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The partial r-Bell polynomials

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Abstract

In this paper, we show that the r-Stirling numbers of both kinds, the r-Whitney numbers of both kinds, the r-Lah numbers and the r-Whitney-Lah numbers form particular cases of a family of polynomials forming a generalization of the partial Bell polynomials. We deduce the generating functions of several restrictions of these numbers. In addition, a new combinatorial interpretations is presented for the r-Whitney numbers and the r-Whitney-Lah numbers.

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References

  1. Bell, E.T.: Exponential polynomials. Ann. Math. 35, 258–277 (1934)

    Article  MATH  MathSciNet  Google Scholar 

  2. Broder, A.Z.: The \(r\)-Stirling numbers. Discrete Math. 49, 241–259 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chrysaphinou, O.: On Touchard polynomials. Discrete Math. 54(2), 143–152 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  4. Cheon, G.S., Jung, J.H.: r-Whitney numbers of Dowling lattices. Discrete Math. 308, 2450–2459 (2012)

    MATH  MathSciNet  Google Scholar 

  5. Comtet, L.: Advanced Combinatorics. D. Reidel Publishing Company, Dordrecht (1974)

    Book  MATH  Google Scholar 

  6. Corcino, C.B., Hsu, L.C., Tan, E.L.: Asymptotic approximations of \(r\)-Stirling numbers. Approx. Theory Appl. (N.S.) 15, 13–25 (1999)

    MATH  MathSciNet  Google Scholar 

  7. Maamra, M.S., Mihoubi, M.: Note on some restricted Stirling numbers of the second kind. C. R. Math. Acad. Sci. 354, 231–234 (2016)

    Article  MathSciNet  Google Scholar 

  8. Mező, I.: A new formula for the Bernoulli polynomials. Results. Math. 58, 329–335 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  9. Mező, I.: On the maximum of \(r\)-Stirling numbers. Adv. Appl. Math. 41, 293–306 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  10. Mező, I.: New properties of \(r\)-Stirling series. Acta Math. Hungar. 119, 341–358 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  11. Mező, I.: The \(r\)-Bell numbers. J. Integer Seq. 14, Article 11.1.1, p 14 (2011)

  12. Mihoubi, M.: Bell polynomials and binomial type sequences. Discrete Math. 308, 2450–2459 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  13. Mihoubi, M.: Partial Bell Polynomials and Inverse Relations. J. Integer Seq., 13, Article 10.4.5. (2010)

  14. Mihoubi, M.: Polynômes multivariés de Bell et polynômes de type binomial. Thesis (Ph. D.) Université des sciences et de la technologie Houari-Boumediene, Alger (2008)

  15. Mihoubi, M., Maamra, M.S.: The \((r_1,\dots, r_p)\)-Stirling numbers of the second kind. Integers 12(5), 1047–1059 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  16. Mihoubi, M., Reggane, L.: The \(s\)-degenerate \(r\)-Lah numbers. Ars Combin. 120, 333–340 (2015)

    MATH  MathSciNet  Google Scholar 

  17. Port, D.: Polynomial maps with applications to combinatorics and probability theory. Thesis (Ph. D.) Massachusetts Institute of Technology, Dept. of Mathematics (1994)

  18. Pitman, J.: Combinatorial stochastic processes. Lecture Notes in Mathematics, vol. 1875. Springer-Verlag, Berlin (2006)

  19. Rahmani, M.: Some results on Whitney numbers of Dowling lattices. Arab J. Math. Sci. 20(1), 11–27 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  20. Rahmani, M.: Generalized Stirling transform. Miskolc. Math. Notes 15(2), 677–690 (2014)

    MATH  MathSciNet  Google Scholar 

  21. Wang, W., Wang, T.: General identities on Bell polynomials. Comput. Math. Appl. 58(1), 104–118 (2009)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgements

We are very grateful to the referees for pointing out some errors in the previous version, as well as their help in the revision of this manuscript.

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  1. Faculty of Mathematics, USTHB, P.O. Box 32, El Alia, 16111, Algiers, Algeria

    Miloud Mihoubi & Mourad Rahmani

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  1. Miloud Mihoubi
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  2. Mourad Rahmani
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Correspondence to Mourad Rahmani.

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Mihoubi, M., Rahmani, M. The partial r-Bell polynomials. Afr. Mat. 28, 1167–1183 (2017). https://doi.org/10.1007/s13370-017-0510-z

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  • DOI: https://doi.org/10.1007/s13370-017-0510-z

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