q-Dedekind type sums related to q-zeta function and basic L-series☆

By Yilmaz Simsek. 2006

57 citations recorded by the source corpus.

Abstract The main purpose of this paper is to define new generating functions. By applying the Mellin transformation formula to these generating functions, we define q-analogue of Riemann zeta function, q-analogue Hurwitz zeta function, q-analogue Dirichlet L-function and two-variable q–L-function. In particular, by using these generating functions, we will construct new generating functions which produce q-Dedekind type sums and q-Dedekind type sums attached to Dirichlet character. We also give the relations between these sums and Dedekind sums. Furthermore, by using *-product which is given in this paper, we will give the relation between Dedekind sums and q–L function as well.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/a71f0e5a1aad4f849234c2b7446a380b9df39952

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In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

Citations in the corpus

Twisted Dedekind Type Sums Associated with Barnes' Type Multiple Frobenius-Euler l-Functions (2007, background): “Simsek [38], [40], [44], [47] considered Dedekind sums.”
On the analogs of Euler numbers and polynomials associated with p-adic q-integral on Zp at q=−1 (2007, unknown): “[6,19,20], (3) we note that, by substituting x = 0 into (3), Bn(0) = Bn are the nth Bernoulli numbers.”
Euler Numbers and Polynomials Associated with Zeta Functions (2008, background): “n n! , |t| f.[1 -33]. These numbers arise in the series expansions of trigonometric functions, and are extremely important in number theory and analysis.”
q-Extension of the Euler formula and trigonometric functions (2007, background): “When speaking about q-extensions, one can regard q as an indeterminate, as a complex number q ∈ C ,o r ap-adic number q ∈ Cp .T heq -basic numbers are def ined ([1, 3, 4, 6, 7 ]) by [n]q = 1 − q”
Symmetry of power sum polynomials and multivariate fermionic p-adic invariant integral on ℤp (2009, unknown): “[1–22]).”
Note on the Euler Numbers and Polynomials (2008, background): “where again we must first expand and then replace Bi by Bi, cf. [6,8,9,10, 11 ].”
Special functions related to Dedekind-type DC-sums and their applications (2009, background): “n=0 Bn(z)t /n!, |t| < 2π, where Bn(0) = Bn stands for the Bernoulli numbers ([1–64]).”
New approach to the complete sum of products of the twisted (h, q)-Bernoulli numbers and polynomials (2007, background): “In [24], the first author defined generating functions of q-Bernoulli numbers and polynomials.”
Twisted (h,q)-Bernoulli numbers and polynomials related to twisted (h,q)-zeta function and L-function☆ (2006, unknown): “He also gave classical formulae related to Hurwitz zeta function and Lerch’s expression lnΓ (s) = ζ ′(0, x) − ζ ′(0) (see also [14,25,27]).”
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): “When one talks of q-extension, q is variously considered as an indeterminate, a complex q ∈ C, or a p-adic number q ∈ Cp, see [9-22].”