Skip to main content
Log in

On Some Degenerate Differential and Degenerate Difference Operators

  • Research Articles
  • Published:
Russian Journal of Mathematical Physics Aims and scope Submit manuscript

Abstract

The aim of this paper is to make use of certain degenerate differential and degenerate difference operators in order to study some identities involving the degenerate harmonic numbers, certain finite sums of general nature, the sums of the values of the generalized falling factorials at consecutive positive integers, and the degenerate Laguerre polynomials.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. G. E. Andrews, R. Askey, and R. Roy, Special Functions. Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999.

    Google Scholar 

  2. S. Araci, “A New Class of Bernoulli Polynomials Attached to Polyexponential Functions and Related Identities”, Adv. Stud. Contemp. Math. (Kyungshang), 31:2 (2021), 195–204.

    Google Scholar 

  3. K. N. Boyadzhiev, “Harmonic Number Identities via Euler’s Transform”, J. Integer Seq., 12:6 (2009).

    MathSciNet  MATH  Google Scholar 

  4. L. Carlitz, “Degenerate Stirling, Bernoulli and Eulerian numbers”, Utilitas Math., 15 (1979), 51–88.

    MathSciNet  MATH  Google Scholar 

  5. L. Carlitz, “A Degenerate Staudt-Clausen Theorem”, Arch. Math. (Basel), 7 (1956), 28–33.

    Article  MathSciNet  Google Scholar 

  6. L. Comtet, Advanced Combinatorics, The art of Finite and Infinite Expansions, Revised and enlarged edition. D. Reidel Publishing Co., Dordrecht, 1974.

    MATH  Google Scholar 

  7. D. S. Kim and T. Kim, “A Note on a New Type of Degenerate Bernoulli Numbers”, Russ. J. Math. Phys., 27:2 (2020), 227–235.

    Article  MathSciNet  Google Scholar 

  8. T. Kim, “A Note on Degenerate Stirling Polynomials of the Second Kind”, Proc. Jangjeon Math. Soc., 20:3 (2017), 319–331.

    MathSciNet  MATH  Google Scholar 

  9. T. Kim, “Sums of Powers of Consecutive \(q\)-Integers”, Adv. Stud. Contemp. Math. (Kyungshang), 9:1 (2004), 15-тАУ18.

    MathSciNet  MATH  Google Scholar 

  10. T. Kim, D. V. Dolgy, D. S. Kim, H. K. Kim, and S. H. Park, “A Note on Degenerate Generalized Laguerre Polynomials and Lah Numbers”, Adv. Difference Equ, :421 (2021).

    MathSciNet  Google Scholar 

  11. T. Kim and D. S. Kim, “Some Identities on Truncated Polynomials Associated With Degenerate Bell Polynomials”, Russ. J. Math. Phys, 28:3 (2021), 342–355.

    Article  MathSciNet  Google Scholar 

  12. T. Kim and D. S. Kim, “Note on the Degenerate Gamma Function”, Russ. J. Math. Phys., 27:3 (2020), 352–358.

    Article  MathSciNet  Google Scholar 

  13. T. Kim, D. S. Kim, H. Lee, S. H. Park, and J. Kwon, “New Properties on Degenerate Bell Polynomials”, Complexity, (2021) Article ID 7648994.

  14. 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.

  15. S. Roman, The Umbral Calculus, Pure and Applied Mathematics, 111. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1984.

    MATH  Google Scholar 

  16. Y. Simsek, “Construction of Generalized Leibnitz Type Numbers and Their Properties”, Adv. Stud. Contemp. Math. (Kyungshang), 31:3 (2021), 311–323.

    Google Scholar 

  17. T. Usman, M. Aman, O. Khan, K. S. Nisar, and S. Araci, “Construction of Partially Degenerate Laguerre-Genocchi Polynomials with Their Applications”, AIMS Math., 5:5 (2020), 4399-тАУ4411.

    Article  MathSciNet  Google Scholar 

  18. E. T. Whittaker and G. N. Watson, “A Course of Modern Analysis, An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions”, Reprint of the fourth (1927) edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, (1996).

    MATH  Google Scholar 

Download references

Acknowledgments

The work reported in this paper was conducted during the sabbatical year of Kwangwoon University in 2022.

Author information

Authors and Affiliations

Authors

Corresponding authors

Correspondence to T. Kim or D. S. Kim.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kim, T., Kim, D.S. On Some Degenerate Differential and Degenerate Difference Operators. Russ. J. Math. Phys. 29, 37–46 (2022). https://doi.org/10.1134/S1061920822010046

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1061920822010046

Navigation