##
**Euler numbers and polynomials associated with zeta functions.**
*(English)*
Zbl 1145.11019

By using Euler zeta and Hurwitz-Euler zeta functions, several relations have been obtained between Euler numbers and zeta functions, and some infinite sums and trigonometric identities have been introduced in terms of Euler numbers.

Reviewer: Mehmet Cenkci (Antalya)

### MSC:

11B68 | Bernoulli and Euler numbers and polynomials |

11M41 | Other Dirichlet series and zeta functions |

PDF
BibTeX
XML
Cite

\textit{T. Kim}, Abstr. Appl. Anal. 2008, Article ID 581582, 11 p. (2008; Zbl 1145.11019)

### References:

[1] | T. Kim, “A note on p-adic q-integral on \Bbb Zp associated with q-Euler numbers,” Advanced Studies in Contemporary Mathematics, vol. 15, no. 2, 133 pages, 2007. · Zbl 1132.11369 |

[2] | I. N. Cangül, V. Kurt, Y. Simsek, H. K. Pak, and S.-H. Rim, “An invariant p-adic q-integral associated with q-Euler numbers and polynomials,” Journal of Nonlinear Mathematical Physics, vol. 14, no. 1, 8 pages, 2007. · Zbl 1159.11008 |

[3] | M. Cenkci, “The p-adic generalized twisted (h,q)-Euler-l-function and its applications,” Advanced Studies in Contemporary Mathematics, vol. 15, no. 1, 37 pages, 2007. · Zbl 1173.11008 |

[4] | H. Ozden, Y. Simsek, and I. N. Cangul, “Euler polynomials associated with p-adic q-Euler measure,” General Mathematics, vol. 15, no. 2, 24 pages, 2007. · Zbl 1155.11056 |

[5] | H. Ozden and Y. Simsek, “A new extension of q-Euler numbers and polynomials related to their interpolation functions,” Applied Mathematics Letters. In press. · Zbl 1152.11009 |

[6] | M. Cenkci, M. Can, and V. Kurt, “p-adic interpolation functions and Kummer-type congruences for q-twisted and q-generalized twisted Euler numbers,” Advanced Studies in Contemporary Mathematics, vol. 9, no. 2, 203 pages, 2004. · Zbl 1083.11016 |

[7] | M. Cenkci and M. Can, “Some results on q-analogue of the Lerch zeta function,” Advanced Studies in Contemporary Mathematics, vol. 12, no. 2, 213 pages, 2006. · Zbl 1098.11016 |

[8] | A. S. Hegazi and M. Mansour, “A note on q-Bernoulli numbers and polynomials,” Journal of Nonlinear Mathematical Physics, vol. 13, no. 1, 9 pages, 2006. · Zbl 1109.33024 |

[9] | T. Kim, “On p-adic q-l-functions and sums of powers,” Journal of Mathematical Analysis and Applications, vol. 329, no. 2, 1472 pages, 2007. · Zbl 1154.11310 |

[10] | T. Kim, L. C. Jang, S. H. Rim, et al., “Introduction to Non-Archimedean Integrals and Their Applications,” Kyo Woo Sa, 2007. |

[11] | T. Kim, “q-Volkenborn integration,” Russian Journal of Mathematical Physics, vol. 9, no. 3, 288 pages, 2002. · Zbl 1092.11045 |

[12] | T. Kim, “On the analogs of Euler numbers and polynomials associated with p-adic q-integral on \Bbb Zp at q= - 1,” Journal of Mathematical Analysis and Applications, vol. 331, no. 2, 779 pages, 2007. · Zbl 1120.11010 |

[13] | T. Kim, “q-Extension of the Euler formula and trigonometric functions,” Russian Journal of Mathematical Physics, vol. 14, no. 3, 275 pages, 2007. · Zbl 1188.33001 |

[14] | 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, 186 pages, 2005. · Zbl 1190.11049 |

[15] | T. Kim, “Non-Archimedean q-integrals associated with multiple Changhee q-Bernoulli polynomials,” Russian Journal of Mathematical Physics, vol. 10, no. 1, 91 pages, 2003. · Zbl 1072.11090 |

