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Extended Stirling numbers of the first kind associated with Daehee numbers and polynomials

  • Taekyun KimEmail author
  • Dae San Kim
Original Paper

Abstract

In this paper, we study the extended Stirling numbers of the first kind which have close connection with the r-Stirling numbers of the first kind and generalize the usual Stirling numbers of the first kind. We derive recurrence relations for those numbers and show their connections with the Stirling numbers of the first kind, higher-order Daehee polynomials and Bell polynomials. In addition, we introduce Daehee–Stirling numbers of the first kind and deduce their connections with the extended Stirling numbers of the first kind and r-Whitney numbers.

Keywords

Extended Stirling numbers of the first kind Daehee numbers Daehee polynomials Daehee–Stirling numbers of the first kind 

Mathematics Subject Classification

11B73 11B83 05A19 

Notes

Acknowledgements

The authors would like to thank the referee for his valuable suggestions and comments which improved the original manuscript greatly in its present form.

References

  1. 1.
    Aigner, M.: Whitney numbers. Combinatorial geometries. Encyclopedia Math. Appl., 29, 139–160. Cambridge Univ. Press, Cambridge (1987)Google Scholar
  2. 2.
    Broder, A.Z.: The \(r\)-Stirling numbers. Discrete Math. 49, 241–259 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Carlitz, L.: Weighted Stirling numbers of the first and second kind II. Fibonacci Quart. 18(3), 242–257 (1980)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Carlitz, L.: Weighted Stirling numbers of the first and second kind I. Fibonacci Quart. 18(2), 147–162 (1980)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Carlitz, L.: Some numbers related to the Stirling numbers of the first and second kind. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 544–576, 49–55 (1976)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Carlitz, L.: A note on Stirling numbers of the first kind. Math. Mag. 37(5), 318–321 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    El-Desouky, B.S., Mustafa, A.: New results on higher-order Daehee and Bernoulli numbers and polynomials. Adv. Differ. Equ. 2016, 32, 21 pp (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gould, H.W.: The \(q\)-Stirling numbers of first kind and second kinds. Duke Math. J. 28, 281–289 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hansen, E.R.: A table of series and products. Prentice-Hall Inc, London (1975)zbMATHGoogle Scholar
  10. 10.
    He, Y.: Summation formula of products of the Apostol–Bernoulli and Apostol–Euler polynomials. Ramanujan J. 43(2), 447–464 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Jang, L.-C., Kim, T.: Some identities of Bell polynomials. J. Comput. Anal. Appl. 20(3), 584–589 (2016)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Khidr, A.M., El-Desouky, B.S.: A symmetric sum involving the Stirling numbers of the first kind. Eur. J. Combin. 5(1), 51–54 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kim, D.S., Kim, T.: Some identities of Bell polynomials. Sci. China Math. 58(10), 2095–2104 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kim, D.S., Kim, T.: Daehee numbers and polynomials. Appl. Math. Sci. (Ruse) 7(120), 5969–5976 (2013)MathSciNetGoogle Scholar
  15. 15.
    Kim, T.: Some identities on the \(q\)-Euler polynomials of higher order and \(q\)-Stirling numbers by the fermionic \(p\)-adic integral on \(\mathbb{Z}_p\), Russ. J. Math. Phys. 16(4), 484–491 (2009)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Kim, T., Kim, D.S.: On \(\lambda \)-Bell polynomials associated with umbral calculus. Russ. J. Math. Phys. 24(1), 69–78 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kim, T., Kim, D.S., Jang, L.-C., Kwon, H.-I., Seo, J.J.: Differential equations arising from Stirling polynomials and applications. Proc. Jangjeon Math. Soc. 19(2), 199–212 (2016)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Mezö, I.: On the maximum of \(r\)-Stirmilg numbers. Adv. Appl. Math. 41, 293–306 (2008)CrossRefzbMATHGoogle Scholar
  19. 19.
    Moser, L., Wyman, M.: Asymptotic development of the Stirling numbers of the first kind. J. Lond. Math. Soc. 33, 133–146 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Park, C.J.: Zero-one sequences and Stirling numbers of the first kind. Fibonacci Quart. 15(3), 231–232 (1977)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Park, J.-W.: On a \(q\)-analogue of \((h, q)\)-Dahee numbers and polynomials of higher order. J. Comput. Anal. Appl. 21(4), 769–776 (2016)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Park, J.-W., Rim, S.-H., Kwon, J.: The twisted Daehee numbers and polynomials. Adv. Differ. Equ. 2014, 1, 9 pp (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Roman, S.: The umbral calculus, Pure and Applied Mathematics, 111. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York (1984). x+193 pp. ISBN: 0-12-594380-6Google Scholar
  24. 24.
    Simsek, Y.: Identities on the Changhee numbers and Apostol-type Daehee polynomials. Adv. Stud. Contemp. Math. (Kyungshang) 27(2), 199–212 (2017)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Italia S.r.l., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsKwangwoon UniversitySeoulRepublic of Korea
  2. 2.Department of MathematicsSogang UniversitySeoulRepublic of Korea

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