Skip to main content
SpringerLink
Log in

Construction and computation of unified Stirling-type numbers emerging from p-adic integrals and symmetric polynomials

  • Original Paper
  • Published:
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

The aim of this paper is to give construction and computation methods for generalized and unified representations of Stirling-type numbers and Bernoulli-type numbers and polynomials. Firstly, we define generalized and unified representations of the falling factorials. By using these new representations as components of the generating functions, we also construct generalized and unified representations of Stirling-type numbers. By making use of the symmetric polynomials, we give computational formulas and algorithm for these numbers. Applying Riemann integral to the unified falling factorials, we introduce new families of Bernoulli-type numbers and polynomials of the second kind by their computation formulas and plots drawn by the Wolfram programming language in Mathematica. Applying p-adic integrals to the unified falling factorials, we construct two new sequences that involve some well-known special numbers such as the Stirling numbers, the Bernoulli numbers and the Euler numbers. Finally, we give not only further remarks and observations, but also some open questions regarding the potential applications and relations of our results.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Alayont, F., Krzywonos, N.: Rook polynomials in three and higher dimensions. Involve 6(1), 35–52 (2013)

    MathSciNet  MATH  Google Scholar 

  2. Alayont, F., Moger-Reischer, R., Swift, R.: Rook number interpretations of generalized central factorial and Genocchi numbers. Preprint https://faculty.gvsu.edu/alayontf/notes/generalized_central_genocchi_numbers_preprint.pdf

  3. Bayad, A., Simsek, Y., Srivastava, H.M.: Some array type polynomials associated with special numbers and polynomials. Appl. Math. Comput. 244, 149–157 (2014)

    MathSciNet  MATH  Google Scholar 

  4. Belbachir, H., Belkhir, A., Bousbaa, I.E.: Combinatorial approach of certain generalized Stirling numbers. arXiv:1411.6271v1

  5. Belbachir, H., Djemmada, Y.: The \((l, r)\)-Stirling numbers: a combinatorial approach. arXiv:2101.11039v1

  6. Bona, M.: Introduction to Enumerative Combinatorics. The McGraw-Hill Companies Inc., New York (2007)

    MATH  Google Scholar 

  7. Butzer, P.L., Schmidt, K., Stark, E.L., Vogt, L.: Central factorial numbers; their main properties and some applications. Numer. Funct. Anal. Optim. 10(5 & 6), 419–488 (1989)

    MathSciNet  MATH  Google Scholar 

  8. Cangul, I.N., Cevik, A.S., Simsek, Y.: A new approach to connect algebra with analysis: relationships and applications between presentations and generating functions. Bound. Value Probl. 2013(51), 1–17 (2013)

    MathSciNet  MATH  Google Scholar 

  9. Carlitz, L.: Some numbers related to the Stirling numbers of the first and second kind. Univ. Beograd Publ. Elektr. Fak. Ser. Mat. Fiz. 544–576, 49–55 (1976)

    MathSciNet  MATH  Google Scholar 

  10. Carlitz, L.: Degenerate Stirling numbers, Bernoulli and Eulerian numbers. Utilitas Math. 15, 51–88 (1979)

    MathSciNet  MATH  Google Scholar 

  11. Carlitz, L.: Weighted Stirling numbers of the first and second kind I, II. Fib. Q. 18, 147–162 & 242–257 (1980)

  12. Cevik, A.S., Cangul, I.N., Simsek, Y.: Analysis approach to finite monoids. Fixed Point Theory Appl. 2013(15), 1–18 (2013)

    MathSciNet  MATH  Google Scholar 

  13. Cevik, A.S., Das, K.C., Simsek, Y., Cangul, I.N.: Some array polynomials over special monoid presentations. Fixed Point Theory Appl. 2013(44), 1–14 (2013)

    MathSciNet  MATH  Google Scholar 

  14. Charalambides, C.A.: Enumerative Combinatorics. Chapman&Hall/Crc Press Company, London (2002)

    MATH  Google Scholar 

  15. Cigler, J.: Fibonacci Polynomials and Central Factorial Numbers. Preprint https://homepage.univie.ac.at/johann.cigler/preprints/central-factorial.pdf

  16. Comtet, L.: Numbers de Stirling generaux et fonctions symetriques. C. R. Acad. Sci. Paris (Ser. A) 275(16), 747–750 (1972)

