Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ON FINITE TIMES DEGENERATE HIGHER-ORDER CAUCHY NUMBERS AND POLYNOMIALS

  • Jeong, Joohee (Department of Mathematics Education Kyungpook National University) ;
  • Rim, Seog-Hoon (Department of Mathematics Education Kyungpook National University)
  • Received : 2015.09.08
  • Published : 2016.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

Cauchy polynomials are also called Bernoulli polynomials of the second kind and these polynomials are very important to study mathematical physics. D. S. Kim et al. have studied some properties of Bernoulli polynomials of the second kind associated with special polynomials arising from umbral calculus. T. Kim introduced the degenerate Cauchy numbers and polynomials which are derived from the degenerate function $e^t$. Recently J. Jeong, S. H. Rim and B. M. Kim studied on finite times degenerate Cauchy numbers and polynomials. In this paper we consider finite times degenerate higher-order Cauchy numbers and polynomials, and give some identities and properties of these polynomials.

Keywords

References

  1. A. Bayad and T. Kim, Identities involving values of Bernstein, q-Bernoulli, and q-Euler polynomials, Russ. J. Math. Phys. 18 (2011), no. 2, 133-143. https://doi.org/10.1134/S1061920811020014
  2. L. Comtet, Advanced Combinatorics, Reidel, Doredecht, 1974.
  3. J. Jeong, S. H. Rim, and B. M. Kim, On finite times degenerate Cauchy numbers and polynomials, Adv. Difference Equ. 2015 (2015), 321, 12 pp. https://doi.org/10.1186/s13662-014-0332-3
  4. D. S. Kim and T. Kim, Higher-order Cauchy of the first kind and poly-Cauchy of the first kind mixed type polynomials, Adv. Stud. Contemp. Math. 24 (2013), no. 4,621-636.
  5. D. S. Kim and T. Kim, A note on poly-Bernoulli and higher-order poly-Bernoulli polynomials, Russ. J. Math. Phys. 22 (2015), no. 1, 26-33. https://doi.org/10.1134/S1061920815010057
  6. D. S. Kim, T. Kim, and S. H. Lee, Poly-Cauchy numbers and polynomials with umbral calculus viewpoint, Int. J. Math. Anal. 7 (2013), no. 45-48, 2235-2253. https://doi.org/10.12988/ijma.2013.37185
  7. T. Kim, Identities involving Laguerre polynomials derived from unbral calculus, Russ. J. Math. Phys. 21 (2014), no. 1, 36-45. https://doi.org/10.1134/S1061920814010038
  8. T. Kim, On the degenerate Cauchy numbers and polynomials, Proc. Jangjeon Math. Soc. 18 (2015), no. 3, 307-312.
  9. T. Kim, Barnes' type multiple degenerate Bernoulli and Euler polynomials, Appl. Math. Comput. 258 (2015), 556-564.
  10. T. Kim, On the degenerate higher-order Cauchy numbers and polynomials, Adv. Studies in Contemp. Math. 25 (2015), no. 3, 417-421.
  11. T. Kim, Degenerate Euler zeta function, Russ. J. Math. Phys. 22 (2015), no. 4, 469-472. https://doi.org/10.1134/S1061920815040068
  12. T. Kim, D. S. Kim, I. H. Kwon, and J. J. Seo, Higher-order Cauchy numbers and polynomials, Appl. Math. Sci. 9 (2015), no. 40, 1989-2004.
  13. T. Kim and T. Mansour, Umbral calculus associated with Frobenius-type Eulerian polynomials, Russ. J. Math. Phys. 21 (2014), no. 4, 484-493. https://doi.org/10.1134/S1061920814040062
  14. D. Merlini, R. Sprugnoli, and M. C. Verri, The Cauchy numbers, Discrete Math. 306 (2006), no. 16, 1906-1920. https://doi.org/10.1016/j.disc.2006.03.065
  15. H. L. Saad, On the Cauchy polynomials, Int. Math. Forum 9 (2014), no. 3, 1695-1706. https://doi.org/10.12988/imf.2014.410172