Riordan arrays and \(r\)-Stirling number identities. (English) Zbl 1502.05019
Summary: By using the theory of Riordan arrays, we establish four pairs of general \(r\)-Stirling number identities, which reduce to various identities on harmonic numbers, hyperharmonic numbers, the Stirling numbers of the first and second kind, the \(r\)-Stirling numbers of the first and second kind, and the \(r\)-Lah numbers. We further discuss briefly the connections between the \(r\)-Stirling numbers and the Cauchy numbers, the generalized hyperharmonic numbers, and the poly-Bernoulli polynomials. Many known identities are shown to be special cases of our results, and the combinatorial interpretations of several particular identities are also presented as supplements.
MSC:
| 05A19 | Combinatorial identities, bijective combinatorics |
| 05A15 | Exact enumeration problems, generating functions |
| 11B73 | Bell and Stirling numbers |
| 15B36 | Matrices of integers |
Keywords:
Riordan arrays; \(r\)-Stirling numbers; hyperharmonic numbers; combinatorial identities; combinatorial interpretationsReferences:
| [1] | Bayad, A.; Hamahata, Y., Polylogarithms and poly-Bernoulli polynomials, Kyushu J. Math., 65, 1, 15-24 (2011) · Zbl 1268.11030 |
| [2] | Barry, P., Riordan Arrays: A Primer (2017), Logic Press: Logic Press Raleigh |
| [3] | Benjamin, A. T.; Gaebler, D.; Gaebler, R., A combinatorial approach to hyperharmonic numbers, Integers, 3, Article A15 pp. (2003) · Zbl 1128.11309 |
| [4] | Belbachir, H.; Belkhir, A., Cross recurrence relations for r-Lah numbers, Ars Comb., 110, 199-203 (2013) · Zbl 1313.11034 |
| [5] | Belbachir, H.; Bousbaa, I. E., Combinatorial identities for the r-Lah numbers, Ars Comb., 115, 453-458 (2014) · Zbl 1340.05010 |
| [6] | Boutiche, M. A.; Rahmani, M.; Srivastava, H. M., Explicit formulas associated with some families of generalized Bernoulli and Euler polynomials, Mediterr. J. Math., 14, 2, Article 89 pp. (2017) · Zbl 1402.11034 |
| [7] | Broder, A. Z., The r-Stirling numbers, Discrete Math., 49, 3, 241-259 (1984) · Zbl 0535.05006 |
| [8] | Can, M.; Dağlı, M. C., Extended Bernoulli and Stirling matrices and related combinatorial identities, Linear Algebra Appl., 444, 114-131 (2014) · Zbl 1285.05013 |
| [9] | Cheon, G.-S.; El-Mikkawy, M. E.A., Generalized harmonic numbers with Riordan arrays, J. Number Theory, 128, 2, 413-425 (2008) · Zbl 1131.05011 |
| [10] | Cheon, G.-S.; Hwang, S.-G.; Lee, S.-G., Several polynomials associated with the harmonic numbers, Discrete Appl. Math., 155, 18, 2573-2584 (2007) · Zbl 1130.11011 |
| [11] | Cheon, G.-S.; Jung, J.-H., r-Whitney numbers of Dowling lattices, Discrete Math., 312, 15, 2337-2348 (2012) · Zbl 1246.05009 |
| [12] | Comtet, L., Advanced Combinatorics (1974), D. Reidel Publishing Co.: D. Reidel Publishing Co. Dordrecht · Zbl 0283.05001 |
| [13] | Conway, J. H.; Guy, R. K., The Book of Numbers (1996), Copernicus: Copernicus New York · Zbl 0866.00001 |
| [14] | Dil, A.; Mező, I.; Cenkci, M., Evaluation of Euler-like sums via Hurwitz zeta values, Turk. J. Math., 41, 6, 1640-1655 (2017) · Zbl 1424.11130 |
| [15] | Graham, R. L.; Knuth, D. E.; Patashnik, O., Concrete Mathematics (1994), Addison-Wesley Publishing Company: Addison-Wesley Publishing Company Reading, MA · Zbl 0836.00001 |
| [16] | Guo, B.-N.; Mező, I.; Qi, F., An explicit formula for Bernoulli polynomials in terms of r-Stirling numbers of the second kind, Rocky Mt. J. Math., 46, 6, 1919-1923 (2016) · Zbl 1371.11045 |
| [17] | He, T.-X.; Shapiro, L. W., Fuss-Catalan matrices, their weighted sums, and stabilizer subgroups of the Riordan group, Linear Algebra Appl., 532, 25-42 (2017) · Zbl 1441.05016 |
