Abstract and Applied Analysis

On the q -Extension of Apostol-Euler Numbers and Polynomials

Young-Hee Kim, Wonjoo Kim, and Lee-Chae Jang

Full-text: Open access

Abstract

Recently, Choi et al. (2008) have studied the q -extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials of order n and multiple Hurwitz zeta function. In this paper, we define Apostol's type q -Euler numbers E n , q , ξ and q -Euler polynomials E n , q , ξ ( x ) . We obtain the generating functions of E n , q , ξ and E n , q , ξ ( x ) , respectively. We also have the distribution relation for Apostol's type q -Euler polynomials. Finally, we obtain q -zeta function associated with Apostol's type q -Euler numbers and Hurwitz's type q -zeta function associated with Apostol's type q -Euler polynomials for negative integers.

Article information

Source
Abstr. Appl. Anal., Volume 2008 (2008), Article ID 296159, 10 pages.

Dates
First available in Project Euclid: 10 February 2009

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1234298999

Digital Object Identifier
doi:10.1155/2008/296159

Mathematical Reviews number (MathSciNet)
MR2466221

Zentralblatt MATH identifier
1247.11028

Citation

Kim, Young-Hee; Kim, Wonjoo; Jang, Lee-Chae. On the $q$ -Extension of Apostol-Euler Numbers and Polynomials. Abstr. Appl. Anal. 2008 (2008), Article ID 296159, 10 pages. doi:10.1155/2008/296159. https://projecteuclid.org/euclid.aaa/1234298999


