JOURNAL BROWSE
Search
Advanced SearchSearch Tips
IDENTITIES OF SYMMETRY FOR THE HIGHER ORDER q-BERNOULLI POLYNOMIALS
facebook(new window)  Pirnt(new window) E-mail(new window) Excel Download
 Title & Authors
IDENTITIES OF SYMMETRY FOR THE HIGHER ORDER q-BERNOULLI POLYNOMIALS
Son, Jin-Woo;
  PDF(new window)
 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
p-adic analysis;higher order q-Bernoulli polynomials;power sums;Volkenborn integral;q-extension of the power sums;identities of symmetry;
 Language
English
 Cited by
 References
1.
T. M. Apostol, On the Lerch zeta function, Pacific J. Math. 1 (1951), 161-167. crossref(new window)

2.
A. Bayad, Fourier expansions for Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials, Math. Comp. 80 (2011), no. 276, 2219-2221. crossref(new window)

3.
J. G. F. Belinfante, Problems and Solutions: Elementary Problems: E3237-E3242, Amer. Math. Monthly 94 (1987), no. 10, 995-996. crossref(new window)

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. crossref(new window)

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. crossref(new window)

7.
F. T. Howard, Applications of a recurrence for the Bernoulli numbers, J. Number Theory 52 (1995), no. 1, 157-172. crossref(new window)

8.
D. S. Kim, Identities of symmetry for q-Bernoulli polynomials, Comput. Math. Appl. 60 (2010), no. 8, 2350-2359. crossref(new window)

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. crossref(new window)

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. crossref(new window)

12.
M.-S. Kim and S. Hu, Sums of products of Apostol-Bernoulli numbers, Ramanujan J. 28 (2012), no. 1, 113-123. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

19.
Q.-M. Luo, Apostol-Euler polynomials of higher order and Gaussian hypergeometric functions, Taiwanese J. Math. 10 (2006), no. 4, 917-925.

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. crossref(new window)

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. crossref(new window)

22.
V. Namias, A simple derivation of Stirling's asymptotic series, Amer. Math. Monthly 93 (1986), no. 1, 25-29. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

34.
S.-l. Yang, An identity of symmetry for the Bernoulli polynomials, Discrete Math. 308 (2008), no. 4, 550-554. crossref(new window)

35.
P.-T. Young, Degenerate Bernoulli polynomials, generalized factorial sums, and their applications, J. Number Theory 128 (2008), no. 4, 738-758. crossref(new window)