[16] | Y. Simsek, “On twisted q-Hurwitz zeta function and q-two-variable L-function,” Applied Mathematics and Computation, vol. 187, no. 1, 466 pages, 2007. · Zbl 1143.11032 |

[17] | Y. Simsek, “On p-adic twisted q-L-functions related to generalized twisted Bernoulli numbers,” Russian Journal of Mathematical Physics, vol. 13, no. 3, 340 pages, 2006. · Zbl 1163.11312 |

[18] | 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, 790 pages, 2006. · Zbl 1139.11051 |

[19] | Y. Simsek, “Theorems on twisted L-function and twisted Bernoulli numbers,” Advanced Studies in Contemporary Mathematics, vol. 11, no. 2, 205 pages, 2005. · Zbl 1178.11058 |

[20] | 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, 333 pages, 2006. · Zbl 1149.11054 |

[21] | L. Carlitz, “q-Bernoulli numbers and polynomials,” Duke Mathematical Journal, vol. 15, no. 4, 987 pages, 1948. · Zbl 0032.00304 |

[22] | L. Carlitz, “Expansions of q-Bernoulli numbers,” Duke Mathematical Journal, vol. 25, no. 2, 355 pages, 1958. · Zbl 0102.03201 |

[23] | L. Carlitz, “q-Bernoulli and Eulerian numbers,” Transactions of the American Mathematical Society, vol. 76, no. 2, 332 pages, 1954. · Zbl 0058.01204 |

[24] | M. Cenkci, Y. Simsek, and V. Kurt, “Further remarks on multiple p-adic q-L-function of two variables,” Advanced Studies in Contemporary Mathematics, vol. 14, no. 1, 49 pages, 2007. · Zbl 1143.11043 |

[25] | E. Y. Deeba and D. M. Rodriguez, “Stirling’s series and Bernoulli numbers,” The American Mathematical Monthly, vol. 98, no. 5, 423 pages, 1991. · Zbl 0743.11012 |

[26] | B. A. Kupershmidt, “Reflection symmetries of q-Bernoulli polynomials,” Journal of Nonlinear Mathematical Physics, vol. 12, supplement 1, 412 pages, 2005. · Zbl 1362.33021 |

[27] | H. Ozden, Y. Simsek, S.-H. Rim, and I. N. Cangul, “A note on p-adic q-Euler measure,” Advanced Studies in Contemporary Mathematics, vol. 14, no. 2, 233 pages, 2007. · Zbl 1143.11008 |

[28] | C. S. Ryoo, “The zeros of the generalized twisted Bernoulli polynomials,” Advances in Theoretical and Applied Mathematics, vol. 1, no. 2-3, 143 pages, 2006. · Zbl 1133.11014 |

[29] | M. Schork, “Ward’s “calculus of sequences”, q-calculus and the limit q\rightarrow - 1,” Advanced Studies in Contemporary Mathematics, vol. 13, no. 2, 131 pages, 2006. · Zbl 1111.05010 |

[30] | M. Schork, “Combinatorial aspects of normal ordering and its connection to q-calculus,” Advanced Studies in Contemporary Mathematics, vol. 15, no. 1, 49 pages, 2007. · Zbl 1141.05019 |

[31] | K. Shiratani and S. Yamamoto, “On a p-adic interpolation function for the Euler numbers and its derivatives,” Memoirs of the Faculty of Science. Kyushu University. Series A, vol. 39, no. 1, 113 pages, 1985. · Zbl 0574.12017 |

[32] | H. J. H. Tuenter, “A symmetry of power sum polynomials and Bernoulli numbers,” The American Mathematical Monthly, vol. 108, no. 3, 258 pages, 2001. · Zbl 0983.11008 |

[33] | J. C. Baez, “The Riemannn zeta function,” preprint. |

[34] | R. Apery, “Irrationalite de \zeta (2) et \zeta (3),” Asterisque, vol. 61, 11 pages, 1979. · Zbl 0401.10049 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.