# The Lucas congruence for Stirling numbers of the second kind

Acta Arithmetica (2000)

• Volume: 94, Issue: 1, page 41-52
• ISSN: 0065-1036

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## Abstract

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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: $\genfrac{}{}{0pt}{}{\left(a}{\genfrac{}{}{0pt}{}{b\right)\equiv {\prod }_{i=0}^{n}\left({a}_{i}}{{b}_{i}\right)modp}}$ with $a={\sum }_{i=0}^{n}{a}_{i}{p}^{i}$, $b={\sum }_{i=0}^{n}{b}_{i}{p}^{i}$; $0\le {a}_{i}\le p-1$, $0\le {b}_{i}\le p-1$. 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.

## How to cite

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Sánchez-Peregrino, Roberto. "The Lucas congruence for Stirling numbers of the second kind." Acta Arithmetica 94.1 (2000): 41-52. <http://eudml.org/doc/207424>.

@article{Sánchez2000,
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.},
author = {Sánchez-Peregrino, Roberto},
journal = {Acta Arithmetica},
keywords = {Stirling numbers},
language = {eng},
number = {1},
pages = {41-52},
title = {The Lucas congruence for Stirling numbers of the second kind},
url = {http://eudml.org/doc/207424},
volume = {94},
year = {2000},
}

TY - JOUR
AU - Sánchez-Peregrino, Roberto
TI - The Lucas congruence for Stirling numbers of the second kind
JO - Acta Arithmetica
PY - 2000
VL - 94
IS - 1
SP - 41
EP - 52
AB - 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.
LA - eng
KW - Stirling numbers
UR - http://eudml.org/doc/207424
ER -

## References

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1. [1] H. W. Becker and J. Riordan, The arithmetic of Bell and Stirling numbers, Amer. J. Math. 70 (1948), 385-394. Zbl0036.02302
2. [2] L. Carlitz, Weighted Stirling numbers of the first and second kind, I, Fibonacci Quart. 18 (1970), 147-162. Zbl0428.05003
3. [3] L. Comtet, Analyse Combinatoire, Tomes i, ii, Presses Univ. de France, Paris, 1970.
4. [4] A. E. Fekete, Apropos two notes on notation, Amer. Math. Monthly 99 (1992), 403-422.
5. [5] F. T. Howard, Congruences for the Stirling numbers and associated Stirling numbers, Acta Arith. 55 (1990), 29-41. Zbl0648.10008
6. [6] C. Jordan, Calculus of Finite Differences, Chelsea, New York, 1960.
7. [7] M. Kaneko, Poly-Bernoulli numbers, J. Théor. Nombres Bordeaux 9 (1997), 221-228.
8. [8] N. Koblitz, p-adic Numbers, p-adic Analysis, and Zeta-Functions, Grad. Texts in Math. 58, Springer, Berlin, 1977.
9. [9] R. Sánchez-Peregrino, Identité de Bernstein pour une fonction homogène à singularité isolée, Rend. Sem. Mat. Univ. Padova 81 (1989), 221-227. Zbl0684.32007
10. [10] R. Sánchez-Peregrino, Another proof of Kaneko's recurrence formula for the Bernoulli numbers, pre- print.
11. [11] J. Stirling, Methodus differentialis, sive tractatus de summatione et interpolazione serierum infinitarum, Londini, 1730.

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