References
- T. M. Apostol, On the Lerch zeta function, Pacific J. Math. 1 (1951), 161-167. https://doi.org/10.2140/pjm.1951.1.161
- 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
- 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
- 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
- H. Cohen, Number Theory Vol. II: Analytic and Modern Tools, Graduate Texts in Mathematics, 240, Springer, New York, 2007.
- 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
- 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
- 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
- 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
-
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. - 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
- 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
- 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
- T. Kim, On the symmetries of the q-Bernoulli polynomials, Abstr. Appl. Anal. 2008 (2008), Art. ID 914367, 7 pp.
- 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
-
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 - 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.
- 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
- 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
- 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
- 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
- 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
- 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
- N. E. Norlund, Vorlesungen uber Differenzenrechnung, Berlin, 1924.
- Ju. V. Osipov, p-adic zeta functions, (Russian), Uspekhi Mat. Nauk 34 (1979), no. 3, 209-210.
- W. H. Schikhof, Ultrametric Calculus: An Introduction to p-Adic Analysis, Cambridge University Press, 2006.
- 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
- 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
- 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
- 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
- L. Tao and Z. W. Sun, A reciprocity law for uniform functions, Nanjing Univ. J. Math. Biquarterly 21 (2004), no. 2, 201-205.
- 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
- 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
- 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
- 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