Movatterモバイル変換


[0]ホーム

URL:


Skip to main content
Springer Nature Link
Log in

New Computational Formulas for Special Numbers and Polynomials Derived from Applying Trigonometric Functions to Generating Functions

  • Published:
Milan Journal of Mathematics Aims and scope Submit manuscript

Abstract

The aim of this paper is to apply trigonometric functions with functional equations of generating functions. Using the resulted new equations and formulas from this application, we obtain many special numbers and polynomials such as the Stirling numbers, Bernoulli and Euler type numbers, the array polynomials, the Catalan numbers, and the central factorial numbers. We then introduce combinatorial sums related to these special numbers and polynomials. Moreover, we gave some remarks that relates our new findings from this paper to the relations found in earlier studies.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Japan)

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. 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 (2007).https://doi.org/10.1080/01630568908816313

    Article MathSciNet MATH  Google Scholar 

  2. Brychkov, YuA: On multiple sums of special functions. Integral Transforms Spec. Funct.21(12), 877–884 (2010)

    Article MathSciNet  Google Scholar 

  3. Carlitz, L.: Recurrences for the Bernoulli and Euler Numbers. II. Math. Nachr.29, 151–160 (1965)

    Article MathSciNet  Google Scholar 

  4. Cigler, J.: Fibonacci polynomials and central factorial numbers. Preprint.https://homepage.univie.ac.at/johann.cigler/preprints/central-factorial.pdf

  5. Comtet, L.: Advanced Combinatorics. D. Reidel Publishing Company, Dordrecht (1974)

    Book  Google Scholar 

  6. Djordjević, G.B., Milovanović, G.V.: Special Classes of Polynomials. University of Niš, Faculty of Technology, Leskovac (2014)

    Google Scholar 

  7. Gould, H.W.:Combinatorial Identities: Table III: Binomial Identities Derived from Trigonometric and Exponential Series. May 3, 2010, Vol.6,https://math.wvu.edu/~hgould/Vol.6.PDF

  8. Gun, D., Simsek, Y.: Some new identities and inequalities for Bernoulli polynomials and numbers of higher order related to the Stirling and Catalan numbers. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat.114, 1–12 (2020).https://doi.org/10.1007/s13398-020-00899-z

    Article MathSciNet MATH  Google Scholar 

  9. Kilar, N., Simsek, Y.: Formulas and relations of special numbers and polynomials arising from functional equations of generating functions. Montes Taurus J. Pure Appl. Math.3(1), 106–123 (2021)

    Google Scholar 

  10. Kilar, N., Simsek, Y.: Two parametric kinds of Eulerian-type polynomials associated with Euler’s formula. Symmetry11, 1–19 (2019)

    Article  Google Scholar 

  11. Kilar, N., Simsek, Y.: Identities and relations for special numbers and polynomials: an approach to trigonometric functions. Filomat34(2), 535–542 (2020)

    Article MathSciNet  Google Scholar 

  12. Kilar, N., Simsek, Y.: A special approach to derive new formulas for some special numbers and polynomials. Turk. J. Math.44, 2217–2240 (2020)

    Article MathSciNet  Google Scholar 

  13. Kilar, N., Simsek, Y.: Computational formulas and identities for new classes of Hermite-based Milne–Thomson type polynomials: Analysis of generating functions with Euler’s formula. Math. Methods Appl. Sci. (2021).https://doi.org/10.1002/mma.7220

    Article MATH  Google Scholar 

  14. Kim, T., Ryoo, C.S.: Some identities for Euler and Bernoulli polynomials and their zeros. Axioms7, 1–19 (2018)

    Article  Google Scholar 

  15. Luo, Q.-M., Srivastava, H.M.: Some generalizations of the Apostol–Bernoulli and Apostol–Euler polynomials. J. Math. Anal. Appl.308, 290–302 (2005)

    Article MathSciNet  Google Scholar 

  16. Masjed-Jamei, M., Beyki, M.R., Koepf, W.: A new type of Euler polynomials and numbers. Mediterr. J. Math.15, 1–17 (2018)

