Journal of Integer Sequences, Vol. 19 (2016), Article 16.5.5

On Binomial Identities in Arbitrary Bases


Lin Jiu
Department of Mathematics
Tulane University
New Orleans, LA 70118
USA

Christophe Vignat
Department of Mathematics
Tulane University
New Orleans, LA 70118
USA
and
Department of Physics
Université Orsay Paris Sud
91405 Orsay cedex
France

Abstract:

We first extend the digital binomial identity as given by Nguyen et al. to an identity in an arbitrary base b, by introducing the b-ary binomial coefficients. Then, we study the properties of these coefficients such as their orthogonality, their link with Lucas theorem and their extension to multinomial coefficients. Finally, we analyze the structure of the corresponding b-ary Pascal-like triangles.

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Received February 12 2016; revised versions received April 15 2016; May 15 2016. Published in Journal of Integer Sequences, June 2 2016. Minor revision, November 13 2016. New corrected version and corrigendum posted, September 25 2019.

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