    MATH  Google Scholar 

  17. Comtet, L.: Advanced Combinatorics. D. Reidel Publication Company, Dordrecht (1974)

    MATH  Google Scholar 

  18. Dolgy, D.V., Jang, G.-W., Kim, D.S., Kim, T.: Explicit expressions for Catalan-Daehee numbers. Proc. Jangjeon Math. Soc. 20(3), 1–9 (2017)

    MathSciNet  MATH  Google Scholar 

  19. El-Desouky, B.S.: The multiparameter noncentral Stirling numbers. Fibonacci Q. 32(3), 218–225 (1994)

    MathSciNet  MATH  Google Scholar 

  20. Gould, H., Quaintance, J.: Double fun with double factorials. Math. Mag. 85(3), 177–192 (2012)

    MathSciNet  MATH  Google Scholar 

  21. Hauss, M.: Rapidly converging series representations for zeta-type functions. Contemp. Math. 190, 143–163 (1995)

    MathSciNet  MATH  Google Scholar 

  22. Howard, F.T.: Degenerate weighted Stirling numbers. Discrete Math. 57, 45–58 (1985)

    MathSciNet  MATH  Google Scholar 

  23. Hsu, L.C., Shiue, P.J.-S.: A unified approach to generalized Stirling numbers. Adv. Appl. Math. 20(AM980586), 366–384 (1998)

  24. Jang, L.C., Kim, T.: A new approach to \(q\)-Euler numbers and polynomials. J. Concr. Appl. Math. 6, 159–168 (2008)

    MathSciNet  MATH  Google Scholar 

  25. Jordan, C.: Calculus of Finite Differences, 2nd edn. Chelsea, New York (1950)

    MATH  Google Scholar 

  26. Kac, V., Cheung, P.: Quantum Calculus. Springer, New York (2001)

    MATH  Google Scholar 

  27. Kim, M.-S., Kim, D.: The \(q\)-Stirling numbers of the second kind and its applications. J. Nonlinear Sci. Appl. 11, 971–983 (2018)

    MathSciNet  MATH  Google Scholar 

  28. Kim, T.: \(q\)-Volkenborn integration. Russ. J. Math. Phys. 19, 288–299 (2002)

    MathSciNet  MATH  Google Scholar 

  29. Kim, T.: \(q\)-Euler numbers and polynomials associated with \(p\)-adic \(q\)-integral and basic \(q\)-zeta function. Trends Int. Math. Sci. Stud. 9, 7–12 (2006)

    Google Scholar 

  30. Kim, T.: A Note on Catalan numbers associated with \(p\)-adic integral on \({\mathbb{Z}}_p\). Proc. Jangjeon Math. Soc. 19(3), 493–501 (2016)

    MathSciNet  MATH  Google Scholar 

  31. Kim, T.: An invariant \(p\)-adic \(q\)-integral on \({\mathbb{Z}}_p\). Appl. Math. Lett. 21, 105–108 (2008)

    MathSciNet  Google Scholar 

  32. Kim, T., Rim, S.-H., Dolgy, D.V., Pyo, S.-S.: Explicit expression for symmetric identities of \(w\)-Catalan-Daehee polynomials. Notes Numb. Theory Discrete Math. 24(4), 99–111 (2018)

    Google Scholar 

  33. Kim, T., Kim, D.S., Kim, H.Y., Kwon, J.: Degenerate Stirling polynomials of the second kind and some applications. Symmetry 11(8), 1046 (2019)

    Google Scholar 

  34. Koepf, W.: Hypergeometric Summation, An Algorithmic Approach to Summation and Special Function Identities. Vieweg, Braunschweig (1998)

  35. Koshy, T.: Catalan Numbers with Applications. Oxford University Press, Oxford (2009)

    MATH  Google Scholar 

  36. Kucukoglu, I., Simsek, B., Simsek, Y.: New classes of Catalan-type numbers and polynomials with their applications related to \(p\)-adic integrals and computational algorithms. Turk. J. Math. 44, 2337–2355 (2020)

    MathSciNet  MATH  Google Scholar 

  37. Luo, Q.M., Srivastava, H.M.: Some generalizations of the Apostol-Genocchi polynomials and the Stirling numbers of the second kind. Appl. Math. Comput. 217, 5702–5728 (2011)

