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Journals / Hacettepe Journal of Mathematics and Statistics / 2026 / Volume: 54 Issue: 2

On arithmetical properties of degenerate Cauchy polynomials and numbers

Mehmet Cenkci(AKDENIZ UNIVERSITY)0000-0002-2867-3448
Takao Komatsu(Henan Academy of Sciences)0000-0001-6204-5368
Pages
—
Publication Date
2025-04-28
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

These results are particularly useful to deduce interesting and new divisibility properties for degenerate Cauchy polynomials and numbers.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:In this paper, we study the arithmetical properties of degenerate Cauchy polynomials and numbers.
📊
Key Findings & Data:We present some basic properties and recurrence relations, determine all the coefficients of degenerate Cauchy polynomials, and give some convolution identities.
💡
Conclusion & Impact:These results are particularly useful to deduce interesting and new divisibility properties for degenerate Cauchy polynomials and numbers.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:degenerate Cauchy polynomials · degenerate Cauchy numbers · recurrences

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.

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Cite This Article

Cenkci, M., Komatsu, T. (2025). On arithmetical properties of degenerate Cauchy polynomials and numbers. Hacettepe Journal of Mathematics and Statistics, Volume: 54 Issue: 2. https://doi.org/10.15672/hujms.1354679

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References (24)

  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.