| [18] | Jagerman, D. L., Difference Equations with Applications to Queues (2000), Marcel Dekker, Inc.: Marcel Dekker, Inc. New York · Zbl 0963.39001 |
| [19] | Jordan, C., Calculus of Finite Differences (1965), Chelsea Publishing Co.: Chelsea Publishing Co. New York · Zbl 0154.33901 |
| [20] | Kargın, L.; Can, M., Harmonic number identities via polynomials with r-Lah coefficients, C. R. Math. Acad. Sci. Paris, 358, 5, 535-550 (2020) · Zbl 1478.11032 |
| [21] | Kargın, L.; Cenkci, M.; Dil, A.; Can, M., Generalized harmonic numbers via poly-Bernoulli polynomials, Publ. Math. Debrecen, 100, 3-4, 365-386 (2022) · Zbl 1499.11145 |
| [22] | Komatsu, T., Recurrence relations of poly-Cauchy numbers by the r-Stirling transform, Mediterr. J. Math., 19, 1, Article 37 pp. (2022) · Zbl 1512.11030 |
| [23] | Komatsu, T.; Mező, I., Several explicit formulae of Cauchy polynomials in terms of r-Stirling numbers, Acta Math. Hung., 148, 2, 522-529 (2016) · Zbl 1374.05018 |
| [24] | Merlini, D.; Sprugnoli, R.; Verri, M. C., The Cauchy numbers, Discrete Math., 306, 16, 1906-1920 (2006) · Zbl 1098.05008 |
| [25] | Merris, R., The p-Stirling numbers, Turk. J. Math., 24, 4, 379-399 (2000) · Zbl 0970.05003 |
| [26] | Mező, I., A new formula for the Bernoulli polynomials, Results Math., 58, 3-4, 329-335 (2010) · Zbl 1237.11010 |
| [27] | Mező, I., The r-Bell numbers, J. Integer Seq., 14, 1, Article 11.1.1 pp. (2011) · Zbl 1205.05017 |
| [28] | Mihoubi, M.; Rahmani, M., The partial r-Bell polynomials, Afr. Math., 28, 7-8, 1167-1183 (2017) · Zbl 1387.05018 |
| [29] | Mihoubi, M.; Tiachachat, M., Some applications of the r-Whitney numbers, C. R. Math. Acad. Sci. Paris, 352, 12, 965-969 (2014) · Zbl 1333.11023 |
| [30] | Nörlund, N. E., Vorlesungen über Differenzenrechnung (1924), Springer: Springer Berlin · JFM 50.0315.02 |
| [31] | Nyul, G.; Rácz, G., The r-Lah numbers, Discrete Math., 338, 10, 1660-1666 (2015) · Zbl 1315.05018 |
| [32] | Ohno, Y.; Sasaki, Y., Recursion formulas for poly-Bernoulli numbers and their applications, Int. J. Number Theory, 17, 1, 175-189 (2021) · Zbl 1478.11028 |
| [33] | Rahmani, M., On p-Bernoulli numbers and polynomials, J. Number Theory, 157, 350-366 (2015) · Zbl 1332.11027 |
| [34] | Roman, S., The Umbral Calculus (1984), Academic Press, Inc.: Academic Press, Inc. New York · Zbl 0536.33001 |
| [35] | Shapiro, L. W.; Getu, S.; Woan, W. J.; Woodson, L. C., The Riordan group, Discrete Appl. Math., 34, 1-3, 229-239 (1991) · Zbl 0754.05010 |
| [36] | Shapiro, L. W.; Sprugnoli, R.; Barry, P.; Cheon, G.-S.; He, T.-X.; Merlini, D.; Wang, W., The Riordan Group and Applications, Springer Monographs in Mathematics (2022), Springer: Springer Cham · Zbl 1498.05002 |
| [37] | Sprugnoli, R., Riordan arrays and combinatorial sums, Discrete Math., 132, 1-3, 267-290 (1994) · Zbl 0814.05003 |
| [38] | Sprugnoli, R., Riordan arrays and the Abel-Gould identity, Discrete Math., 142, 1-3, 213-233 (1995) · Zbl 0832.05007 |
| [39] | Spieß, J., Some identities involving harmonic numbers, Math. Comput., 55, 192, 839-863 (1990) · Zbl 0724.05005 |
| [40] | Wang, C.; Miska, P.; Mező, I., The r-derangement numbers, Discrete Math., 340, 7, 1681-1692 (2017) · Zbl 1415.05007 |
| [41] | Wang, W., Generalized higher order Bernoulli number pairs and generalized Stirling number pairs, J. Math. Anal. Appl., 364, 1, 255-274 (2010) · Zbl 1221.11061 |
| [42] | Wang, W., Riordan arrays and harmonic number identities, Comput. Math. Appl., 60, 5, 1494-1509 (2010) · Zbl 1201.11028 |
| [43] | Zave, D. A., A series expansion involving the harmonic numbers, Inf. Process. Lett., 5, 3, 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.
