Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

IDENTITIES OF SYMMETRY FOR THE HIGHER ORDER q-BERNOULLI POLYNOMIALS

  • Son, Jin-Woo (Department of Mechanical Engineering Kyungnam University)
  • Received : 2014.02.26
  • Published : 2014.09.01
Download PDF
⟨ Previous Next ⟩

Abstract

The study of the identities of symmetry for the Bernoulli polynomials arises from the study of Gauss's multiplication formula for the gamma function. There are many works in this direction. In the sense of p-adic analysis, the q-Bernoulli polynomials are natural extensions of the Bernoulli and Apostol-Bernoulli polynomials (see the introduction of this paper). By using the N-fold iterated Volkenborn integral, we derive serval identities of symmetry related to the q-extension power sums and the higher order q-Bernoulli polynomials. Many previous results are special cases of the results presented in this paper, including Tuenter's classical results on the symmetry relation between the power sum polynomials and the Bernoulli numbers in [A symmetry of power sum polynomials and Bernoulli numbers, Amer. Math. Monthly 108 (2001), no. 3, 258-261] and D. S. Kim's eight basic identities of symmetry in three variables related to the q-analogue power sums and the q-Bernoulli polynomials in [Identities of symmetry for q-Bernoulli polynomials, Comput. Math. Appl. 60 (2010), no. 8, 2350-2359].

