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Closed formulas and determinantal expressions for higher-order Bernoulli and Euler polynomials in terms of Stirling numbers

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

Abstract

In this paper, the author presents several closed forms and determinantal expressions involving Stirling numbers of the second kind for higher-order Bernoulli and Euler polynomials by applying the Faà di Bruno formula and some properties of Bell polynomials.

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References

  1. Borwein, J.M., Crandall, R.E.: Closed forms: what they are and why we care. Notices Amer. Math. Soc. 60(1), 50–65 (2013). https://doi.org/10.1090/noti936

    Article  MathSciNet  MATH  Google Scholar 

  2. Bourbaki, N.: Functions of a real variable, elementary theory, translated from the 1976 French original by Philip Spain. Elements of mathematics (Berlin). Springer, Berlin (2004). https://doi.org/10.1007/978-3-642-5

    Book  Google Scholar 

  3. Comtet, L.: Advanced combinatorics: the art of finite and infinite expansions, revised and enlarged edition. D. Reidel Publishing Co., Dordrecht (1974). https://doi.org/10.1007/978-94-010-2196-8

    Book  MATH  Google Scholar 

  4. Dai, L., Pan, H.: Closed forms for degenerate Bernoulli polynomials. Bull. Aust. Math. Soc. 101(2), 207–217 (2020). https://doi.org/10.1017/S0004972719001266

    Article  MathSciNet  MATH  Google Scholar 

  5. 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. RACSAM, 114(4) (2020) Paper no. 167, 12 pp; https://doi.org/10.1007/s13398-020-00899-z

  6. Guo, B.-N., Qi, F.: An explicit formula for Bernoulli numbers in terms of Stirling numbers of the second kind. J. Anal. Number Theory 3(1), 27–30 (2015)

    Google Scholar 

  7. Hu, S., Kim, M.-S.: Two closed forms for the Apostol-Bernoulli polynomials. Ramanujan J. 46(1), 103–117 (2018). https://doi.org/10.1007/s11139-017-9907-4

    Article  MathSciNet  MATH  Google Scholar 

  8. Kim, T., Kim, D.S.: Extended Stirling numbers of the first kind associated with Daehee numbers and polynomials. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 113(2), 1159–1171 (2019). https://doi.org/10.1007/s13398-018-0540-x

    Article  MathSciNet  MATH  Google Scholar 

  9. Nörlund, N.E.: Vorlesungen über Differenzenrechnung. Springer-Verlag, Berlin (1924)

    Book  Google Scholar 

  10. Qi, F., Niu, D.-W., Lim, D., Yao, Y.-H.: Special values of the Bell polynomials of the second kind for some sequences and functions. J. Math. Anal. Appl. 491(2), 31 (2020). https://doi.org/10.1016/j.jmaa.2020.124382. (Article 124382)

    Article  MathSciNet  MATH  Google Scholar 

  11. Qi, F., Kouba, O., Kaddoura, I.: Computation of several Hessenberg determinants. Mathematica Slovaca 71, (2021). in press

  12. Qi, F., Dağlı, M.C., Du, W.-S.: Determinantal forms and recursive relations of the Delannoy two-functional sequence. Adv. Theory Nonlinear Anal. Appl. 4(3), 184–193 (2020). https://doi.org/10.31197/atnaa.772734

    Article  Google Scholar 

  13. Qi, F., Lim, D., Yao, Y.-H.: Notes on two kinds of special values for the Bell polynomials of the second kind. Miskolc Math. Notes 20(1), 465–474 (2019). https://doi.org/10.18514/MMN.2019.2635

    Article  MathSciNet  MATH  Google Scholar 

  14. Qi, F., Chapman, R.J.: Two closed forms for the Bernoulli polynomials. J. Number Theory 159, 89–100 (2016). https://doi.org/10.1016/j.jnt.2015.07.021

    Article  MathSciNet  MATH  Google Scholar 

  15. Qi, F., Guo, B.-N.: Some determinantal expressions and recurrence relations of the Bernoulli polynomials. Mathematics 4(4), 1–11 (2016). https://doi.org/10.3390/math4040065

    Article  MATH  Google Scholar 

  16. Qi, F.: Derivatives of tangent function and tangent numbers. Appl. Math. Comput. 268, 844–858 (2015). https://doi.org/10.1016/j.amc.2015.06.123

    Article  MathSciNet  MATH  Google Scholar 

  17. Qi, F., Zheng, M.-M.: Explicit expressions for a family of the Bell polynomials and applications. Appl. Math. Comput. 258, 597–607 (2015). https://doi.org/10.1016/j.amc.2015.02.027

    Article  MathSciNet  MATH  Google Scholar 

  18. Qi, F., Guo, B.-N.: Explicit formulas for special values of the Bell polynomials of the second kind and for the Euler numbers and polynomials. Mediterr. J. Math. 14(3), 14 (2017). https://doi.org/10.1007/s00009-017-0939-1. (Article 140)

    Article  MathSciNet  MATH  Google Scholar 

  19. Qi, F., Lim, D., Guo, B.-N.: Explicit formulas and identities for the Bell polynomials and a sequence of polynomials applied to differential equations. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 113, 1–9 (2019). https://doi.org/10.1007/s13398-017-0427-2

    Article  MathSciNet  MATH  Google Scholar 

  20. Qi, F.: An explicit formula for the Bell numbers in terms of the Lah and Stirling numbers. Mediterr. J. Math. 13, 2795–2800 (2016). https://doi.org/10.1007/s00009-015-0655-7

    Article  MathSciNet  MATH  Google Scholar 

  21. Wei, C.-F., Qi, F.: Several closed expressions for the Euler numbers. J. Inequal. Appl. 219, 2015 (2015). https://doi.org/10.1186/s13660-015-0738-9

    Article  MathSciNet  MATH  Google Scholar 

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

    Muhammet Cihat Dağlı

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Dağlı, M.C. Closed formulas and determinantal expressions for higher-order Bernoulli and Euler polynomials in terms of Stirling numbers. RACSAM 115, 32 (2021). https://doi.org/10.1007/s13398-020-00970-9

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