Abstract and Applied Analysis

On Multiple Interpolation Functions of the Nörlund-Type q -Euler Polynomials

Mehmet Acikgoz and Yilmaz Simsek

Full-text: Open access

Abstract

In (2006) and (2009), Kim defined new generating functions of the Genocchi, Nörlund-type q -Euler polynomials and their interpolation functions. In this paper, we give another definition of the multiple Hurwitz type q -zeta function. This function interpolates Nörlund-type q -Euler polynomials at negative integers. We also give some identities related to these polynomials and functions. Furthermore, we give some remarks about approximations of Bernoulli and Euler polynomials.

Article information

Source
Abstr. Appl. Anal., Volume 2009 (2009), Article ID 382574, 14 pages.

Dates
First available in Project Euclid: 16 March 2010

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1268745567

Digital Object Identifier
doi:10.1155/2009/382574

Mathematical Reviews number (MathSciNet)
MR2516009

Zentralblatt MATH identifier
1194.11027

Citation

Acikgoz, Mehmet; Simsek, Yilmaz. On Multiple Interpolation Functions of the Nörlund-Type $q$ -Euler Polynomials. Abstr. Appl. Anal. 2009 (2009), Article ID 382574, 14 pages. doi:10.1155/2009/382574. https://projecteuclid.org/euclid.aaa/1268745567


