## The Lucas congruence for Stirling numbers of the second kind

### Volume 94 / 2000

#### Abstract

0. Introduction. The numbers introduced by Stirling in 1730 in his Methodus differentialis [11], subsequently called "Stirling numbers" of the first and second kind, are of the greatest utility in the calculus of finite differences, in number theory, in the summation of series, in the theory of algorithms, in the calculation of the Bernstein polynomials [9]. In this study, we demonstrate some properties of Stirling numbers of the second kind similar to those satisfied by binomial coefficients; in particular we show that they satisfy a congruence analogous to that of Lucas, that is to: $(a \atop b) ≡ ∏_{i=0}^{n} (a_{i} \atop b_{i}) mod p$ with $a = ∑_{i=0}^{n} a_{i} p^i$, $b = ∑_{i=0}^{n} b_{i} p^{i}$; $0 ≤ a_i ≤ p-1$, $0 ≤ b_i ≤ p-1$. Using Proposition 4.1 we give another proof for Kaneko's recurrence formula for poly-Bernoulli numbers [10]. Some of the results are similar to those of Howard [5]. In conclusion, I wish to give my best thanks to the Geometry Group of the Dipartimento di Matematica Pura ed Applicata and Dipartimento di Metodi Matematici per le Scienze Applicate of the University of Padova, for support and help given during the preparation of this work. In particular, I wish to thank Frank Sullivan for his precious advice and suggestions.