On Cauchy Numbers and Their Generalizations

Year 2020, , 456 - 474, 01.06.2020

Levent Kargın

https://doi.org/10.35378/gujs.604550

Cited By: 4

Abstract

This paper is concerned with both kinds of the Cauchy numbers and their generalizations. Taking into account Mellin derivative, we relate p-Cauchy numbers of the second kind with shifted Cauchy numbers of the first kind, which yields new explicit formulas for the Cauchy numbers of the both kind. We introduce a generalization of the Cauchy numbers and investigate several properties, including recurrence relations, convolution identities and generating functions. In particular, these results give rise to new identities for Cauchy numbers. 

Keywords

Cauchy numbers, Poly-Cauchy numbers, p-Cauchy numbers, Generating function, Recurrence relations

Project Number

FBA-2018-3723.

References

• [1] Comtet, L., Advanced Combinatorics, Reidel, Dordrecht, (1974).
• [2] Agoh, T. and Dilcher, K., “Recurrence relations for Nörlund numbers and Bernoulli numbers of the second kind”, Fibonacci Q., 48: 4-12, (2010).
• [3] Young, P.T., “A 2-adic formula for Bernoulli numbers of the second kind and for the Nörlund numbers”, J. Number Theory, 128: 2951-2962, (2008).
• [4] Nörlund, N. E.,Vorlesungen Äuber Direrenzenrechnung, Springer-Verlag, Berlin, (1924).
• [5] Cenkci, M. and Young, P.T., “Generalizations of poly-Bernoulli and poly-Cauchy numbers”, Eur. J. Math., 1:799-828, (2015).
• [6] Komatsu, T., “Hypergeometric Cauchy numbers”, Int. J. Number Theory, 9: 545-560, (2013).
• [7] Komatsu, T., Laohakosol,V., and Liptai, K., “A generalization of poly-Cauchy numbers and their properties”, Abstr. Appl. Anal., 2013: Article ID 179841, (2013).
• [8] Komatsu, T., “Poly-Cauchy numbers”, Kyushu J. Math., 67: 143-153, (2013).
• [9] Komatsu, T., “Poly-Cauchy numbers with a q parameter”, Raman. J., 31: 353-371, (2013).
• [10] Komatsu, T., “Incomplete poly-Cauchy numbers”, Monatsh. Math., 180: 271-288, (2016).
• [11] Komatsu, T., Mezö, I. and Szalay, L., “Incomplete Cauchy numbers”, Acta Math. Hungar., 149: 306-323, (2016).
• [12] Komatsu, T. and Young, P.T., “Generalized Stirling numbers with poly-Bernoulli and poly-Cauchy numbers”, Int. J. Number Theory, 14(05): 1211-1222, (2018).
• [13] Boyadzhiev, K.N., “Polyexponentials”, available from: http://arxiv.org/pdf/0710.1332v1.pdf.
• [14] Komatsu, T. and Szalay, L., “Shifted poly-Cauchy numbers”, Lith. Math. J., 54: 166-181, (2014).
• [15] Rahmani, M., “On p-Cauchy numbers”, Filomat, 30(10): 2731-2742, (2016).
• [16] Lah, I., “A new kind of numbers and its application in the actuarial mathematics”, Bol. Inst. Actuár. Port., 9: 7-15, (1954).
• [17] Rahmani, M., “Generalized Stirling transform”, Miskolc Math. Notes, 15: 677-690, (2014).
• [18] Komatsu, T., “Sums of products of Cauchy numbers, including poly-Cauchy numbers”, J. Discrete Math., 2013: Article ID373927, (2013).
• [19] Howard, F. T., Nörlund’s number B_n^n, Applications of Fibonacci Numbers, Vol. 5, Kluwer Acad. Publ., Dordrecht, (1993).
• [20] Zhao, F.Z., “Sums of products of Cauchy numbers”, Discrete Mathematics, 309(12): 3830-3842, (2009).