    Article MathSciNet  Google Scholar 

  17. Masjed-Jamei, M., Beyki, M.R., Koepf, W.: An extension of the Euler–Maclaurin quadrature formula using a parametric type of Bernoulli polynomials. Bull. Sci. Math.156, 1–26 (2019)

    Article MathSciNet  Google Scholar 

  18. Milovanović, G.V., Stanić, M.P., Tomović, T.V.: Trigonometric multiple orthogonal polynomials of semi-integer degree and the corresponding quadrature formulas. Publ. Inst. Math. Nouv. Sér.96(110), 211–226 (2014)

    Article MathSciNet  Google Scholar 

  19. Milovanović, G.V., Simsek, Y., Stojanović, V.S.: A class of polynomials and connections with Bernoulli’s numbers. J. Anal.27, 709–726 (2019)

    Article MathSciNet  Google Scholar 

  20. Ozdemir, G., Simsek, Y., Milovanović, G.V.: Generating functions for special polynomials and numbers including Apostol-type and Humbert-type polynomials. Mediterr. J. Math.14, 1–17 (2017)

    Article MathSciNet  Google Scholar 

  21. Qi, F., Guo, B.-N.: Sums of infinite power series whose coefficients involve products of the Catalan–Qi numbers. Montes Taurus J. Pure Appl. Math.1(2), 1–12 (2019)

    MathSciNet  Google Scholar 

  22. Roman, S.: An Introduction to Catalan Numbers. Birkhauser, Basel (2015)

    Book  Google Scholar 

  23. Simsek, Y.: Special functions related to Dedekind-type DC-sums and their applications. Russ. J. Math. Phys.17, 495–508 (2010)

    Article MathSciNet  Google Scholar 

  24. Simsek, Y.: Generating functions for generalized Stirling type numbers, array type polynomials, Eulerian type polynomials and their applications. Fixed Point Theory Appl.87, 343–1355 (2013)

    MathSciNet MATH  Google Scholar 

  25. Simsek, Y.: Special numbers on analytic functions. Appl. Math.5, 1091–1098 (2014)

    Article  Google Scholar 

  26. Simsek, Y.: Analysis of the Bernstein basis functions: an approach to combinatorial sums involving binomial coefficients and Catalan numbers. Math. Method. Appl. Sci.38, 3007–3021 (2015)

    Article MathSciNet  Google Scholar 

  27. Simsek, Y.: Computation methods for combinatorial sums and Euler-type numbers related to new families of numbers. Math. Methods Appl. Sci.40(7), 2347–2361 (2017)

    Article MathSciNet  Google Scholar 

  28. Simsek, Y.: New families of special numbers for computing negative order Euler numbers and related numbers and polynomials. Appl. Anal. Discret. Math.12, 1–35 (2018).https://doi.org/10.2298/AADM1801001S

    Article MathSciNet  Google Scholar 

  29. Simsek, Y.: Combinatorial identities and sums for special numbers and polynomials. Filomat32(20), 6869–6877 (2018)

    Article MathSciNet  Google Scholar 

  30. Stanley, R.P.: Catalan Numbers. Cambridge University Press, New York (2015)

    Book  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

Download references

Acknowledgements

The present paper was supported by Scientific Research Project Administration of Akdeniz University with Project Number: FDK-2020-5276. The authors would like to thank the referees for their valuable comments on this present paper.

Author information

Authors and Affiliations

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

    Neslihan Kilar & Yilmaz Simsek

Authors
  1. Neslihan Kilar

    You can also search for this author inPubMed Google Scholar

  2. Yilmaz Simsek

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence toYilmaz Simsek.

Rights and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kilar, N., Simsek, Y. New Computational Formulas for Special Numbers and Polynomials Derived from Applying Trigonometric Functions to Generating Functions.Milan J. Math.89, 217–239 (2021). https://doi.org/10.1007/s00032-021-00333-y

Download citation

Keywords

Mathematics Subject Classification

Access this article

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Japan)

Instant access to the full article PDF.

Advertisement


[8]ページ先頭

©2009-2025 Movatter.jp