Abstract
By using the restricted and associated Stirling numbers of the first kind and by generalizing the (unsigned) Stirling numbers of the first kind, we define the generalized incomplete poly-Cauchy numbers by combining the generalized and the incomplete poly-Cauchy numbers, and study their arithmetical and combinatorial properties. We also study the corresponding generalized incomplete poly-Bernoulli numbers.
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References
M. Cenkci, T. Komatsu, Poly-Bernoulli numbers and polynomials with a \(q\) parameter. J. Number Theory 152, 38–54 (2015)
C.A. Charalambides, Enumerative Combinatorics (Discrete Mathematics and Its Applications) (Chapman and Hall/CRC, London, 2002)
L. Comtet, Advanced Combinatorics (Reidel, Dordrecht, 1974)
F.T. Howard, Associated Stirling numbers. Fibonacci Quart. 18, 303–315 (1980)
Ch. Jordan, Calculus of Finite Differences (Chelsea Publ. Co., New York, 1950)
M. Kaneko, Poly-Bernoulli numbers. J. Th. Nombres Bordeaux 9, 221–228 (1997)
T. Komatsu, Poly-Cauchy numbers. Kyushu J. Math. 67, 143–153 (2013)
T. Komatsu, Poly-Cauchy numbers with a \(q\) parameter. Ramanujan J. 31, 353–371 (2013)
T. Komatsu, Hypergeometric Cauchy numbers. Int. J. Number Theory 9, 545–560 (2013)
T. Komatsu, Incomplete poly-Cauchy numbers. Monatsh. Math. 180, 271–288 (2016)
T. Komatsu, V. Laohakosol, K. Liptai, A generalization of poly-Cauchy numbers and their properties, Abstr. Appl. Anal. 2013(2013), Article ID 179841, 8 pages
T. Komatsu, K. Liptai, I. Mező, Incomplete poly-Bernoulli numbers associated with incomplete Stirling numbers. Publ. Math. Debrecen 88, 357–368 (2016)
T. Komatsu, I. Mező, L. Szalay, Incomplete Cauchy numbers. Acta Math. Hungar. 149, 306–323 (2016)
T. Komatsu, L. Szalay, Shifted poly-Cauchy numbers. Lith. Math. J. 54, 166–181 (2014)
D. Merlini, R. Sprugnoli, M.C. Verri, The Cauchy numbers. Discret. Math. 306, 1906–1920 (2006)
I. Mező, Periodicity of the last digits of some combinatorial sequences, J. Integer Seq. 17 (2014), Article 14.1.1
J. Riordan, Combinatorial Identities (Wiley, New York, 1968)
Y. Sasaki, On generalized poly-Bernoulli numbers and related \(L\)-functions. J. Number Theory 132, 156–170 (2012)
F.-Z. Zhao, Some properties of associated Stirling numbers, J. Integer Seq. 11 (2008), Article 08.1.7, 9 pages
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The author was supported in part by the grant of Wuhan University and by the grant of Hubei Provincial Experts Program.
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Komatsu, T. Generalized incomplete poly-Bernoulli and poly-Cauchy numbers. Period Math Hung 75, 96–113 (2017). https://doi.org/10.1007/s10998-016-0167-7
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DOI: https://doi.org/10.1007/s10998-016-0167-7