Two types of hypergeometric degenerate Cauchy numbers. (English) Zbl 1477.11046
Summary: In 1985, Howard introduced degenerate Cauchy polynomials together with degenerate Bernoulli polynomials. His degenerate Bernoulli polynomials have been studied by many authors, but his degenerate Cauchy polynomials have been forgotten. In this paper, we introduce some kinds of hypergeometric degenerate Cauchy numbers and polynomials from the different viewpoints. By studying the properties of the first one, we give their expressions and determine the coefficients. Concerning the second one, called H-degenerate Cauchy polynomials, we show several identities and study zeta functions interpolating these polynomials.
MSC:
| 11B75 | Other combinatorial number theory |
| 11B37 | Recurrences |
| 11C20 | Matrices, determinants in number theory |
| 11M41 | Other Dirichlet series and zeta functions |
| 15A15 | Determinants, permanents, traces, other special matrix functions |
| 33C05 | Classical hypergeometric functions, \({}_2F_1\) |
Keywords:
hypergeometric Cauchy numbers; degenerate Cauchy numbers; hypergeometric functions; zeta functions; determinantsReferences:
| [1] | F. T. Howard, Degenerate weighted Stirling numbers, Discrete Math. 57 (1985), 45-58. · Zbl 0606.10009 |
| [2] | L. Carlitz, Degenerate Stirling, Bernoulli, and Eulerian numbers, Utilitas Math. 15 (1979), 51-88. · Zbl 0404.05004 |
| [3] | F. T. Howard, Explicit formulas for degenerate Bernoulli numbers, Discrete Math. 162 (1996), 175-185. · Zbl 0873.11016 |
| [4] | P. T. Young, Degenerate Bernoulli polynomials, generalized factorial sums, and their applications, J. Number Theory 128 (2008), 738-758. · Zbl 1204.11053 |
| [5] | T. Komatsu, Hypergeometric degenerate Bernoulli polynomials and numbers, Ars Math. Contemp. (to appear), . · Zbl 1476.11047 · doi:10.26493/1855-3974.1907.3c2 |
| [6] | M. Cenkci and F. T. Howard, Notes on degenerate numbers, Discrete Math. 307 (2007), 2359-2375. · Zbl 1145.11018 |
| [7] | M. Aoki and T. Komatsu, Remarks on hypergeometric Cauchy numbers, Math. Rep. (Bucur), (to appear), arXiv:1802.05455. |
| [8] | T. Komatsu, Hypergeometric Cauchy numbers, Int. J. Number Theory 9 (2013), 545-560. · Zbl 1301.11030 |
| [9] | F. T. Howard, A sequence of numbers related to the exponential function, Duke Math. J. 34 (1967), 599-615. · Zbl 0189.04204 |
| [10] | F. T. Howard, Some sequences of rational numbers related to the exponential function, Duke Math. J. 34 (1967), 701-716. · Zbl 0189.04205 |
| [11] | K. Kamano, Sums of products of hypergeometric Bernoulli numbers, J. Number Theory 130 (2010), 2259-2271. · Zbl 1261.11021 |
| [12] | T. Komatsu, Complementary Euler numbers, Period. Math. Hungar. 75 (2017), 302-314. · Zbl 1413.11126 |
| [13] | T. Komatsu and H. Zhu, Hypergeometric Euler numbers, AIMS Math. 5 (2020), 1284-1303. · Zbl 1484.11077 |
| [14] | J. W. L. Glaisher, Expressions for Laplace’s coefficients, Bernoullian and Eulerian numbers etc. as determinants, Messenger 6 (1875), no. 2, 49-63. · JFM 08.0306.01 |
| [15] | M. Aoki, T. Komatsu, and G. K. Panda, Several properties of hypergeometric Bernoulli numbers, J. Inequal. Appl. 2019 (2019), 113, . · Zbl 1498.11071 · doi:10.1186/s13660-019-2066-y |
| [16] | Y. Komori and A. Yoshihara, Cauchy numbers and polynomials associated with hypergeometric Bernoulli numbers, J. Comb. Number Theory 9 (2017), no. 2, 123-142. · Zbl 1411.11025 |
| [17] | S. Hu and T. Komatsu, Explicit expressions for the related numbers of higher order Appell polynomials, Queast. Math. (2019), . · Zbl 1476.11039 · doi:10.2989/16073606.2019.1596174 |
| [18] | M. Rahmani, On p-Cauchy numbers, Filomat 30 (2016), 2731-2742. · Zbl 1459.11054 |
| [19] | L. Comtet, Advanced Combinatorics, Reidel, Doredecht, 1974. |
| [20] | D. Merlini, R. Sprugnoli, and M. C. Verri, The Cauchy numbers, Discrete Math. 306 (2006), 1906-1920. · Zbl 1098.05008 |
| [21] | M. Kuba, On functions on Arakawa and Kaneko and multiple zeta values, Appl. Anal. Discrete Math. 4 (2010), 45-53. · Zbl 1299.11064 |
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.