Export citation

References

  • T. Kim, ``$q$-Volkenborn integration,'' Russian Journal of Mathematical Physics, vol. 9, no. 3, pp. 288--299, 2002.
  • T. Kim, ``On $p$-adic $q$-$L$-functions and sums of powers,'' Discrete Mathematics, vol. 252, no. 1--3, pp. 179--187, 2002.
  • T. Kim, ``On Euler-Barnes multiple zeta functions,'' Russian Journal of Mathematical Physics, vol. 10, no. 3, pp. 261--267, 2003.
  • T. Kim, ``Sums of powers of consecutive $q$-integers,'' Advanced Studies in Contemporary Mathematics, vol. 9, no. 1, pp. 15--18, 2004.
  • T. Kim, ``Analytic continuation of multiple $q$-zeta functions and their values at negative integers,'' Russian Journal of Mathematical Physics, vol. 11, no. 1, pp. 71--76, 2004. \setlengthemsep1.5pt
  • T. Kim, ``$q$-Riemann zeta function,'' International Journal of Mathematics and Mathematical Sciences, vol. 2004, no. 12, pp. 599--605, 2004.
  • T. Kim, ``Power series and asymptotic series associated with the $q$-analog of the two-variable $p$-adic $L$-function,'' Russian Journal of Mathematical Physics, vol. 12, no. 2, pp. 186--196, 2005.
  • T. Kim, ``$q$-generalized Euler numbers and polynomials,'' Russian Journal of Mathematical Physics, vol. 13, no. 3, pp. 293--298, 2006.
  • T. Kim, ``Multiple $p$-adic $L$-function,'' Russian Journal of Mathematical Physics, vol. 13, no. 2, pp. 151--157, 2006.
  • T. Kim, ``On the analogs of Euler numbers and polynomials associated with $p$-adic $q$-integral on $\mathbbZ_p$ at $q=-1$,'' Journal of Mathematical Analysis and Applications, vol. 331, no. 2, pp. 779--792, 2007.
  • L. Carlitz, ``$q$-Bernoulli numbers and polynomials,'' Duke Mathematical Journal, vol. 15, no. 4, pp. 987--1000, 1948.
  • I. N. Cangul, H. Ozden, and Y. Simsek, ``Generating functions of the $(h,q)$ extension of twisted Euler polynomials and numbers,'' Acta Mathematica Hungarica, vol. 120, no. 3, pp. 281--299, 2008.
  • L. Carlitz, ``$q$-Bernoulli and Eulerian numbers,'' Transactions of the American Mathematical Society, vol. 76, no. 2, pp. 332--350, 1954.
  • M. Cenkci, ``The $p$-adic generalized twisted $(h,q)$-Euler-$l$-function and its applications,'' Advanced Studies in Contemporary Mathematics, vol. 15, no. 1, pp. 37--47, 2007.
  • M. Cenkci and M. Can, ``Some results on $q$-analogue of the Lerch zeta function,'' Advanced Studies in Contemporary Mathematics, vol. 12, no. 2, pp. 213--223, 2006.
  • 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,'' Applied Mathematics and Computation, vol. 199, no. 2, pp. 723--737, 2008.
  • M. Garg, K. Jain, and H. M. Srivastava, ``Some relationships between the generalized Apostol-Bernoulli polynomials and Hurwitz-Lerch zeta functions,'' Integral Transforms and Special Functions, vol. 17, no. 11, pp. 803--815, 2006.
  • L.-C. Jang, ``Multiple twisted $q$-Euler numbers and polynomials associated with $p$-adic $q$-integrals,'' Advances in Difference Equations, vol. 2008, Article ID 738603, 11 pages, 2008.
  • T. Kim, ``On the $q$-extension of Euler and Genocchi numbers,'' Journal of Mathematical Analysis and Applications, vol. 326, no. 2, pp. 1458--1465, 2007.
  • T. Kim, ``$q$-extension of the Euler formula and trigonometric functions,'' Russian Journal of Mathematical Physics, vol. 14, no. 3, pp. 275--278, 2007.
  • T. Kim, ``On the symmetries of the $q$-Bernoulli polynomials,'' Abstract and Applied Analysis, vol. 2008, Article ID 914367, 7 pages, 2008.
  • T. Kim, ``The modified $q$-Euler numbers and polynomials,'' Advanced Studies in Contemporary Mathematics, vol. 16, no. 2, pp. 161--170, 2008.
  • T. Kim, ``$q$-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients,'' Russian Journal of Mathematical Physics, vol. 15, no. 1, pp. 51--57, 2008.
  • T. Kim, ``On the multiple $q$-Genocchi and Euler numbers,'' Russian Journal of Mathematical Physics, vol. 15, no. 4, pp. 481--486, 2008.
  • T. Kim, J. Y. Choi, and J. Y. Sug, ``Extended $q$-Euler numbers and polynomials associated with fermionic $p$-adic $q$-integral on $\mathbbZ_p$,'' Russian Journal of Mathematical Physics, vol. 14, no. 2, pp. 160--163, 2007.
  • T. Kim, M.-S. Kim, L. Jang, and S.-H. Rim, ``New $q$-Euler numbers and polynomials associated with $p$-adic $q$-integrals,'' Advanced Studies in Contemporary Mathematics, vol. 15, no. 2, pp. 243--252, 2007.
  • Y. H. Kim, W. J. Kim, and C. S. Ryoo, ``On the twisted $q$-Euler zeta function čommentComment on ref. [28?]: Please update the information of this reference, if possible. associated with twisted $q$-Euler numbers,'' communicated.
  • T. Kim, S.-H. Rim, and Y. Simsek, ``A note on the alternating sums of powers of consecutive $q$-integers,'' Advanced Studies in Contemporary Mathematics, vol. 13, no. 2, pp. 159--164, 2006.
  • T. Kim and Y. Simsek, ``Analytic continuation of the multiple Daehee $q$-$l$-functions associated with Daehee numbers,'' Russian Journal of Mathematical Physics, vol. 15, no. 1, pp. 58--65, 2008.
  • S.-D. Lin and H. M. Srivastava, ``Some families of the Hurwitz-Lerch zeta functions and associated fractional derivative and other integral representations,'' Applied Mathematics and Computation, vol. 154, no. 3, pp. 725--733, 2004.
  • S.-D. Lin, H. M. Srivastava, and P.-Y. Wang, ``Some expansion formulas for a class of generalized Hurwitz-Lerch zeta functions,'' Integral Transforms and Special Functions, vol. 17, no. 11, pp. 817--827, 2006.
  • Q.-M. Luo, ``Apostol-Euler polynomials of higher order and Gaussian hypergeometric functions,'' Taiwanese Journal of Mathematics, vol. 10, no. 4, pp. 917--925, 2006.
  • Q.-M. Luo and H. M. Srivastava, ``Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials,'' Journal of Mathematical Analysis and Applications, vol. 308, no. 1, pp. 290--302, 2005.
  • H. Ozden, I. N. Cangul, and Y. Simsek, ``Multivariate interpolation functions of higher-order $q$-Euler numbers and their applications,'' Abstract and Applied Analysis, vol. 2008, Article ID 390857, 16 pages, 2008.
  • H. Ozden, I. N. Cangul, and Y. Simsek, ``Remarks on sum of products of $(h,q)$-twisted Euler polynomials and numbers,'' Journal of Inequalities and Applications, vol. 2008, Article ID 816129, 8 pages, 2008.
  • H. Ozden and Y. Simsek, ``A new extension of $q$-Euler numbers and polynomials related to their interpolation functions,'' Applied Mathematics Letters, vol. 21, no. 9, pp. 934--939, 2008.
  • Y. Simsek, ``On $p$-adic twisted $q$-$L$-functions related to generalized twisted Bernoulli numbers,'' Russian Journal of Mathematical Physics, vol. 13, no. 3, pp. 340--348, 2006.
  • Y. Simsek, ``Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions,'' Advanced Studies in Contemporary Mathematics, vol. 16, no. 2, pp. 251--278, 2008.
  • H. M. Srivastava, T. Kim, and Y. Simsek, ``$q$-Bernoulli numbers and polynomials associated with multiple $q$-zeta functions and basic $L$-series,'' Russian Journal of Mathematical Physics, vol. 12, no. 2, pp. 241--268, 2005.
  • T. M. Apostol, ``On the Lerch zeta function,'' Pacific Journal of Mathematics, vol. 1, pp. 161--167, 1951.
  • W. Wang, C. Jia, and T. Wang, ``Some results on the Apostol-Bernoulli and Apostol-Euler polynomials,'' Computers & Mathematics with Applications, vol. 55, no. 6, pp. 1322--1332, 2008. \endthebibliography