Keywords

References

  1. T. M. Apostol, On the Lerch zeta function, Pacific J. Math. 1 (1951), 161-167. https://doi.org/10.2140/pjm.1951.1.161
  2. A. Bayad, Fourier expansions for Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials, Math. Comp. 80 (2011), no. 276, 2219-2221. https://doi.org/10.1090/S0025-5718-2011-02476-2
  3. J. G. F. Belinfante, Problems and Solutions: Elementary Problems: E3237-E3242, Amer. Math. Monthly 94 (1987), no. 10, 995-996. https://doi.org/10.2307/2322611
  4. J. G. F. Belinfante and I. Gessel, Problems and Solutions: Solutions of Elementary Problems: E3237, Amer. Math. Monthly 96 (1989), no. 4, 364-365. https://doi.org/10.2307/2324102
  5. H. Cohen, Number Theory Vol. II: Analytic and Modern Tools, Graduate Texts in Mathematics, 240, Springer, New York, 2007.
  6. E. Y. Deeba and D. M. Rodriguez, Stirling's series and Bernoulli numbers, Amer. Math. Monthly 98 (1991), no. 5, 423-426. https://doi.org/10.2307/2323860
  7. F. T. Howard, Applications of a recurrence for the Bernoulli numbers, J. Number Theory 52 (1995), no. 1, 157-172. https://doi.org/10.1006/jnth.1995.1062
  8. D. S. Kim, Identities of symmetry for q-Bernoulli polynomials, Comput. Math. Appl. 60 (2010), no. 8, 2350-2359. https://doi.org/10.1016/j.camwa.2010.08.028
  9. D. S. Kim, N. Lee, J. Na, and K. H. Park, Identities of symmetry for higher-order Euler polynomials in three variables (II), J. Math. Anal. Appl. 379 (2011), no. 1, 388-400. https://doi.org/10.1016/j.jmaa.2011.01.034
  10. D. S. Kim and K. H. Park, Identities of symmetry for Euler polynomials arising from quotients of fermionic integrals invariant under $S_3$, J. Inequal. Appl. 2010 (2010), Art. ID 851521, 16 pp.
  11. M.-S. Kim, On Euler numbers, polynomials and related p-adic integrals, J. Number Theory 129 (2009), no. 9, 2166-2179. https://doi.org/10.1016/j.jnt.2008.11.004
  12. M.-S. Kim and S. Hu, Sums of products of Apostol-Bernoulli numbers, Ramanujan J. 28 (2012), no. 1, 113-123. https://doi.org/10.1007/s11139-011-9340-z
  13. T. Kim, On the analogs of Euler numbers and polynomials associated with p-adic q-integral on Zp at q = -1, J. Math. Anal. Appl. 331 (2007), no. 2, 779-792. https://doi.org/10.1016/j.jmaa.2006.09.027
  14. T. Kim, On the symmetries of the q-Bernoulli polynomials, Abstr. Appl. Anal. 2008 (2008), Art. ID 914367, 7 pp.
  15. T. Kim, Symmetry p-adic invariant integral on Zp for Bernoulli and Euler polynomials, J. Difference Equ. Appl. 14 (2008), no. 12, 1267-1277. https://doi.org/10.1080/10236190801943220
  16. T. Kim, Symmetry of power sum polynomials and multivariate fermionic p-adic invariant integral on ${\mathbb{Z}}_p$, Russ. J. Math. Phys. 16 (2009), no. 1, 93-96. https://doi.org/10.1134/S1061920809010063
  17. N. Koblitz, p-adic Analysis: a Short Course on Resent Work, London Mathematical Society Lecture Note Series, 46, Cambridge University Press, Cambridge-New York, 1980.
  18. H. Liu and W. Wang, Some identities on the Bernoulli, Euler and Genocchi polynomials via power sums and alternate power sums, Discrete Math. 309 (2009), no. 10, 3346-3363. https://doi.org/10.1016/j.disc.2008.09.048
  19. Q.-M. Luo, Apostol-Euler polynomials of higher order and Gaussian hypergeometric functions, Taiwanese J. Math. 10 (2006), no. 4, 917-925. https://doi.org/10.11650/twjm/1500403883
  20. Q.-M. Luo, Fourier expansions and integral representations for the Apostol-Bernoulli and Apostol-Euler polynomials, Math. Comp. 78 (2009), no. 268, 2193-2208. https://doi.org/10.1090/S0025-5718-09-02230-3
  21. Q.-M. Luo and H. M. Srivastava, Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials, J. Math. Anal. Appl. 308 (2005), no. 1, 290-302. https://doi.org/10.1016/j.jmaa.2005.01.020
  22. V. Namias, A simple derivation of Stirling's asymptotic series, Amer. Math. Monthly 93 (1986), no. 1, 25-29. https://doi.org/10.2307/2322540
  23. L. M. Navas, F. J. Ruiz, and J. L. Varona, Asymptotic estimates for Apostol-Bernoulli and Apostol-Euler polynomials, Math. Comp. 81 (2012), no. 279, 1707-1722. https://doi.org/10.1090/S0025-5718-2012-02568-3
  24. N. E. Norlund, Vorlesungen uber Differenzenrechnung, Berlin, 1924.
  25. Ju. V. Osipov, p-adic zeta functions, (Russian), Uspekhi Mat. Nauk 34 (1979), no. 3, 209-210.
  26. W. H. Schikhof, Ultrametric Calculus: An Introduction to p-Adic Analysis, Cambridge University Press, 2006.
  27. Y. Simsek, Twisted (h, q)-Bernoulli numbers and polynomials related to twisted (h, q)-zeta function and L-function, J. Math. Anal. Appl. 324 (2006), no. 2, 790-804. https://doi.org/10.1016/j.jmaa.2005.12.057
  28. Y. Simsek, Complete sum of products of (h, q)-extension of Euler polynomials and numbers, J. Difference Equ. Appl. 16 (2010), no. 11, 1331-1348. https://doi.org/10.1080/10236190902813967
  29. Z. W. Sun, Introduction to Bernoulli and Euler polynomials, A Lecture Given in Taiwan on June 6, 2002. http://math.nju.edu.cn/.zwsun/BerE.pdf
  30. B. A. Tangedal and P. T. Young, On p-adic multiple zeta and log gamma functions, J. Number Theory 131 (2011), no. 7, 1240-1257. https://doi.org/10.1016/j.jnt.2011.01.010
  31. L. Tao and Z. W. Sun, A reciprocity law for uniform functions, Nanjing Univ. J. Math. Biquarterly 21 (2004), no. 2, 201-205.
  32. H. J. H. Tuenter, A symmetry of power sum polynomials and Bernoulli numbers, Amer. Math. Monthly 108 (2001), no. 3, 258-261. https://doi.org/10.2307/2695389
  33. W. Wang and W. Wang, Some results on power sums and Apostol-type polynomials, Integral Transforms Spec. Funct. 21 (2010), no. 3-4, 307-318. https://doi.org/10.1080/10652460903169288
  34. S.-l. Yang, An identity of symmetry for the Bernoulli polynomials, Discrete Math. 308 (2008), no. 4, 550-554. https://doi.org/10.1016/j.disc.2007.03.030
  35. P.-T. Young, Degenerate Bernoulli polynomials, generalized factorial sums, and their applications, J. Number Theory 128 (2008), no. 4, 738-758. https://doi.org/10.1016/j.jnt.2007.02.007