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Journal of Nonlinear Mathematical Physics Volume 12, 2005 - Issue sup1: Special Issue in Honour of Francesco Calogero on the Occasion of His 70th Birthday
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Original Articles

Reflection Symmetries of q-Bernoulli Polynomials

Boris A Kupershmidt The University of Tennessee Space Institute, Tullahoma, TN, 37388, USA E-mail: [email protected]
Pages 412-422 | Published online: 21 Jan 2013

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Yilmaz Simsek. (2010) Complete sum of products of (h, q)-extension of Euler polynomials and numbers. Journal of Difference Equations and Applications 16:11, pages 1331-1348.
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Taekyun Kim. (2008) Symmetry p-adic invariant integral on ℤ for Bernoulli and Euler polynomials. Journal of Difference Equations and Applications 14:12, pages 1267-1277.
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Taekyun Kim, Daeyeoul Kim & Ja Kyung Koo. (2007) p-adic interpolating function associated with Euler numbers. Journal of Nonlinear Mathematical Physics 14:2, pages 250-257.
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Taekyun Kim. (2006) A note on q-Bernoulli numbers and polynomials. Journal of Nonlinear Mathematical Physics 13:3, pages 315-322.
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A S Hegazi & M Mansour. (2006) A Note on q−Bernoulli Numbers and Polynomials. Journal of Nonlinear Mathematical Physics 13:1, pages 9-18.
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Articles from other publishers (42)

