Catalan Numbers and Related Divisibility Questions

Christian Ballot Département de Mathématiques et Informatique Université de Caen-Normandie France

Christian Ballot
Département de Mathématiques et Informatique
Université de Caen-Normandie
France