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Some identities of Bell polynomials

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Abstract

We investigate Bell polynomials, also called Touchard polynomials or exponential polynomials, by using and without using umbral calculus. We use three different formulas in order to express various known families of polynomials such as Bernoulli polynomials, poly-Bernoulli polynomials, Cauchy polynomials and falling factorials in terms of Bell polynomials and vice versa. In addition, we derive several properties of Bell polynomials along the way.

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References

  1. Andrews G E. The Theory of Partitions. Cambridge: Cambridge University Press, 1998

    MATH  Google Scholar 

  2. Araci S. Novel identities involving Genocchi numbers and polynomials arising from applications of umbral calculus. Appl Math Comput, 2014, 233: 599–607

    Article  MathSciNet  Google Scholar 

  3. Araci S, Kong X, Acikgoz M, et al. A new approach to multivariate q-Euler polynomials using the umbral calculus. J Integer Seq, 2014, 17: 1–8

    MathSciNet  Google Scholar 

  4. Bayad A. Modular properties of elliptic Bernoulli and Euler functions. Adv Stud Contemp Math, 2010, 20: 389–401

    MathSciNet  MATH  Google Scholar 

  5. Bell E T. Exponential polynomials. Ann of Math, 1934, 35: 258–277

    Article  MathSciNet  Google Scholar 

  6. Comtet L. Advanced Combinatorics. Reidel: Kluwer, 1974

    Book  MATH  Google Scholar 

  7. Dere R, Simsek Y. Applications of umbral algebra to some special polynomials. Adv Stud Contemp Math, 2012, 22: 433–438

    MathSciNet  MATH  Google Scholar 

  8. Di Bucchianico A, Loeb D. A selected survey of umbral calculus. Electron J Combin, 1995, 2: 28pp

  9. Gaboury S, Tremblay R, Fugère B-J. Some explicit formulas for certain new classes of Bernoulli, Euler and Genocchi polynomials. Proc Jangjeon Math Soc, 2014, 17: 115–123

    MathSciNet  MATH  Google Scholar 

  10. Gould H W, He T. Characterization of (c)-Riordan arrays, Gegenbauer-Humbert-type polynomial sequences, and (c)-Bell polynomials. J Math Res Appl, 2013, 33: 505–527

    MathSciNet  MATH  Google Scholar 

  11. Herscovici O, Mansour T. Identities involving Touchard polynomials derived from umbral calculus. Adv Stud Contemp Math, 2015, 25: 39–46

    Google Scholar 

  12. Kim D S, Dolgy D V, Kim T, et al. Identities involving Bernoulli and Euler polynomials arising from Chebyshev polynomials. Proc Jangjeon Math Soc, 2012, 15: 361–370

    MathSciNet  MATH  Google Scholar 

  13. Kim D S, Kim T. Higher-order Cauchy of the first kind and poly-Cauchy of the first kind mixed type polynomials. Adv Stud Contemp Math, 2013, 23: 621–636

    MATH  Google Scholar 

  14. Kim D S, Kim T. Higher-order Cauchy of the second kind and poly-Cauchy of the second kind mixed type polynomials. Ars Combin, 2014, 115: 435–451

    MathSciNet  Google Scholar 

  15. Kim D S, Kim T, Ryoo C S. Sheffer sequences for the powers of Sheffer pairs under umbral composition. Adv Stud Contemp Math, 2013, 23: 275–285

    MathSciNet  MATH  Google Scholar 

  16. Kim D S, Lee N, Na J, et al. Abundant symmetry for higher-order Bernoulli polynomials (I). Adv Stud Contemp Math, 2013, 23: 461–482

    MathSciNet  MATH  Google Scholar 

  17. Kim T. Identities involving Laguerre polynomials derived from umbral calculus. Russ J Math Phys, 2014, 21: 36–45

    Article  MathSciNet  Google Scholar 

  18. Kim T, Dolgy D V, Kim D S, et al. A note on the identities of special polynomials. Ars Combin Ser A, 2014, 113: 97–106

    MathSciNet  Google Scholar 

  19. Kim T, Kim D S, Mansour T, et al. Umbral calculus and Sheffer sequences of polynomials. J Math Phys, 2013, 54: 083504

    Article  MathSciNet  Google Scholar 

  20. Luo Q M, Qi F. Relationships between generalized Bernoulli numbers and polynomials and generalized Euler numbers and polynomials. Adv Stud Contemp Math, 2003, 7: 11–18

    MathSciNet  MATH  Google Scholar 

  21. Mansour T. Combinatorics of Set Partitions. Boca Raton: CRC Press, 2013

    Google Scholar 

  22. Mansour T, Shattuck M. A recurrence related to the Bell numbers. Integers, 2012, 12: 373–384

    MathSciNet  MATH  Google Scholar 

  23. Riordan J. An Introduction to Combinatorial Analysis. Princeton: Princeton University Press, 1980

    Book  MATH  Google Scholar 

  24. Roman S. The Umbral Calculus. New York: Academic Press, 1984

    MATH  Google Scholar 

  25. Roman S. More on the umbral calculus, with emphasis on the q-umbral calculus. J Math Anal Appl, 1985, 107: 222–254

    Article  MathSciNet  MATH  Google Scholar 

  26. Zhang Z, Yang H. Some closed formulas for generalized Bernoulli-Euler numbers and polynomials. Proc Jangjeon Math Soc, 2008, 11: 191–198

    MathSciNet  MATH  Google Scholar 

  27. Zhang Z, Yang J. Notes on some identities related to the partial Bell polynomials. Tamsui Oxf J Inf Math Sci, 2012, 28: 39–48

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Sogang University, Seoul, 121-742, Republic of Korea

    Dae San Kim

  2. Department of Mathematics, Kwangwoon University, Seoul, 139-701, Republic of Korea

    Taekyun Kim

Authors
  1. Dae San Kim
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  2. Taekyun Kim
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Correspondence to Taekyun Kim.

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Kim, D.S., Kim, T. Some identities of Bell polynomials. Sci. China Math. 58, 1–10 (2015). https://doi.org/10.1007/s11425-015-5006-4

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  • DOI: https://doi.org/10.1007/s11425-015-5006-4

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