Abstract and Applied Analysis

Euler Numbers and Polynomials Associated with Zeta Functions

Taekyun Kim

Abstract

For $s\in{}\mathbb{C}$, the Euler zeta function and the Hurwitz-type Euler zeta function are defined by ${\zeta{}}_{E}(s)=2{\sum{}}_{n=1}^{\infty{}}({(-1)}^{n}/{n}^{s})$, and ${\zeta{}}_{E}(s,x)=2{\sum{}}_{n=0}^{\infty{}}({(-1)}^{n}/{(n+x)}^{s})$. Thus, we note that the Euler zeta functions are entire functions in whole complex $s$-plane, and these zeta functions have the values of the Euler numbers or the Euler polynomials at negative integers. That is, ${\zeta{}}_{E}(-k)={E}_{k}^{\ast}$, and ${\zeta{}}_{E}(-k,x)={E}_{k}^{\ast}(x)$. We give some interesting identities between the Euler numbers and the zeta functions. Finally, we will give the new values of the Euler zeta function at positive even integers.

Article information

Source
Abstr. Appl. Anal., Volume 2008 (2008), Article ID 581582, 11 pages.

Dates
First available in Project Euclid: 9 September 2008

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1220969166

Digital Object Identifier
doi:10.1155/2008/581582

Mathematical Reviews number (MathSciNet)
MR2407279

Zentralblatt MATH identifier
1145.11019

Citation

Kim, Taekyun. Euler Numbers and Polynomials Associated with Zeta Functions. Abstr. Appl. Anal. 2008 (2008), Article ID 581582, 11 pages. doi:10.1155/2008/581582. https://projecteuclid.org/euclid.aaa/1220969166

References

• 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.
• 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, pp. 8--14, 2007.
• 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.
• H. Ozden, Y. Simsek, and I. N. Cangul, Euler polynomials associated with $p$-adic $q$-Euler measure,'' General Mathematics, vol. 15, no. 2, pp. 24--37, 2007.
• H. Ozden and Y. Simsek, A new extension of $q$-Euler numbers and polynomials related to their interpolation functionsčommentPlease update the information of this reference, if possible.,'' Applied Mathematics Letters. In press.
• 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, pp. 203--216, 2004.
• 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.
• A. S. Hegazi and M. Mansour, A note on $q$-Bernoulli numbers and polynomials,'' Journal of Nonlinear Mathematical Physics, vol. 13, no. 1, pp. 9--18, 2006.
• T. Kim, On $p$-adic $q$-$l$-functions and sums of powers,'' Journal of Mathematical Analysis and Applications, vol. 329, no. 2, pp. 1472--1481, 2007. \setlengthemsep2.35pt
• T. Kim, L. C. Jang, S. H. Rim, et al., Introduction to Non-Archimedean Integrals and Their Applications,'' Kyo Woo Sa, 2007.
• T. Kim, $q$-Volkenborn integration,'' Russian Journal of Mathematical Physics, vol. 9, no. 3, pp. 288--299, 2002.
• 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, $q$-Extension of the Euler formula and trigonometric functions,'' Russian Journal of Mathematical Physics, vol. 14, no. 3, pp. 275--278, 2007.
• 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, Non-Archimedean $q$-integrals associated with multiple Changhee $q$-Bernoulli polynomials,'' Russian Journal of Mathematical Physics, vol. 10, no. 1, pp. 91--98, 2003.
• Y. Simsek, On twisted $q$-Hurwitz zeta function and $q$-two-variable $L$-function,'' Applied Mathematics and Computation, vol. 187, no. 1, pp. 466--473, 2007.
• 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, 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, Theorems on twisted $L$-function and twisted Bernoulli numbers,'' Advanced Studies in Contemporary Mathematics, vol. 11, no. 2, pp. 205--218, 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.
• L. Carlitz, $q$-Bernoulli numbers and polynomials,'' Duke Mathematical Journal, vol. 15, no. 4, pp. 987--1000, 1948.
• L. Carlitz, Expansions of $q$-Bernoulli numbers,'' Duke Mathematical Journal, vol. 25, no. 2, pp. 355--364, 1958.
• L. Carlitz, $q$-Bernoulli and Eulerian numbers,'' Transactions of the American Mathematical Society, vol. 76, no. 2, pp. 332--350, 1954.
• 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, pp. 49--68, 2007.
• E. Y. Deeba and D. M. Rodriguez, Stirling's series and Bernoulli numbers,'' The American Mathematical Monthly, vol. 98, no. 5, pp. 423--426, 1991.
• B. A. Kupershmidt, Reflection symmetries of $q$-Bernoulli polynomials,'' Journal of Nonlinear Mathematical Physics, vol. 12, supplement 1, pp. 412--422, 2005.
• 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, pp. 233--239, 2007.
• C. S. Ryoo, The zeros of the generalized twisted Bernoulli polynomials,'' Advances in Theoretical and Applied Mathematics, vol. 1, no. 2-3, pp. 143--148, 2006.
• M. Schork, Ward's calculus of sequences'', $q$-calculus and the limit $qarrow-1$,'' Advanced Studies in Contemporary Mathematics, vol. 13, no. 2, pp. 131--141, 2006.
• M. Schork, Combinatorial aspects of normal ordering and its connection to $q$-calculus,'' Advanced Studies in Contemporary Mathematics, vol. 15, no. 1, pp. 49--57, 2007.
• 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, pp. 113--125, 1985.
• H. J. H. Tuenter, A symmetry of power sum polynomials and Bernoulli numbers,'' The American Mathematical Monthly, vol. 108, no. 3, pp. 258--261, 2001.
• J. C. Baez, The Riemannn zeta functiončommentPlease update the information of this reference, if possible.,'' preprint.
• R. Apery, Irrationalite de $\zeta(2)$ et $\zeta(3)$,'' Asterisque, vol. 61, pp. 11--13, 1979.