Identities of Symmetry for q-Euler Polynomials

Author(s)
Dae San Kim

ABSTRACT

In this paper, we derive eight basic identities of symmetry in three variables related to q-Euler polynomials and the q -analogue of alternating power sums. These and most of their corollaries are new, since there have been results only about identities of symmetry in two variables. These abundance of symmetries shed new light even on the existing identities so as to yield some further interesting ones. The derivations of identities are based on the p-adic integral expression of the generating function for the q -Euler polynomials and the quotient of integrals that can be expressed as the exponential generating function for the q -analogue of alternating power sums.

In this paper, we derive eight basic identities of symmetry in three variables related to q-Euler polynomials and the q -analogue of alternating power sums. These and most of their corollaries are new, since there have been results only about identities of symmetry in two variables. These abundance of symmetries shed new light even on the existing identities so as to yield some further interesting ones. The derivations of identities are based on the p-adic integral expression of the generating function for the q -Euler polynomials and the quotient of integrals that can be expressed as the exponential generating function for the q -analogue of alternating power sums.

KEYWORDS

q -Euler Polynomial, q -Analogue of Alternating Power sum, Fermionic Integral, Identities of Symmetry

q -Euler Polynomial, q -Analogue of Alternating Power sum, Fermionic Integral, Identities of Symmetry

Cite this paper

nullD. Kim, "Identities of Symmetry for q-Euler Polynomials,"*Open Journal of Discrete Mathematics*, Vol. 1 No. 1, 2011, pp. 22-31. doi: 10.4236/ojdm.2011.11003.

nullD. Kim, "Identities of Symmetry for q-Euler Polynomials,"

References

[1] E. Deeba and D. Rodriguez, Stirling’s and Bernoulli numbers, Amer. Math. Monthly 98 (1991), 423-426. doi:10.2307/2323860

[2] F. T. Howard, Applications of a recurrence for the Bernoulli numbers, J. Number Theory 52(1995), 157-172. doi:10.1006/jnth.1995.1062

[3] D. S. Kim, Identities of symmetry for q-Bernoulli polynomials, submitted. doi:10.1080/10236190801943220

[4] D. S. Kim and K. H. Park, Identities of symmetry for Bernoulli polynomials arising from quotients of Volkenborn integrals invariant under S3, submitted.

[5] T. Kim, Symmetry p-adic invariant integral on for Bernoulli and Euler polynomials, J. Difference Equ. Appl. 14 (2008), 1267-1277.

[6] T. Kim, On the symmetries of the q-Bernoulli polynomials, Abstr. Appl. Anal. 2008(2008), 7 pages(Article ID 914367).

[7] T. Kim, K. H. Park, and K. W. Hwang, On the identities of symmetry for the ζ-Euler polynomials of higher-order , Adv. Difference Equ. 2009(2009), 9 pages (Article ID 273545).

[8] H. Tuenter, A symmetry of power sum polynomials and Bernoulli numbers, Amer. Math. Monthly 108 (2001), 258-261. doi:10.2307/2695389

[9] S. Yang, An identity of symmetry for the Bernoulli polynomials, Discrete Math. 308 (2008), 550-554. doi:10.1016/j.disc.2007.03.030

[1] E. Deeba and D. Rodriguez, Stirling’s and Bernoulli numbers, Amer. Math. Monthly 98 (1991), 423-426. doi:10.2307/2323860

[2] F. T. Howard, Applications of a recurrence for the Bernoulli numbers, J. Number Theory 52(1995), 157-172. doi:10.1006/jnth.1995.1062

[3] D. S. Kim, Identities of symmetry for q-Bernoulli polynomials, submitted. doi:10.1080/10236190801943220

[4] D. S. Kim and K. H. Park, Identities of symmetry for Bernoulli polynomials arising from quotients of Volkenborn integrals invariant under S3, submitted.

[5] T. Kim, Symmetry p-adic invariant integral on for Bernoulli and Euler polynomials, J. Difference Equ. Appl. 14 (2008), 1267-1277.

[6] T. Kim, On the symmetries of the q-Bernoulli polynomials, Abstr. Appl. Anal. 2008(2008), 7 pages(Article ID 914367).

[7] T. Kim, K. H. Park, and K. W. Hwang, On the identities of symmetry for the ζ-Euler polynomials of higher-order , Adv. Difference Equ. 2009(2009), 9 pages (Article ID 273545).

[8] H. Tuenter, A symmetry of power sum polynomials and Bernoulli numbers, Amer. Math. Monthly 108 (2001), 258-261. doi:10.2307/2695389

[9] S. Yang, An identity of symmetry for the Bernoulli polynomials, Discrete Math. 308 (2008), 550-554. doi:10.1016/j.disc.2007.03.030