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The Ramanujan Journal

, Volume 46, Issue 1, pp 103–117 | Cite as

Two closed forms for the Apostol–Bernoulli polynomials

  • Su Hu
  • Min-Soo Kim
Article
  • 206 Downloads

Abstract

In this note, we shall obtain two closed forms for the Apostol–Bernoulli polynomials.

Keywords

Stirling numbers Apostol–Bernoulli polynomials Closed forms 

Mathematics Subject Classification

11B68 05A19 

Notes

Acknowledgements

The authors are enormously grateful to the anonymous referee whose comments and suggestions lead to a large improvement of the paper. The first author is supported by the National Natural Science Foundation of China (Grant No. 11501212). The second author is supported by the Kyungnam University Foundation Grant, 2016.

References

  1. 1.
    Apostol, T.M.: On the Lerch zeta function. Pac. J. Math. 1, 161–167 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Apostol, T.M.: Addendum to “On the Lerch zeta function”. Pac. J. Math. 2, 10 (1952)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bayad, A.: Fourier expansions for Apostol–Bernoulli, Apostol–Euler and Apostol–Genocchi polynomials. Math. Comp. 80, 2219–2221 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    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)Google Scholar
  5. 5.
    Comtet, L.: Advanced Combinatorics: The Art of Finite and Infinite Expansions, Revised and, Enlarged edn. D. Reidel Publishing Co., Dordrecht (1974)CrossRefzbMATHGoogle Scholar
  6. 6.
    Cheon, G.-S.: A note on the Bernoulli and Euler polynomials. Appl. Math. Lett. 16, 365–368 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Choi, J., Jang, D.S., Srivastava, H.M.: A generalization of the Hurwitz–Lerch zeta function. Integral Transform. Spec. Funct. 19(1–2), 65–79 (2008)MathSciNetzbMATHGoogle Scholar
  8. 8.
  9. 9.
    Deeba, E., Rodriguez, D.: Stirlings series and Bernoulli numbers. Am. Math. Mon. 98, 423–426 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Euler, L.: Institutiones calculi differentialis, (II) 487–491Google Scholar
  11. 11.
    Garg, M., Jain, K., Srivastava, H.M.: Some relationships between the generalized Apostol–Bernoulli polynomials and Hurwitz–Lerch Zeta functions. Integral Transform. Spec. Funct. 17(11), 803–815 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Howard, F.T.: Applications of a recurrence for Bernoulli numbers. J. Number Theory 52, 157–172 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kim, M.-S., Hu, S.: Sums of products of Apostol–Bernoulli numbers. Ramanujan J. 28, 113–123 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lu, D.-Q., Luo, Q.-M.: Some properties of the generalized Apostol type polynomials. Bound. Value Probl. 2013, 64 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lu, D.-Q., Luo, Q.-M.: Some unified formulas and representations for the Apostol-type polynomials. Adv. Differ. Equ. 2015, 137 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Luke, Y.L.: The Special Functions and Their Applications, vol. I. Academic Press, New York (1969)zbMATHGoogle Scholar
  17. 17.
    Luo, Q.-M.: On the Apostol–Bernoulli polynomials. Cent. Eur. J. Math. 2(4), 509–515 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Luo, Q.-M.: Apostol–Euler polynomials of higher order and Gaussian hypergeometric functions. Taiwan. J. Math. 10(4), 917–925 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Luo, Q.-M.: The multiplication formulas for the Apostol–Bernoulli and Apostol–Euler polynomials of higher order. Integral Transform. Spec. Funct. 20, 377–391 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Luo, Q.-M.: Fourier expansions and integral representations for the Apostol–Bernoulli and Apostol–Euler polynomials. Math. Comput. 78, 2193–2208 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Luo, Q.-M.: An explicit relationship between the generalized Apostol–Bernoulli and Apostol–Euler polynomials associated with \(\lambda \)-Stirling numbers of the second kind. Houston J. Math. 36, 1159–1171 (2010)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Luo, Q.-M.: Extension for the Genocchi polynomials and its Fourier expansions and integral representations. Osaka J. Math. 48, 291–309 (2011)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Luo, Q.-M.: q-Extensions of some results involving the Luo–Srivastava generalizations of the Apostol–Bernoulli and Apostol-Euler polynomials. Filomat 28, 329–351 (2014)Google Scholar
  24. 24.
    Luo, Q.-M.: q-Apostol–Euler polynomials and q-alternating sums. Ukr. Math. J. 65, 1231–1246 (2014)Google Scholar
  25. 25.
    Luo, Q.-M.: Elliptic extensions of the Apostol–Bernoulli and Apostol–Euler polynomials. Appl. Math. Comput. 261, 156–166 (2015)MathSciNetGoogle Scholar
  26. 26.
    Luo, Q.-M., Srivastava, H.M.: Some generalizations of the Apostol–Bernoulli and Apostol–Euler polynomials. J. Math. Anal. Appl. 308, 290–302 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Luo, Q.-M., Srivastava, H.M.: Some relationships between the Apostol–Bernoulli and Apostol–Euler polynomials. Comput. Math. Appl. 51, 631–642 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    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)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Navas, L.M., Ruiz, F.J., Varona, J.L.: Asymptotic estimates for Apostol–Bernoulli and Apostol–Euler polynomials. Math. Comput. 81, 1707–1722 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Qi, F., Chapman, R.J.: Two closed forms for the Bernoulli polynomials. J. Number Theory 159, 89–100 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Saalschütz, L.: Vorlesungen über die Bernoullischen Zahlen, ihren Zusammenhang mit den Secanten-Coefficienten und ihre wichtigeren Anwendungen, Berlin, 1893 (Available since 1964 in Xerographed form from University Microfilms, Ann Arbor, Michigan, Order No. OP-17136)Google Scholar
  32. 32.
    Srivastava, H.M.: Some formulae for the Bernoulli and Euler polynomials at rational arguments. Math. Proc. Camb. Philos. Soc. 129, 77–84 (2000)CrossRefzbMATHGoogle Scholar
  33. 33.
    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)MathSciNetGoogle Scholar
  34. 34.
    Srivastava, H.M., Pintér, A.: Remarks on some relationships between the Bernoulli and Euler polynomials. Appl. Math. Lett. 17, 375–380 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Srivastava, H.M., Kurt, B., Simsek, Y.: Some families of Genocchi type polynomials and their interpolation functions. Integral Transform. Spec. Funct. 23, 919–938. See also Corrigendum. Integral Transforms Spec. Funct. 23, 939–940 (2012)Google Scholar
  36. 36.
    Vandiver, H.S.: An arithmetical theory of the Bernoulli numbers. Trans. Am. Math. Soc. 51, 502–531 (1942)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Xu, A.-M., Cen, Z.-D.: Some identities involving exponential functions and Stirling numbers and applications. J. Comput. Appl. Math. 260, 201–207 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Zhang, Z.-Z., Yang, J.-Z.: Notes on some identities related to the partial Bell polynomials. Tamsui Oxf. J. Inf. Math. Sci. 28, 39–48 (2012)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Zorich, V.A.: Mathematical Analysis I, Translated from the 2002 Fourth Russian Edition by Roger Cooke. Universitext. Springer, Berlin (2009)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2018

Authors and Affiliations

  1. 1.Department of MathematicsSouth China University of TechnologyGuangzhouChina
  2. 2.Center for General EducationKyungnam UniversityChangwon-siRepublic of Korea

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