Skip to main content

Hermite base Bernoulli type polynomials on the umbral algebra


The aim of this paper is to construct new generating functions for Hermite base Bernoulli type polynomials, which generalize not only the Milne-Thomson polynomials but also the two-variable Hermite polynomials. We also modify the Milne-Thomson polynomials, which are related to the Bernoulli polynomials and the Hermite polynomials. Moreover, by applying the umbral algebra to these generating functions, we derive new identities for the Bernoulli polynomials of higher order, the Hermite polynomials and numbers of higher order, and the Stirling numbers of the second kind.

This is a preview of subscription content, access via your institution.


  1. P. Blasiak, G. Dattoli, A. Horzela, and K. A. Penson, “Representations of Monomiality Principle with Sheffer-Type Polynomials and Boson Normal Ordering,” Phys. Lett. A 352, 7–12 (2006).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  2. G. Bretti and P. E. Ricci, “Multidimensional Extensions of the Bernoulli and Appell Polynomials,” Taiwanese J. Math. 8, 415–428 (2004).

    MATH  MathSciNet  Google Scholar 

  3. G. Dattoli, M. Migliorati, and H. M. Srivastava, “Sheffer Polynomials, Monomiality Principle, Algebraic Methods and the Theory of Classical Polynomials,” Math. Comput. Modelling 45, 1033–1041 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  4. R. Dere and Y. Simsek, “Genocchi Polynomials Associated with the Umbral Algebra,” Appl. Math. Comput. 218(3), 756–761 (2011).

    Article  MATH  MathSciNet  Google Scholar 

  5. L. M. Milne-Thomson, “Two Classes of Generalized Polynomials,” Proc. London Math. Soc. s2-35(1), 514–522 (1933).

    Article  MathSciNet  Google Scholar 

  6. M.-S. Kim, “A Note on Sums of Products of Bernoulli Numbers,” Appl. Math. Lett. 24, 55–61 (2011).

    Article  MATH  MathSciNet  Google Scholar 

  7. S. Roman, The Umbral Calculus (Dover Publ. Inc. New York, 2005).

    Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to R. Dere.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Dere, R., Simsek, Y. Hermite base Bernoulli type polynomials on the umbral algebra. Russ. J. Math. Phys. 22, 1–5 (2015).

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI:


  • Mathematical Physic
  • Hermite Polynomial
  • Bernoulli Polynomial
  • Stirling Number
  • Quantum Harmonic Oscillator