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

The p-Adic Valuation of Lucasnomials When p is a Special Prime

Christian Ballot
Normandie Université
Laboratoire de Mathématiques Nicolas Oresme
14000 Caen


To determine the p-adic valuation of a Lucasnomial $\binom{m}{n}_U$, 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 $\gcd(P,Q)$. 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.

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

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