Journal of Integer Sequences, Vol. 24 (2021), Article 21.1.4 |

Normandie Université

UNICAEN, CNRS

Laboratoire de Mathématiques Nicolas Oresme

14000 Caen

France

**Abstract:**

To determine the *p*-adic valuation of a Lucasnomial
,
i.e., of a generalized binomial coefficient with respect to a fundamental
Lucas sequence *U*=*U*(*P*,*Q*), there is an adequate Kummer rule if *p* is
a regular prime. No such rule exists if *p* is a special prime, i.e.,
if *p* divides .
We provide a complete description of the
*p*-adic valuation of Lucasnomials when *p* is special, with some
numerical examples. Applications to the integrality of generalized
Lucasnomial Fuss-Catalan numbers, Lucasnomial ballot and Lucasnomial
Lobb numbers are also given.

Received July 3 2020; revised versions received December 12 2020; December 31 2020.
Published in *Journal of Integer Sequences*,
December 31 2020.

Return to