Export citation

References

  • T. M. Apostol, Mathematical Analysis: A Modern Approach to Advanced Calculus, Addison-Wesley, Reading, Mass, USA, 1957.
  • R. G. Bartle, The Elements of Real Analysis, John Wiley & Sons, New York, NY, USA, 2nd edition, 1976.
  • 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.
  • F. A. Costabile and F. Dell'Accio, ``Polynomial approximation of CM functions by means of boundary values and applications: a survey,'' Journal of Computational and Applied Mathematics, vol. 210, no. 1-2, pp. 116--135, 2007.
  • I. N. Cangul, V. Kurt, H. Ozden, and Y. Simsek, ``On higher order $w$-$q$ Genocchi numbers,'' Advanced Studies in Contemporary Mathematics, vol. 19, no. 1, 2009.
  • I. N. Cangul, H. Ozden, and Y. Simsek, ``A new approach to $q$-Genocchi numbers and their interpolation functions,'' Nonlinear Analysis: Theory, Methods & Applications. In press.
  • 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.
  • J. Guillera and J. Sondow, ``Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent,'' Ramanujan Journal, vol. 16, no. 3, pp. 247--270, 2008.
  • L.-C. Jang, S.-D. Kim, D. W. Park, and Y. S. Ro, ``A note on Euler number and polynomials,'' Journal of Inequalities and Applications, vol. 2006, Article ID 34602, 5 pages, 2006.
  • L. Jang and T. Kim, ``$q$-Genocchi numbers and polynomials associated with fermionic $p$-adic invariant integrals on $\mathbbZ_p$,'' Abstract and Applied Analysis, vol. 2008, Article ID 232187, 8 pages, 2008.
  • T. Kim, L.-C. Jang, and C.-S. Ryoo, ``Note on $q$-extensions of Euler numbers and polynomials of higher order,'' Journal of Inequalities and Applications, vol. 2008, Article ID 371295, 9 pages, 2008.
  • T. Kim, ``$q$-Volkenborn integration,'' Russian Journal of Mathematical Physics, vol. 9, no. 3, pp. 288--299, 2002.
  • T. Kim, ``Non-Archimedean $q$-integrals associated with multiple Changhee $q$-Bernoulli polynomials,'' Russian Journal of Mathematical Physics, vol. 10, no. 1, pp. 91--98, 2003.
  • T. Kim, ``On Euler-Barnes multiple zeta functions,'' Russian Journal of Mathematical Physics, vol. 10, no. 3, pp. 261--267, 2003.
  • T. Kim, ``A note on the $q$-multiple zeta function,'' Advanced Studies in Contemporary Mathematics, vol. 8, no. 2, pp. 111--113, 2004.
  • T. Kim, ``Analytic continuation of multiple $q$-zeta fand their values at negative integers,'' Russian Journal of Mathematical Physics, vol. 11, no. 1, pp. 71--76, 2004.
  • T. Kim, ``$q$-generalized Euler numbers and polynomials,'' Russian Journal of Mathematical Physics, vol. 13, no. 3, pp. 293--298, 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.
  • 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, ``New approach to $q$-Genocch, Euler numbers and polynomials and their interpolation functions,'' Advanced Studies in Contemporary Mathematics, vol. 18, no. 2, pp. 105--112, 2009.
  • T. Kim, ``Some identities on the $q$-Euler polynomials of higher order and $q$-Stirling numbers by the fermionic $p$-adic integral on $\mathbbZ_p$,'' to appear in Russian Journal of Mathematical Physics.
  • T. Kim, ``$q$-Euler numbers and polynomials associated with $p$-adic $q$-integrals,'' Journal of Nonlinear Mathematical Physics, vol. 14, no. 1, pp. 15--27, 2007.
  • T. Kim, ``A note on $p$-adic $q$-integral on $\mathbbZ_p$ associated with $q$-Euler numbers,'' Advanced Studies in Contemporary Mathematics, vol. 15, no. 2, pp. 133--137, 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, ``The modified $q$-Euler numbers and polynomials,'' Advanced Studies in Contemporary Mathematics, vol. 16, no. 2, pp. 161--170, 2008.
  • T. Kim, ``Euler numbers and polynomials associated with zeta functions,'' Abstract and Applied Analysis, vol. 2008, Article ID 581582, 11 pages, 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, ``$q$-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients,'' Russian Journal of Mathematical Physics, vol. 15, no. 1, pp. 51--57, 2008.
  • T. Kim and J.-S. Cho, ``A note on multiple Dirichlet's $q$-$L$-function,'' Advanced Studies in Contemporary Mathematics, vol. 11, no. 1, pp. 57--60, 2005.
  • T. Kim and S.-H. Rim, ``On Changhee-Barnes' $q$-Euler numbers and polynomials,'' Advanced Studies in Contemporary Mathematics, vol. 9, no. 2, pp. 81--86, 2004.
  • T. Kim, Y.-H. Kim, and K.-W. Hwang, ``Note on the generalization of the higher order $q$-Genocchi numbers and $q$-Euler numbers,'' preprint, http://arxiv.org/abs/0901.1697.
  • 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.
  • 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.
  • J. L. López and N. M. Temme, ``Uniform approximations of Bernoulli and Euler polynomials in terms of hyperbolic functions,'' Studies in Applied Mathematics, vol. 103, no. 3, pp. 241--258, 1999.
  • 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, 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 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.
  • S.-H. Rim and T. Kim, ``A note on $q$-Euler numbers associated with the basic $q$-zeta function,'' Applied Mathematics Letters, vol. 20, no. 4, pp. 366--369, 2007.
  • S. Ostrovska, ``On the $q$-Bernstein polynomials,'' Advanced Studies in Contemporary Mathematics, vol. 11, no. 2, pp. 193--204, 2005.
  • Y. Simsek, ``$q$-analogue of twisted $l$-series and $q$-twisted Euler numbers,'' Journal of Number Theory, vol. 110, no. 2, pp. 267--278, 2005.
  • Y. Simsek, ``$q$-Dedekind type sums related to $q$-zeta function and basic $L$-series,'' Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 333--351, 2006.
  • Y. Simsek, ``Twisted ($h,q$)-Bernoulli numbers and polynomials related to twisted ($h,q$)-zeta function and $L$-function,'' Journal of Mathematical Analysis and Applications, vol. 324, no. 2, pp. 790--804, 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.
  • Y. Simsek, ``$q$-Hardy-Berndt type sums associated with $q$-Genocchi type zeta and $q$-$l$-functions,'' Nonlinear Analysis: Theory, Methods & Applications. In press.
  • Y. Simsek, ``Complete sums of products of ($h,q$)-extension of Euler numbers and polynomials,'' to appear in Journal of Difference Equations and Applications http://arxiv.org/abs/0707.2849.
  • Y. Simsek, ``Special functions related to dedekind type DC-sums and their applications,'' preprint, http://arxiv.org/abs/0902.0380.
  • 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.
  • H. M. Srivastava, ``Remarks on some relationships between the Bernoulli and Euler polynomials,'' Applied Mathematics Letters, vol. 17, no. 4, pp. 375--380, 2004.
  • H. M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001.
  • R. Sitaramachandrarao, ``Dedekind and Hardy sums,'' Acta Arithmetica, vol. 48, pp. 325--340, 1978.
  • J. Zhao, ``Multiple $q$-zeta functions and multiple $q$-polylogarithms,'' Ramanujan Journal, vol. 14, no. 2, pp. 189--221, 2007.