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On arithmetical properties of degenerate Cauchy polynomials and numbers

Year 2025, Volume: 54 Issue: 2, 457 - 469, 28.04.2025

Mehmet Cenkci , Takao Komatsu

https://doi.org/10.15672/hujms.1354679

Abstract

In this paper, we study the arithmetical properties of degenerate Cauchy polynomials and numbers. We present some basic properties and recurrence relations, determine all the coefficients of degenerate Cauchy polynomials, and give some convolution identities. These results are particularly useful to deduce interesting and new divisibility properties for degenerate Cauchy polynomials and numbers.

Keywords

degenerate Cauchy polynomials degenerate Cauchy numbers recurrences convolutions congruences

References

  • [1] T. Agoh, Convolution identities for Bernoulli and Genocchi polynomials, The Electronic Journal of Combinatorics 21, #P1.65, 2014.
  • [2] T. Arakawa, T. Ibuyikama and M. Kaneko, Bernoulli Numbers and Zeta Functions, Springer Verlag, New York, 2014.
  • [3] L. Carlitz, A degenerate Staudt-Clausen theorem, Archiv der Mathematik 7, 28–33, 1956.
  • [4] L. Carlitz, Degenerate Stirling, Bernoulli, and Eulerian numbers, Utilitas Mathematica 15, 51–88, 1979.
  • [5] M. Cenkci and F.T. Howard, Notes on degenerate numbers, Discrete Mathematics 307, 2359–2375, 2007.
  • [6] M. Cenkci and P.T. Young, Generalizations of poly-Bernoulli and poly-Cauchy numbers, European Journal of Mathematics 1, 799–828, 2015.
  • [7] L. Comtet, Advanced Combinatorics, Reidel, Dordrecht, 1974.
  • [8] R.L. Graham, D.E. Knuth and O. Patashnik, Concrete Mathematics, 2nd ed., Addison-Wesley, Reading, 1994.
  • [9] F.T. Howard, Degenerate weighted Stirling numbers, Discrete Mathematics 57, 45–58, 1985.
  • [10] F.T. Howard, Extensions of congruences of Glaisher and Nielsen concerning Stirling numbers, Fibonacci Quarterly 28, 355–362, 1990.
  • [11] F.T. Howard, Congruences for the Stirling numbers and associated Stirling numbers, Acta Arithmetica, 55, 29–41, 1990.
  • [12] F.T. Howard, Nörlund’s number $B_{n}^{(n)}$, in: Applications of Fibonacci numbers, Vol. 5 (G. E. Bergum, A. N. Philippou and A. F. Horadam, eds.), Springer, Dordrecht, 355–366, 1993.
  • [13] F.T. Howard, Explicit formulas for degenerate Bernoulli numbers, Discrete Mathematics 162, 175–185, 1996.
  • [14] L.C. Hsu and P.J.-S. Shiue, On certain summation problems and generalizations of Eulerian polynomials and numbers, Discrete Mathematics 204, 237–247, 1999.
  • [15] L. Kargın, On Cauchy numbers and their generalizations, Gazi University Journal of Science 33, 456–474, 2020.
  • [16] L. Kargın and B. Çekim, Higher order generalized geometric polynomials, Turkish Journal of Mathematics 42, 887-903, 2018.
  • [17] T. Komatsu, Poly-Cauchy numbers, Kyushu Journal of Mathematics 67, 143–153, 2013.
  • [18] T. Komatsu, Leaping Cauchy numbers, Filomat 32, 6167–6176, 2018.
  • [19] T. Komatsu, Two types of hypergeometric degenerate Cauchy numbers, Open Mathematics 18, 417–433, 2020.
  • [20] T. Komatsu and C. Pita-Ruiz, Shifted Cauchy numbers, Quaestiones Mathematicae 43, 213–226, 2020.
  • [21] D. Merlini, R. Sprugnoli and M.C. Verri, The Cauchy numbers, Discrete Mathematics 306, 1906–1920, 2006.
  • [22] N.E. Nörlund, Vorlesungen über Differenzenrechnung, Springer, Berlin, Germany, 1924, reprinted by Chelsea, Bronx, NY, USA, 1954.
  • [23] M. Rahmani, On p-Cauchy numbers, Filomat 30, 2731–2742, 2016.
  • [24] P.T. Young, Degenerate Bernoulli polynomials, generalized factorial sums, and their applications, Journal of Number Theory 128, 738–758, 2008.

