Abstract and Applied Analysis

Some Identities on the High-Order q -Euler Numbers and Polynomials with Weight 0

Jongsung Choi, Hyun-Mee Kim, and Young-Hee Kim

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We construct the N th order nonlinear ordinary differential equation related to the generating function of q -Euler numbers with weight 0. From this, we derive some identities on q -Euler numbers and polynomials of higher order with weight 0.

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Abstr. Appl. Anal., Volume 2013 (2013), Article ID 459763, 6 pages.

First available in Project Euclid: 27 February 2014

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Choi, Jongsung; Kim, Hyun-Mee; Kim, Young-Hee. Some Identities on the High-Order $q$ -Euler Numbers and Polynomials with Weight 0. Abstr. Appl. Anal. 2013 (2013), Article ID 459763, 6 pages. doi:10.1155/2013/459763. https://projecteuclid.org/euclid.aaa/1393511872

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  • L. Carlitz, “Eulerian numbers and polynomials,” Mathematics Magazine , vol. 32, pp. 247–260, 1959.
  • L. Carlitz, “The product of two Eulerian polynomials,” Mathematics Magazine, vol. 36, no. 1, pp. 37–41, 1963.
  • J. Choi, “A note on Eulerian polynomials of higher order,” Journal of the Chungcheong Mathematical Society, vol. 26, no. 1, pp. 191–196, 2013.
  • J. Choi, T. Kim, and Y. H. Kim, “A recurrence formula for q-Euler numbers of higher order,” Proceedings of the Jangjeon Mathematical Society, vol. 13, no. 3, pp. 321–326, 2010.
  • J. Choi, T. Kim, and Y.-H. Kim, “A note on the $q$-analogues of Euler numbers and polynomials,” Honam Mathematical Journal, vol. 33, no. 4, pp. 529–534, 2011.
  • D. S. Kim, “Identities of symmetry for generalized Euler polynomials,” International Journal of Combinatorics, vol. 2011, Article ID 432738, 12 pages, 2011.
  • D. S. Kim, T. Kim, J. Choi, and Y. H. Kim, “Identities involving $q$-Bernoulli and $q$-Euler numbers,” Abstract and Applied Analysis, vol. 2012, Article ID 674210, 10 pages, 2012.
  • H.-M. Kim, J. Choi, and T. Kim, “On the extended $q$-Euler numbers and polynomials of higher-order with weight,” Honam Mathematical Journal, vol. 34, no. 1, pp. 1–9, 2012.
  • T. Kim, “Identities involving Frobenius-Euler polynomials arising from non-linear differential equations,” Journal of Number Theory, vol. 132, no. 12, pp. 2854–2865, 2012.
  • T. Kim and J. Choi, “A note on the product of Frobenius-Euler polynomials arising from the $p$-adic integral on ${\mathbb{Z}}_{p}$,” Advanced Studies in Contemporary Mathematics, vol. 22, no. 2, pp. 215–223, 2012.
  • T. Kim and J. Choi, “On the $q$-Euler numbers and polynomials with weight 0,” Abstract and Applied Analysis, vol. 2012, Article ID 795304, 7 pages, 2012.
  • 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.
  • Y. Simsek, “Complete sum of products of $(h,q)$-extension of Euler polynomials and numbers,” Journal of Difference Equations and Applications, vol. 16, no. 11, pp. 1331–1348, 2010.
  • Y. Simsek, “Generating functions for q-Apostol type Frobenius-Euler numbers and polynomials,” Axioms, vol. 1, no. 3, pp. 395–403, 2012.
  • Y. Simsek, “Generating functions for generalized Stirling type numbers, Array type polynomials, Eulerian type polynomials and their applications,” Fixed Point Theory and Applications, vol. 2013, article 87, 2013.
  • C. Zachmanoglou and D. W. Thoe, Introduction to Partial Differntial Equations with Applications, The Williams and Wilkins company, Baltimore, Md, USA, 1976.