    MathSciNet  MATH  Google Scholar 

  38. Mansour, T., Schork, M.: Commutation Relations, Normal Ordering, and Stirling Numbers. Discrete Mathematics and Its Applications. CRC Press, Boca Raton (2016)

    MATH  Google Scholar 

  39. Médicis, A.D., Leroux, P.: Generalized Stirling numbers, convolution formulae and \(p, q\)-analogues. Can. J. Math. 47(3), 474–499 (1995)

    MathSciNet  MATH  Google Scholar 

  40. Merris, R.: Combinatorics, 2nd edn. Wiley, Hoboken (2003)

    MATH  Google Scholar 

  41. Poon, S.S.: Higher Derivatives of the Falling Factorial and Related Generalizations of the Stirling and Harmonic Numbers. arXiv:1401.2737v1

  42. Qi, F.: Explicit formulas for computing Bernoulli numbers of the second kind and Stirling numbers of the first kind. Filomat 28, 319–327 (2014)

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  44. Schikhof, W.H.: Ultrametric Calculus: An Introduction to \(p\)-adic Analysis. In: Cambridge Studies in Advanced Mathematics vol. 4. Cambridge University Press, Cambridge (1984)

  45. Schmidt, M.D.: Combinatorial identities for generalized Stirling numbers expanding \(f\)-factorial functions and the \(f\)-harmonic numbers. J. Integer Seq. 21, article 18.2.7 (2018)

  46. Simsek, Y.: Generating functions for generalized Stirling type numbers, array type polynomials, Eulerian type polynomials and their applications. Fixed Point Theory Appl. 2013(87), 343–355 (2013)

    MathSciNet  MATH  Google Scholar 

  47. Simsek, Y.: Identities on the Changhee numbers and Apostol-type Daehee polynomials. Adv. Stud. Contemp. Math. (Kyungshang) 27(2), 199–212 (2017)

    MATH  Google Scholar 

  48. Simsek, Y.: New families of special numbers for computing negative order Euler numbers and related numbers and polynomials. Appl. Anal. Discrete Math. 12(1), 1–35 (2018)

    MathSciNet  MATH  Google Scholar 

  49. Simsek, Y.: Explicit formulas for \(p\)-adic integrals: Approach to \(p\)-adic distributions and some families of special numbers and polynomials. Montes Taurus J. Pure Appl. Math. 1(1), 1–76 (2019)

    Google Scholar 

  50. Srivastava, H.M.: Some formulas for the Bernoulli and Euler polynomials at rational arguments. Math. Proc. Camb. Philos. Soc. 129, 77–84 (2000)

    MathSciNet  MATH  Google Scholar 

  51. Srivastava, H.M.: Some generalizations and basic (or \(q\)-) extensions of the Bernoulli, Euler and Genocchi polynomials. Appl. Math. Inf. Sci. 5, 390–444 (2011)

    MathSciNet  Google Scholar 

  52. Srivastava, H.M., Choi, J.: Zeta and \(q\)-Zeta Functions and Associated Series and Integrals. Elsevier Science Publishers, Amsterdam (2012)

    MATH  Google Scholar 

  53. Xu, A.: A Newton interpolation approach to generalized Stirling numbers. J. Appl. Math. 2012, article ID: 351935, 1–17 (2012)

  54. Wolfram Research Inc.: Mathematica Online (Wolfram Cloud). Wolfram Research Inc., Champaign (2021). https://www.wolframcloud.com

Download references

Author information

Authors and Affiliations

  1. Department of Engineering Fundamental Sciences, Faculty of Engineering, Alanya Alaaddin Keykubat University, 07425, Antalya, Turkey

    Irem Kucukoglu

  2. Department of Mathematics, Faculty of Science, University of Akdeniz, 07058, Antalya, Turkey

    Yilmaz Simsek

Authors
  1. Irem Kucukoglu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yilmaz Simsek
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Irem Kucukoglu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kucukoglu, I., Simsek, Y. Construction and computation of unified Stirling-type numbers emerging from p-adic integrals and symmetric polynomials. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 115, 167 (2021). https://doi.org/10.1007/s13398-021-01107-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13398-021-01107-2

Keywords

Mathematics Subject Classification

Search

Navigation