Year 2025, Volume: 54 Issue: 2, 457 - 469, 28.04.2025

Mehmet Cenkci , Takao Komatsu

https://doi.org/10.15672/hujms.1354679

Abstract

References

  • [1] T. Agoh, Convolution identities for Bernoulli and Genocchi polynomials, The Electronic Journal of Combinatorics 21, #P1.65, 2014.
  • [2] T. Arakawa, T. Ibuyikama and M. Kaneko, Bernoulli Numbers and Zeta Functions, Springer Verlag, New York, 2014.
  • [3] L. Carlitz, A degenerate Staudt-Clausen theorem, Archiv der Mathematik 7, 28–33, 1956.
  • [4] L. Carlitz, Degenerate Stirling, Bernoulli, and Eulerian numbers, Utilitas Mathematica 15, 51–88, 1979.
  • [5] M. Cenkci and F.T. Howard, Notes on degenerate numbers, Discrete Mathematics 307, 2359–2375, 2007.
  • [6] M. Cenkci and P.T. Young, Generalizations of poly-Bernoulli and poly-Cauchy numbers, European Journal of Mathematics 1, 799–828, 2015.
  • [7] L. Comtet, Advanced Combinatorics, Reidel, Dordrecht, 1974.
  • [8] R.L. Graham, D.E. Knuth and O. Patashnik, Concrete Mathematics, 2nd ed., Addison-Wesley, Reading, 1994.
  • [9] F.T. Howard, Degenerate weighted Stirling numbers, Discrete Mathematics 57, 45–58, 1985.
  • [10] F.T. Howard, Extensions of congruences of Glaisher and Nielsen concerning Stirling numbers, Fibonacci Quarterly 28, 355–362, 1990.
  • [11] F.T. Howard, Congruences for the Stirling numbers and associated Stirling numbers, Acta Arithmetica, 55, 29–41, 1990.
  • [12] F.T. Howard, Nörlund’s number $B_{n}^{(n)}$, in: Applications of Fibonacci numbers, Vol. 5 (G. E. Bergum, A. N. Philippou and A. F. Horadam, eds.), Springer, Dordrecht, 355–366, 1993.
  • [13] F.T. Howard, Explicit formulas for degenerate Bernoulli numbers, Discrete Mathematics 162, 175–185, 1996.
  • [14] L.C. Hsu and P.J.-S. Shiue, On certain summation problems and generalizations of Eulerian polynomials and numbers, Discrete Mathematics 204, 237–247, 1999.
  • [15] L. Kargın, On Cauchy numbers and their generalizations, Gazi University Journal of Science 33, 456–474, 2020.
  • [16] L. Kargın and B. Çekim, Higher order generalized geometric polynomials, Turkish Journal of Mathematics 42, 887-903, 2018.
  • [17] T. Komatsu, Poly-Cauchy numbers, Kyushu Journal of Mathematics 67, 143–153, 2013.
  • [18] T. Komatsu, Leaping Cauchy numbers, Filomat 32, 6167–6176, 2018.
  • [19] T. Komatsu, Two types of hypergeometric degenerate Cauchy numbers, Open Mathematics 18, 417–433, 2020.
  • [20] T. Komatsu and C. Pita-Ruiz, Shifted Cauchy numbers, Quaestiones Mathematicae 43, 213–226, 2020.
  • [21] D. Merlini, R. Sprugnoli and M.C. Verri, The Cauchy numbers, Discrete Mathematics 306, 1906–1920, 2006.
  • [22] N.E. Nörlund, Vorlesungen über Differenzenrechnung, Springer, Berlin, Germany, 1924, reprinted by Chelsea, Bronx, NY, USA, 1954.
  • [23] M. Rahmani, On p-Cauchy numbers, Filomat 30, 2731–2742, 2016.
  • [24] P.T. Young, Degenerate Bernoulli polynomials, generalized factorial sums, and their applications, Journal of Number Theory 128, 738–758, 2008.
There are 24 citations in total.

Details

Primary Language English
Subjects Algebra and Number Theory, Combinatorics and Discrete Mathematics (Excl. Physical Combinatorics)
Journal Section Mathematics
Authors

Mehmet Cenkci 0000-0002-2867-3448

Takao Komatsu 0000-0001-6204-5368

Early Pub Date August 27, 2024
Publication Date April 28, 2025
Published in Issue Year 2025 Volume: 54 Issue: 2

Cite

APA Cenkci, M., & Komatsu, T. (2025). On arithmetical properties of degenerate Cauchy polynomials and numbers. Hacettepe Journal of Mathematics and Statistics, 54(2), 457-469. https://doi.org/10.15672/hujms.1354679
AMA Cenkci M, Komatsu T. On arithmetical properties of degenerate Cauchy polynomials and numbers. Hacettepe Journal of Mathematics and Statistics. April 2025;54(2):457-469. doi:10.15672/hujms.1354679
Chicago Cenkci, Mehmet, and Takao Komatsu. “On Arithmetical Properties of Degenerate Cauchy Polynomials and Numbers”. Hacettepe Journal of Mathematics and Statistics 54, no. 2 (April 2025): 457-69. https://doi.org/10.15672/hujms.1354679.
EndNote Cenkci M, Komatsu T (April 1, 2025) On arithmetical properties of degenerate Cauchy polynomials and numbers. Hacettepe Journal of Mathematics and Statistics 54 2 457–469.
IEEE M. Cenkci and T. Komatsu, “On arithmetical properties of degenerate Cauchy polynomials and numbers”, Hacettepe Journal of Mathematics and Statistics, vol. 54, no. 2, pp. 457–469, 2025, doi: 10.15672/hujms.1354679.
ISNAD Cenkci, Mehmet - Komatsu, Takao. “On Arithmetical Properties of Degenerate Cauchy Polynomials and Numbers”. Hacettepe Journal of Mathematics and Statistics 54/2 (April 2025), 457-469. https://doi.org/10.15672/hujms.1354679.
JAMA Cenkci M, Komatsu T. On arithmetical properties of degenerate Cauchy polynomials and numbers. Hacettepe Journal of Mathematics and Statistics. 2025;54:457–469.
MLA Cenkci, Mehmet and Takao Komatsu. “On Arithmetical Properties of Degenerate Cauchy Polynomials and Numbers”. Hacettepe Journal of Mathematics and Statistics, vol. 54, no. 2, 2025, pp. 457-69, doi:10.15672/hujms.1354679.
Vancouver Cenkci M, Komatsu T. On arithmetical properties of degenerate Cauchy polynomials and numbers. Hacettepe Journal of Mathematics and Statistics. 2025;54(2):457-69.

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