The multiplication formulas for the Apostol-Bernoulli and Apostol-Euler polynomials of higher order

Integral Transforms and Special Functions

Volumn 20, Issue 5, 2009, Pages 377-391

  Luo, Qiu Ming ^a,b  

^a EAST CHINA NORMAL UNIVERSITY   (China)

^b Department of Mathematics   (Taiwan)

Author keywords

multiple power sum and multiple alternating sum; Apostol Bernoulli numbers and polynomials (of higher order); Apostol Euler numbers and polynomials (of higher order); Bernoulli numbers and polynomials (of higher order); Euler numbers and polynomials (of higher order); Generalized multinomial identity; Multinomial identity; Multiplication formula; Power sum and alternating sum; Raabe's multiplication formula

Indexed keywords

EID: 68949114125     PISSN: 10652469     EISSN: 14768291     Source Type: Journal    
DOI: 10.1080/10652460802564324     Document Type: Article

References (22)

1. M. Abramowitz and I.A. Stegun (eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series 55, 4th Printing, Washington, 1965.
2. T.M. Apostol, On the Lerch Zeta function, Pacific J. Math. 1 (1951), pp. 161-167.
3. L. Carlitz, The multiplication formulas for the Bernoulli and Euler polynomials,Math. Mag. 27 (1953/54), pp. 59-64.
4. L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions (Translated from the French by J.W. Nienhuys), Reidel, Dordrecht and Boston, 1974.
5. E. Deeba and D. Rodriguez, Stirlings series and Bernoulli numbers, Amer. Math. Monthly 98 (1991), pp. 423-426.
6. M. Garg, K. Jain, and H.M. Srivastava, Some relationships between the generalized Apostol-Bernoulli polynomials and Hurwitz-Lerch Zeta functions, Integral Transform. Spec. Funct. 17(11) (2006), pp. 803-815.
7. F.T. Howard, Applications of a recurrence for Bernoulli numbers, J. Number Theory 52 (1995), pp. 157-172.
8. T. Kim, Symmetry p-adic invariant integral on Zp for Bernoulli and Euler polynomials, Preprint. Available at http://www.arXiv:0712.0138v1 [math.NT], 2007.
9. p, Preprint. Availabel at http://www.arXiv:0804.4605v1 [math.NT].
10. T. Kim, S.-H. Rim, and Y. Simsek, A note on the alternating sums of powers of consecutive q-integers, Adv. Stud. Contemp. Math. 13(2) (2006), pp. 159-164.
11. Q.-M. Luo, An explicit relationship between the generalized Apostol-Bernoulli and Apostol-Euler polynomials associated with λ-Stirling numbers of the second kind, Houston J. Math. accepted for publication.
12. Q.-M. Luo, Apostol-Euler polynomials of higher order and Gaussian hypergeometric functions, Taiwanese J. Math. 10 (2006), pp. 917-925.
13. Q.-M. Luo and H.M. Srivastava, Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials, J. Math. Anal. Appl. 308 (2005), pp. 290-302.
14. Q.-M. Luo and H.M. Srivastava, Some relationships between the Apostol-Bernoulli and Apostol-Euler polynomials, Comput. Math. Appl. 51 (2006), pp. 631-642.
15. W. Magnus, F. Oberhettinger, and R.P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd Enlarged ed., Springer-Verlag, NewYork, 1966.
16. N.E. Nörlund, Vorlesungen über Differentzenrechnung, Springer-Verlag, Berlin, 1924; Reprinted by Chelsea Publishing Company, Bronx, NewYork, 1954.
17. H. Ozden, I.N. Cangul, and Y. Simsek, Multivariate interpolation functions of higher-order q-Euler numbers and their applications, Abstract. Appl. Anal. (Article No 390857), 2008 Volume 16 pages (2008).
18. H. Ozden, I.N. Cangul, andY. Simsek, Remarks on sum of products of (h; q)-twisted Euler polynomials and numbers, J. Inequal. Appl. 2008 (2008) (Article ID816129), 8 pages.
19. H. Ozden andY. Simsek,Anewextension of q-Euler numbers and polynomials related to their interpolation functions, Appl. Math. Lett. 21 (2008), pp. 934-939.
20. Y. Simsek, V. Kurt, and D. Kim, New approach to the complete sum of products of the twisted (h; q)-Bernoulli numbers and polynomials, J. Nonlinear Math. Phys. 14 (2007), pp. 44-56.
21. H.M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, Boston and London, 2001.
22. W. Wang, C. Jia, and T. Wang, Some results on the Apostol-Bernoulli and Apostol-Euler polynomials, Comput. Math. Appl. 55 (2008), pp. 1322-1332.