Euler Numbers and Polynomials Associated with Zeta Functions

By Taekyun Kim. 2008

107 citations recorded by the source corpus.

For , the Euler zeta function and the Hurwitz-type Euler zeta function are defined by , and . Thus, we note that the Euler zeta functions are entire functions in whole complex -plane, and these zeta functions have the values of the Euler numbers or the Euler polynomials at negative integers. That is, , and . 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.

Subfields: Algebra and Number Theory, Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/73fc7308589aedd108bdc1f84cdd7b9b2c8bfe13

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In Algebra and Number Theory

In Applied Mathematics

Citations in the corpus

Construction of some new families of Apostol-type numbers and polynomials via Dirichlet character and p-adic q-integrals (2018, methodology): “Kim [18] defined the fermionic p -adic integral, which is used to construct generating functions for the Euler, the Genochhi type numbers, and the others, as follows: ∫ Zp f (x) dμ−1 (x) = lim N→∞ p−1 ∑ x=0 (−1) f (x) (1.”
Symmetry of power sum polynomials and multivariate fermionic p-adic invariant integral on ℤp (2009, background): “(6) Kim [5, 6, 8] obtained a symmetry relation between the power sum polynomials and the Euler numbers and also showed that the recurrence relation (5) is a special case of the symmetry relation, which can be stated as identity (7) below (symmetric with respect to w1 and w2).”
Note on the Euler Numbers and Polynomials (2008, background): “es between the Euler numbers and the second kind stirling numbers. §1. Introduction/Deﬁnition The constants Ek in the Taylor series expansion (1) F(t) = 2 et + 1 = X∞ n=0 En tn n! , for |t| ≤ π, (see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, ”
Special functions related to Dedekind-type DC-sums and their applications (2009, methodology): “We establish analytic properties of the sum Tp(h, k).”
An identity of the symmetry for the Frobenius-Euler polynomials associated with the Fermionic $p$-adic invariant $q$-integrals on ${\bf Z}_p$ (2008, background): “0≤j<pN f(j)μq(j + p Zp), representing a q-analogue of Riemann sums for f , see [5, 6].”