Abstract
Let s(n,k) and S(n,k) be the (unsigned) Stirling numbers of the first and second kinds respectively [8, pp. 204–219]. In [1] and [10] congruence formulas (mod p), p prime, are proved for s(n,k) and S(n,k). As applications, the residues (mod p) for p = 2, 3, and 5 are worked out for both kinds of Stirling numbers. In [10] similar formulas are proved for the associated Stirling numbers d(n,k) and b(n,k).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Carlitz, L. “Some Partition Problems Related to the Stirling Numbers of the Second Kind.” Acta Arithmetica 10 (1965): pp. 409–422.
Carlitz, L. “Weighted Stirling Numbers of the First and Second Kind-I.” The Fibonacci Quarterly 18 (1980): pp. 147–162.
Carlitz, L. “Weighted Stirling Numbers of the First and Second Kind-II.” The Fibonacci Quarterly 18 (1980): pp. 242–257.
Carlitz, L. “Degenerate Stirling, Bernoulli and Eulerian Numbers.” Utilitas Math. 15 (1979): pp. 51–88.
Carlitz, L. “Congruences for Generalized Bell and Stirling Numbers.” Duke Math J. 22 (1955): pp. 103–205.
Charalambides, Ch. A. “On Weighted Stirling and Other Related Numbers and Some Combinatorial Applications.” The Fibonacci Quarterly, 22 (1984): pp. 296–309.
Charalambides, Ch. A. and Koutras, M. “On the Differences of the Generalized Factorials at an Arbitrary Point and Their Combinatorial Applications.” Discrete Mathematics, 47 (1983): pp. 183–201.
Comtet, L. Advanced Combinatorics. Dordrecht: Reidel, 1974.
Howard, F. T. “Degenerate Weighted Stirling Numbers.” Discrete Math. 57 (1985): pp. 45–58.
Howard, F. T. “Congruences for the Stirling Numbers and Associated Stirling Numbers.” Acta Arithmetica, to appear.
Howard, F. T. “Weighted Associated Stirling Numbers.” The Fibonacci Quarterly 22 (1984): pp. 156–165.
Kwong, Y. H. H. “Minimum Periods of S(n,k) Modulo M.” The Fibonacci Quarterly, to appear.
Nijenhuis, A. and Wilf, H. S. “Periodicities of Partition Functions and Stirling Numbers Modulo p.” J. Number Theory 25 (1987): pp. 308–312.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1990 Kluwer Academic Publishers
About this chapter
Cite this chapter
Howard, F.T. (1990). Congruences for Weighted and Degenerate Stirling Numbers. In: Bergum, G.E., Philippou, A.N., Horadam, A.F. (eds) Applications of Fibonacci Numbers. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-1910-5_18
Download citation
DOI: https://doi.org/10.1007/978-94-009-1910-5_18
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-7352-3
Online ISBN: 978-94-009-1910-5
eBook Packages: Springer Book Archive