Merlini, Donatella; Sprugnoli, Renzo; Verri, M. Cecilia The Cauchy numbers. (English) Zbl 1098.05008 Discrete Math. 306, No. 16, 1906-1920 (2006). Summary: We study many properties of Cauchy numbers in terms of generating functions and Riordan arrays and find several new identities relating these numbers with Stirling, Bernoulli and harmonic numbers. We also reconsider the Laplace summation formula showing some applications involving the Cauchy numbers. Cited in 61 Documents MSC: 05A15 Exact enumeration problems, generating functions 05A10 Factorials, binomial coefficients, combinatorial functions 11B68 Bernoulli and Euler numbers and polynomials 11B73 Bell and Stirling numbers Keywords:generating functions; Riordan arrays; Laplace summation formula PDFBibTeX XMLCite \textit{D. Merlini} et al., Discrete Math. 306, No. 16, 1906--1920 (2006; Zbl 1098.05008) Full Text: DOI Online Encyclopedia of Integer Sequences: Numerators of Cauchy numbers of second type (= Bernoulli numbers B_n^{(n)}). Denominators of Cauchy numbers of second type (= Bernoulli numbers B_n^{(n)}). Numerators of Cauchy numbers of first type. Denominators of Cauchy numbers of first type. References: [1] Boole, G., An Investigation of the Law of Thought (1958), Dover: Dover New York, (reproduction of 1984 edition) [2] Comtet, L., Advanced Combinatorics (1974), Reidel: Reidel Dordrecht [3] Greene, D. H.; Knuth, D. E., Mathematics for the Analysis of Algorithms (1982), Birkäuser: Birkäuser Boston [4] Henrici, P., Applied and Computational Complex Analysis, I (1988), Wiley: Wiley New York [5] Jagerman, D. L., Difference Equations with Applications to Queues (2000), Marcel Dekker: Marcel Dekker New York · Zbl 0963.39001 [6] D. Merlini, R. Sprugnoli, M.C. Verri, The method of coefficients, Amer. Math. Monthly, accepted for publication.; D. Merlini, R. Sprugnoli, M.C. Verri, The method of coefficients, Amer. Math. Monthly, accepted for publication. · Zbl 1191.05006 [7] Milne-Thomson, L. M., The Calculus of Finite Differences (1951), Macmillan and Co.: Macmillan and Co. London · Zbl 0008.01801 [8] Shapiro, L. W.; Getu, S.; Woan, W.-J.; Woodson, L., The Riordan group, Discrete Appl. Math., 34, 229-239 (1991) · Zbl 0754.05010 [9] Sprugnoli, R., Riordan arrays and combinatorial sums, Discrete Math., 132, 267-290 (1994) · Zbl 0814.05003 [10] Wang, T.; Zhao, X., Some identities related to reciprocal functions, Discrete Math., 265, 323-335 (2003) · Zbl 1017.05022 [11] Wang, T.; Zhao, X.; Ding, S., Some summation rules related to Riordan arrays, Discrete Math., 281, 295-307 (2004) · Zbl 1042.05009 [12] Wilf, H. S., Generating Functionology (1990), Academic Press: Academic Press Boston [13] Yingying, L., On Euler’s constant—calculating sums by integrals, Amer. Math. Monthly, 109, 845-850 (2002) · Zbl 1027.40004 [14] Zave, D. A., A series expansion involving the harmonic numbers, Inform. Process. Lett., 5, 75-77 (1976) · Zbl 0359.65012 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.