Proceedings of the Jangjeon Mathematical Society

Jangjeon Mathematical Society (장전수학회)
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SOME EXPLICIT FORMULAS FOR CERTAIN NEW CLASSES OF BERNOULLI, EULER AND GENOCCHI POLYNOMIALS

  • Gaboury, Sebastien (DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE, UNIVERSITY OF QUEBEC AT CHICOUTIMI) ;
  • Tremblay, Richard (DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE, UNIVERSITY OF QUEBEC AT CHICOUTIMI) ;
  • Fugere, Benoit-Jean (DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE, ROYAL MILITARY COLLEGE)
  • Published : 2014.01.01
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Abstract

In recent years, the subject of Apostol-Bernoulli polynomials, Apostol-Euler polynomials and Apostol-Genocchi polynomials have been studied extensively. Recently, the authors have introduced in [27,28] some new generalized classes of Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials. In this paper, with the help of a result involving an explicit formula for the generalized potential polynomials obtained by Cenkci [5], we develop some explicit formulas related with these new classes of polynomials.

Keywords

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, Pacific J. Math. 1 (1951), 161-167. https://doi.org/10.2140/pjm.1951.1.161
  3. K.N. Boyadzhiev, Apostol-bernoulli function, derivative polynomials and eulerian polynomials, Advances and Applications in Discrete Mathematics 1 (2008), no.2, 109-122.
  4. Yu. A. Brychkov, On multiple sums of special functions, Integral Transform Spec. Funct, 21 (12) (2010), 877-884. https://doi.org/10.1080/10652469.2010.480846
  5. M. Cenkci, An explicit formula for generalized potential polynomials and its applications, Discrete Math. 309 (2009), 1498-1510. https://doi.org/10.1016/j.disc.2008.02.021
  6. J. Choi, P.J. Anderson, and H.M. Srivastava, Some q-extensions of the apostol-bernoulli and the apostol-euler polynomials of order n, and the multiple hurwitz zeta function, Appl. Math. Comput 199 (2008), 723-737. https://doi.org/10.1016/j.amc.2007.10.033
  7. L. Comtet, Advanced combinatorics: The art of finite and infinite expansions, (Translated from french by J.W. Nienhuys),Reidel, Dordrecht, 1974.
  8. A. Erdelyi, W. Magnus, F. Oberhettinger, and F. Tricomi, Higher transcendental functions, vols. 1-3,,1953.
  9. 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. https://doi.org/10.1080/10652460600926907
  10. E.R. Hansen, A table of series and products, Prentice-Hall, Englewood Cliffs, NJ, 1975.
  11. F.T. Howard, A sequence of numbers related to the exponential function, Duke Math.J. 34 (1967), 599-616. https://doi.org/10.1215/S0012-7094-67-03465-5
  12. F.T. Howard, Numbers generated by the reciprocal of $e^x$-x-1, Math. Comput. 31 (1977), 581-598.
  13. F.T. Howard, A theorem relating potential and Bell ploynomials, Discrete Math. 39 (1982), 129-143. https://doi.org/10.1016/0012-365X(82)90136-4
  14. Y. Luke, The special functions and their approximations, vols. 1-2, 1969.
  15. Q.-M. Luo, Apostol-Euler polynomials of higher order and gaussian hypergeometric functions, Taiwanese J. Math. 10 (4) (2006), 917-925. https://doi.org/10.11650/twjm/1500403883
  16. Q.-M. Luo, Fourier expansions and integral representations for the apostol-bernoulli and apostol-euler polynomials, Math. Comp. 78 (2009), 2193-2208. https://doi.org/10.1090/S0025-5718-09-02230-3
  17. Q.-M. Luo, The multiplication fromulas for the apostol-bernoulli and apostol-euler polymomials of higher order, Integral Transform Spec. Funct. 20 (2009), 377-391. https://doi.org/10.1080/10652460802564324
  18. Q.-M. Luo, q-extensions for the Apostol-Genocchi polynomails, Gen. Math. 17 (2) (2009), 113-125.
  19. 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. https://doi.org/10.2298/AADM0902336L
  20. Q.-M. Luo, An explicit relationship between the genaralized apostol-bernoulli and apostol-euler polynomials associated with ${\lambda}$-stirling numbers of the second kind, Houston J. Math. 36 (2010), 1159-1171.
  21. Q.-M. Luo, Extension for the genocchi polynomails and its fourier expansions and integral representations, Osaka J. Math. 48 (2011), 291-310.
  22. 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. https://doi.org/10.1016/j.jmaa.2005.01.020
  23. 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. https://doi.org/10.1016/j.amc.2010.12.048
  24. Q.M. Luo and H.M. Srivastava, Some relationships between the Apoltol-Bernoulli and Apostol-Euler polynomials, Comput. Math. Appl. 51 (2006), 631-642. https://doi.org/10.1016/j.camwa.2005.04.018
  25. F. Magnus, W. Oberhettinger and R.P. Soni, Formulas and theorems for the special functions of mathematical physics, Third enlaged edition, Springer-Verlag, New York, 1966.
  26. M. Prevost, Pade approximation and apostol-bernoulli and apostol-euler polynomials, J. Comput. Appl. Math. 233 (2010), 3005-3017. https://doi.org/10.1016/j.cam.2009.11.050
  27. R. Tremblay, S. Gaboury, and B.J. Fugere, A further generalization of Apostol-Bernoulli polynomials and related polynomials, Honam Mathematical Journal (In press).
  28. R. Tremblay, Some new classes of generalized Apostol-Euler and Apostol-Genocchi polynomials, Int. J. Math. Math. Sci. (In press).