A FURTHER GENERALIZATION OF APOSTOL-BERNOULLI POLYNOMIALS AND RELATED POLYNOMIALS

• Journal title : Honam Mathematical Journal
• Volume 34, Issue 3,  2012, pp.311-326
• Publisher : The Honam Mathematical Society
• DOI : 10.5831/HMJ.2012.34.3.311
Title & Authors
A FURTHER GENERALIZATION OF APOSTOL-BERNOULLI POLYNOMIALS AND RELATED POLYNOMIALS
Tremblay, R.; Gaboury, S.; Fugere, J.;

Abstract
The purpose of this paper is to introduce and investigate two new classes of generalized Bernoulli and Apostol-Bernoulli polynomials based on the definition given recently by the authors [29]. In particular, we obtain a new addition formula for the new class of the generalized Bernoulli polynomials. We also give an extension and some analogues of the Srivastava-Pint$\small{\acute{e}}$r addition theorem [28] for both classes. Finally, by making use of the new adition formula, we exhibit several interesting relationships between generalized Bernoulli polynomials and other polynomials or special functions.
Keywords
Generalized Bernoulli polynomials;Generalized Apostol-Bernoulli polynomials;Generalized Apostol-Euler polynomials;Generating Functions;Addition Theorem;Special Functions;
Language
English
Cited by
1.
SOME EXPLICIT FORMULAS FOR CERTAIN NEW CLASSES OF BERNOULLI, EULER AND GENOCCHI POLYNOMIALS,;;;

Proceedings of the Jangjeon Mathematical Society, 2014. vol.17. 1, pp.115-123
References
1.
M. Abramowitz and I. A. Stegun, Handbook of mathematical functions with formulas, graphs and mathematical tables, National Bureau of Standards, Washington, DC, 1964.

2.
T. M. Apostol, On the Lerch zeta function, Pacic J. Math. 1 (1951), 161-167.

3.
K. N. Boyadzhiev, Apostol-bernoulli functions, derivative polynomials and eulerian polynomials, Advances and Applications in Discrete Mathematics 1 (2008), no.2, 109-122.

4.
J. Choi, P. J. Anderson, and H. M. Srivastava, Some q-extensions of the apostolbernoulli and the apostol-euler polynomials of order n, and the multiple hurwitz zeta function, Appl. Math. Comput 199 (2008), 723-737.

5.
L. Comtet, Advanced combinatorics: The art of nite and innite expansions, (Translated from french by J.W. Nienhuys), Reidel, Dordrecht, 1974.

6.
A. Erdelyi, W. Magnus, F. Oberhettinger, and F. Tricomi, Higher transcendental functions, vols.1-3,, 1953.

7.
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 (2006), no. 11, 803-815.

8.
E. R. Hansen, A table of series and products, Prentice-Hall, Englewood Cliffs, NJ, 1975.

9.
B. Kurt, A further generalization of the Bernoulli polynomials and on the 2D-Bernoulli polynomials $B_{n}^{2}$ (x, y), Appl. Math.Sci. Vol.4 (47) (2010), 2315-2322.

10.
Y. Luke, The special functions and their approximations, vols. 1-2, 1969.

11.
Q.-M. Luo, Apostol-Euler polynomials of higher order and gaussian hypergeometric functions, Taiwanese J. Math. 10 (4) (2006), 917-925.

12.
Q.-M. Luo, Fourier expansions and integral representations for the apostol-bernoulli and apostol-euler polynomials, Math. Comp. 78 (2009), 2193-2208.

13.
Q.-M. Luo, The multiplication formulas for the apostol-bernoulli and apostol-euler polynomials of higher order, Integral Transform Spec. Funct. 20 (2009), 377-391.

14.
Q.-M. Luo, Some formulas for apostol-euler polynomials associated with hurwitz zeta function at rational arguments, Applicable Analysis and Discrete Mathematics 3 (2009), 336-346.

15.
Q.-M. Luo, 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 (2010), 1159-1171.

16.
Q.-M. Luo, Extension for the genocchi polynomials and its fourier expansions and integral representations, Osaka J. Math. 48 (2011), 291-310.

17.
Q.-M. Luo, B.-N. Guo, F. Qui, and L. Debnath, Generalizations of Bernoulli numbers and polynomials, Int. J. Math. Math. Sci. 59 (2003), 3769-3776.

18.
Q.-M. Luo and H.M. Srivastava, Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials, J. Math.Anal.Appl. 308 (1) (2005), 290-302.

19.
Q.-M. Luo and H.M. Srivastava, Some generalizations of the apostol-genocchi polynomials and the stirling numbers of the second kind, Appl. Math. Comput 217 (2011), 5702-5728.

20.
Q.M. Luo and H.M. Srivastava, Some relationships between the Apostol-Bernoulli and Apostol-Euler polynomials, Comput. Math. Appl. 51 (2006), 631-642

21.
F. Magnus, W. Oberhettinger and R. P. Soni, Formulas and theorems for the special functions of mathematical physics, Third enlarged edition, Springer-Verlag, New York, 1966.

22.
P. Natalini and A. Bernardini, A generalization of the Bernoulli polynomials, J. Appl. Math. 3 (2003), 155-163.

23.
M. Prevost, Pade approximation and apostol-bernoulli and apostol-euler polynomials, J. Comput. Appl. Math. 233 (2010), 3005-3017.

24.
E. D. Rainville, Special functions, Macmillan Company, New York, 1960.

25.
H. M. Srivastava and J. Choi, Series associated with zeta and related functions, Kluwer Academin Publishers, Dordrecht, Boston and London, 2001.

26.
H. M. Srivastava, M. Garg, and S. Choudhary, A new generalization of the Bernoulli and related polynomials, Russian J. Math. Phys. 17 (2010), 251-261.

27.
H. M. Srivastava, J.-L. Lavoie, and R. Tremblay, A class of addition theorems, Canad. Math.Bull. 26 (1983), 438-445.

28.
H. M. Srivastava and A. Pinter, Remarks on some relationships between the Bernoulli and Euler polynomials, Appl. Math. Lett. 17 (4) (2004), 375-380.

29.
R. Tremblay, S. Gaboury, and B. J. Fugere, A new class of generalized Apostol- Bernoulli polynomials and some analogues of the Srivastava-Pinter addition theorem, Appl. Math. Lett. 24 (2011), 1888-1893.

30.
W. Wang, C. Jia, and T. Wang, Some results on the Apostol-Bernoulli and Apostol-Euler polynomials, Comput. Math. Appl. 55 (2008), 1322-1332.