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Russian Journal of Mathematical Physics

, Volume 25, Issue 1, pp 44–58 | Cite as

Degenerate r-Stirling Numbers and r-Bell Polynomials

  • T. KimEmail author
  • Y. Yao
  • D. S. Kim
  • G. -W. Jang
Article

Abstract

The purpose of this paper is to exploit umbral calculus in order to derive some properties, recurrence relations, and identities related to the degenerate r-Stirling numbers of the second kind and the degenerate r-Bell polynomials. Especially, we will express the degenerate r-Bell polynomials as linear combinations of many well-known families of special polynomials.

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References

  1. 1.
    L. Carlitz, “Degenerate Stirling, Bernoulli and Eulerian Numbers,” Utilitas Math. 15, 51–88 (1975).MathSciNetzbMATHGoogle Scholar
  2. 2.
    A. Z. Broder, “The r-Stirling Numbers,” Discrete Math. 49, 241–259 (1984).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    D. S. Kim and T. Kim, “Identities Involving r-Stirling Numbers,” J. Comput. Anal. Appl. 17 (4), 674–680 (2014).MathSciNetzbMATHGoogle Scholar
  4. 4.
    D. S. Kim and T. Kim, “Some Identities of Bell Polynomials,” Sci. China Math. 58 (10), 2095–2104 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    D. S. Kim and T. Kim, “A Note on Degenerate Eulerian Numbers and Polynomials,” Adv. Stud. Contemp. Math. (Kyungshang) 27 (4), 431–440 (2017).zbMATHGoogle Scholar
  6. 6.
    T. Kim, “λ-Analogue of Stirling Numbers of the First Kind,” Adv. Stud. Contemp. Math. (Kyungshang) 27 (3), 423–429 (2017).zbMATHGoogle Scholar
  7. 7.
    T. Kim and D. S. Kim, “Extended Stirling Polynomials of the Second Kind and Extended Bell Polynomials,” Proc. Jangjeon Math. Soc. 20 (3), 365–376 (2017).MathSciNetzbMATHGoogle Scholar
  8. 8.
    T. Kim, “A Note on Degenerate Stirling Polynomials of the Second Kind,” Proc. Jangjeon Math. Soc. 20 (3), 319–331 (2017).MathSciNetzbMATHGoogle Scholar
  9. 9.
    T. Kim, D. S. Kim, and G.-W. Jang, “Some Formulas of Ordered Bell Numbers and Polynomials Arising from Umbral Calculus,” Proc. Jangjeon Math. Soc. 20 (4), 659–670 (2017).MathSciNetzbMATHGoogle Scholar
  10. 10.
    I. Mezö, “On the Maximum of r-Stirling Numbers,” Adv. Appl. Math. 41, 293–306 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    M. Mihoubi and H. Belbachir, “Linear Recurrences for r-Bell Polynomials,” J. Integer Seq. 17, Article 14.10.6 (2014).Google Scholar
  12. 12.
    J. Quaintance and H. W. Gould, Combinatorial Identities for Stirling Numbers. The Unpublished Notes of H. W. Gould with a Foreword by George E. Andrews (World Scientific Publishing Co. Pte. Ltd., Singapore, 2016) xv+260 pp. ISBN: 978-981-4725-26-2.zbMATHGoogle Scholar
  13. 13.
    S. Roman, The Umbral Calculus, Pure and Applied Mathematics, Vol. III (Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1984).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Department of Mathematics, College of ScienceTianjin Polytechnic UniversityTianjin CityChina
  2. 2.Department of MathematicsKwangwoon UniversitySeoulRepublic of Korea
  3. 3.Institute of Fundamental SciencesUniversity of Electronic Science and TechnologyChengduChina
  4. 4.Department of MathematicsSogang UniversitySeoulRepublic of Korea

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