Zhihua Chen, Neşe Ömür, Sibel Koparal & Waseem Ahmad Khan. (2023) Some Identities with Multi-Generalized q-Hyperharmonic Numbers of Order r. Symmetry 15:4, pages 917.
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Taekyun Kim, Dae San Kim & Hye Kyung Kim. (2022) λ - q -Sheffer sequence and its applications . Demonstratio Mathematica 55:1, pages 843-865.
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Cheon Seoung Ryoo & Jung Yoog Kang. (2020) Structure of Approximate Roots Based on Symmetric Properties of (p, q)-Cosine and (p, q)-Sine Bernoulli Polynomials. Symmetry 12:6, pages 885.
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Cheon Ryoo. (2018) Symmetric Identities for (P,Q)-Analogue of Tangent Zeta Function. Symmetry 10:9, pages 395.
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Hédi Elmonser. (2017) Symmetric q-Bernoulli numbers and polynomials. Functiones et Approximatio Commentarii Mathematici 56:2.
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Ugur DURAN & Mehmet ACİKGOZ. (2016) New symmetric identities involving generalized Carlitz’s twisted q-Bernoulli polynomials under S_5. Mathematical Sciences and Applications E-Notes 4:2, pages 52-57.
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N.S. JUNG & C.S. RYOO. (2016) SYMMETRIC PROPERTIES FOR GENERALIZED TWISTED q-EULER ZETA FUNCTIONS AND q-EULER POLYNOMIALS. Journal of applied mathematics & informatics 34:1_2, pages 107-114.
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Hui Young Lee & Cheon Seoung Ryoo. (2016) On Polynomials <i>R<sub>n</sub></i>(x) Related to the Stirling Numbers and the Bell Polynomials Associated with the <i>p</i>-Adic Integral on <img src="http://latex.codecogs.com/gif.latex?\mathbb{Z}_{p}" title="\mathbb{Z}_{p}" />. Open Journal of Discrete Mathematics 06:02, pages 89-98.
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N.S. JUNG & C.S. RYOO. (2015) SYMMETRIC IDENTITIES FOR TWISTED q-EULER ZETA FUNCTIONS. Journal of applied mathematics & informatics 33:5_6, pages 649-656.
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Dae San Kim & Taekyun Kim. (2014) Some identities of q-Euler polynomials arising from q-umbral calculus. Journal of Inequalities and Applications 2014:1.
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Dae San Kim & Tae Kyun Kim. (2014) q-Bernoulli polynomials and q-umbral calculus. Science China Mathematics 57:9, pages 1867-1874.
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N. I. Mahmudov & M. Momenzadeh. (2014) On a Class of -Bernoulli, -Euler, and -Genocchi Polynomials . Abstract and Applied Analysis 2014, pages 1-10.
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Daeyeoul Kim, Ja Kyung Koo & Yoon Kyung Park. (2013) New approach to twisted q-Bernoulli polynomials. Advances in Difference Equations 2013:1.
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T. Kim. (2013) On the von Staudt-Clausen theorem for q-Euler numbers. Russian Journal of Mathematical Physics 20:1, pages 33-38.
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H. Y. Lee & C. S. Ryoo. (2013) Some Identities between the Extended -Bernstein Polynomials with Weight and -Bernoulli Polynomials with Weight (,). Journal of Applied Mathematics 2013, pages 1-8.
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Daeyeoul Kim, Burak Kurt & Veli Kurt. (2013) Some Identities on the Generalized q -Bernoulli, q -Euler, and q -Genocchi Polynomials . Abstract and Applied Analysis 2013, pages 1-6.
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Lee-Chae Jang. (2012) Some identities on the twisted q-Bernoulli numbers and polynomials with weight α. Journal of Inequalities and Applications 2012:1.
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H. Y. Lee, N. S. Jung & C. S. Ryoo. (2012) Generalized -Euler Numbers and Polynomials. ISRN Applied Mathematics 2012, pages 1-14.
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H. Y. Lee, N. S. Jung, J. Y. Kang & C. S. Ryoo. (2012) Generalized -Euler Numbers and Polynomials Associated with -Adic -Integral on. International Journal of Mathematics and Mathematical Sciences 2012, pages 1-14.
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J. Y. Kang, H. Y. Lee & N. S. Jung. (2012) On the -Genocchi Numbers and Polynomials with Weight and Weak Weight. Journal of Applied Mathematics 2012, pages 1-8.
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A. Bayad & T. Kim. (2011) Identities involving values of Bernstein, q-Bernoulli, and q-Euler polynomials. Russian Journal of Mathematical Physics 18:2, pages 133-143.
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T. Kim. (2011) A note on q-Bernstein polynomials. Russian Journal of Mathematical Physics 18:1, pages 73-82.
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Taekyun Kim. (2011) An identity of the symmetry for the Frobenius-Euler polynomials associated with the Fermionic $p$-adic invariant $q$-integrals on ${\bf Z}_p$. Rocky Mountain Journal of Mathematics 41:1.
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Taekyun Kim. (2011) Some Identities on the -Integral Representation of the Product of Several -Bernstein-Type Polynomials. Abstract and Applied Analysis 2011, pages 1-11.
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L. C. Jang. (2011) A Note on Some Properties of the Weighted -Genocchi Numbers and Polynomials. Journal of Applied Mathematics 2011, pages 1-10.
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T Kim, A Bayad & Y-H Kim. (2011) A Study on the -Adic -Integral Representation on Associated with the Weighted -Bernstein and -Bernoulli Polynomials. Journal of Inequalities and Applications 2011:1, pages 513821.
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H. Y. Lee, N. S. Jung & C. S. Ryoo. (2011) A Note on the -Euler Numbers and Polynomials with Weak Weight. Journal of Applied Mathematics 2011, pages 1-14.
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Veli Kurt & Mehmet Cenkci. (2010) A NEW APPROACH TO q-GENOCCHI NUMBERS AND POLYNOMIALS. Bulletin of the Korean Mathematical Society 47:3, pages 575-583.
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Eun-Jung Moon, Seog-Hoon Rim, Jeong-Hee Jin & Sun-Jung Lee. (2010) On the Symmetric Properties of Higher-Order Twisted -Euler Numbers and Polynomials. Advances in Difference Equations 2010:1, pages 765259.
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Eun-Jung Moon, Seog-Hoon Rim, Jeong-Hee Jin & Sun-Jung Lee. (2010) On the Symmetric Properties of Higher-Order Twisted q-Euler Numbers and Polynomials. Advances in Difference Equations 2010, pages 1-9.
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T Kim & YH Kim. (2010) Generalized -Euler Numbers and Polynomials of Higher Order and Some Theoretic Identities. Journal of Inequalities and Applications 2010:1, pages 682072.
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T. Kim, J. Choi, Y. H. Kim & L. C. Jang. (2010) On -Adic Analogue of -Bernstein Polynomials and Related Integrals. Discrete Dynamics in Nature and Society 2010, pages 1-9.
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T. Kim. (2009) Symmetry of power sum polynomials and multivariate fermionic p-adic invariant integral on ℤ p. Russian Journal of Mathematical Physics 16:1, pages 93-96.
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Taekyun Kim, Seog-Hoon Rim & Byungje Lee. (2009) Some Identities of Symmetry for the Generalized Bernoulli Numbers and Polynomials. Abstract and Applied Analysis 2009, pages 1-8.
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Taekyun Kim & Byungje Lee. (2009) Some Identities of the Frobenius-Euler Polynomials. Abstract and Applied Analysis 2009, pages 1-7.
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Taekyun Kim, Kyoung Ho Park & Kyung-won Hwang. (2009) On the Identities of Symmetry for the ζ-Euler Polynomials of Higher Order. Advances in Difference Equations 2009, pages 1-9.
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Taekyun Kim & Young-Hee Kim. (2009) On the Symmetric Properties for the Generalized Twisted Bernoulli Polynomials. Journal of Inequalities and Applications 2009:1, pages 164743.
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T. Kim. (2008) q-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients. Russian Journal of Mathematical Physics 15:1, pages 51-57.
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Taekyun Kim. (2008) Euler Numbers and Polynomials Associated with Zeta Functions. Abstract and Applied Analysis 2008, pages 1-11.
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Taekyun Kim & Seog-Hoon Rim. (2007) New Changhee -Euler numbers and polynomials associated with -adic -integrals . Computers & Mathematics with Applications 54:4, pages 484-489.
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C. S. Ryoo. (2007) A numerical investigation of the structure of the roots ofq-Bernoulli polynomials. Journal of Applied Mathematics and Computing 23:1-2, pages 205-214.
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T. Kim. (2006) q-Generalized Euler numbers and polynomials. Russian Journal of Mathematical Physics 13:3, pages 293-298.
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