scholarly journals Reciprocity of poly-Dedekind-type DC sums involving poly-Euler functions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yuankui Ma ◽  
Dae San Kim ◽  
Hyunseok Lee ◽  
Hanyoung Kim ◽  
Taekyun Kim
Keyword(s):  
Modular Group ◽  
Reciprocity Relation ◽  
Dedekind Sums ◽  
Transformation Behavior ◽  
Euler Function ◽  
Dedekind Eta Function ◽  
Reciprocity Relations ◽  
Bernoulli Functions

AbstractThe classical Dedekind sums appear in the transformation behavior of the logarithm of the Dedekind eta-function under substitutions from the modular group. The Dedekind sums and their generalizations are defined in terms of Bernoulli functions and their generalizations, and are shown to satisfy some reciprocity relations. In contrast, Dedekind-type DC (Daehee and Changhee) sums and their generalizations are defined in terms of Euler functions and their generalizations. The purpose of this paper is to introduce the poly-Dedekind-type DC sums, which are obtained from the Dedekind-type DC sums by replacing the Euler function by poly-Euler functions of arbitrary indices, and to show that those sums satisfy, among other things, a reciprocity relation.

scholarly journals Identities on poly-Dedekind sums

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Hyunseok Lee ◽  
Lee-Chae Jang
Keyword(s):  
Modular Group ◽  
Reciprocity Relation ◽  
Dedekind Sums ◽  
Transformation Behavior ◽  
Dedekind Eta Function ◽  
Bernoulli Function ◽  
Bernoulli Functions

Abstract Dedekind sums occur in the transformation behavior of the logarithm of the Dedekind eta-function under substitutions from the modular group. In 1892, Dedekind showed a reciprocity relation for the Dedekind sums. Apostol generalized Dedekind sums by replacing the first Bernoulli function appearing in them by any Bernoulli functions and derived a reciprocity relation for the generalized Dedekind sums. In this paper, we consider the poly-Dedekind sums obtained from the Dedekind sums by replacing the first Bernoulli function by any type 2 poly-Bernoulli functions of arbitrary indices and prove a reciprocity relation for the poly-Dedekind sums.

scholarly journals Some identities and reciprocity relationsof unipoly-Dedekind type DC sums

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Hye Kyung Kim ◽  
Dae Sik Lee
Keyword(s):  
The Other ◽  
Reciprocity Relation ◽  
Dedekind Sums ◽  
Euler Polynomials ◽  
Euler Function ◽  
Polylogarithm Function

AbstractDedekind type DC sums and their generalizations are defined in terms of Euler functions and their generalization. Recently, Ma et al. (Adv. Differ. Equ. 2021:30 2021) introduced the poly-Dedekind type DC sums by replacing the Euler function appearing in Dedekind sums, and they were shown to satisfy a reciprocity relation. In this paper, we consider two kinds of new generalizations of the poly-Dedekind type DC sums. One is a unipoly-Dedekind type DC sum associated with the type 2 unipoly-Euler functions expressed in the type 2 unipoly-Euler polynomials using the modified polyexponential function, and we study some identities and the reciprocity relation for these unipoly-Dedekind type DC sums. The other is a unipoly-Dedekind sums type DC associated with the poly-Euler functions expressed in the unipoly-Euler polynomials using the polylogarithm function, and we derive some identities and the reciprocity relation for those unipoly-Dedekind type DC sums.

scholarly journals Poly-Dedekind sums associated with poly-Bernoulli functions

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Yuankui Ma ◽  
Dae San Kim ◽  
Hyunseok Lee ◽  
Taekyun Kim
Keyword(s):  
Reciprocity Relation ◽  
Dedekind Sums ◽  
Bernoulli Function ◽  
Bernoulli Functions

AbstractApostol considered generalized Dedekind sums by replacing the first Bernoulli function appearing in Dedekind sums by any Bernoulli functions and derived a reciprocity relation for them. Recently, poly-Dedekind sums were introduced by replacing the first Bernoulli function appearing in Dedekind sums by any type 2 poly-Bernoulli functions of arbitrary indices and were shown to satisfy a reciprocity relation. In this paper, we consider other poly-Dedekind sums that are obtained by replacing the first Bernoulli function appearing in Dedekind sums by any poly-Bernoulli functions of arbitrary indices. We derive a reciprocity relation for these poly-Dedekind sums.

scholarly journals Transformation of Some Lambert Series and Cotangent Sums

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 840
Author(s):  
Namhoon Kim
Keyword(s):  
Half Plane ◽  
Modular Group ◽  
Transformation Formula ◽  
Contour Integral ◽  
Dedekind Sums ◽  
Lambert Series ◽  
Simple Derivation ◽  
Special Cases ◽  
Upper Half Plane

By considering a contour integral of a cotangent sum, we give a simple derivation of a transformation formula of the series A ( τ , s ) = ∑ n = 1 ∞ σ s − 1 ( n ) e 2 π i n τ for complex s under the action of the modular group on τ in the upper half plane. Some special cases directly give expressions of generalized Dedekind sums as cotangent sums.

scholarly journals Dedekind sums for a fuchsian group, II

1974 ◽  
Vol 53 ◽  
pp. 171-187 ◽  
Author(s):  
Larry Joel Goldstein
Keyword(s):  
Fuchsian Group ◽  
Modular Group ◽  
Preceding Paper ◽  
Principal Congruence ◽  
Dedekind Sums ◽  
Natural Question ◽  
Congruence Subgroups ◽  
Limit Formula ◽  
Upper Half Plane ◽  
The Subject

In [1] we derived a generalization of Kronecker’s first limit formula. Our generalization was a limit formula for the Eisenstein series for an arbitrary cusp of a Fuchsian group Γ of the first kind operating on the complex upper half-plane H. In that work, we introduced Dedekind sums associated to the principal congruence subgroups Γ(N) of the elliptic modular group. The work of our preceding paper suggests a natural question: Is there a generalization of Kronecker’s second limit formula to the setting of a general Fuchsian group of the first kind? The answer to this question is the subject of this paper.

scholarly journals Dedekind Sums for a Fuchsian Group, I

1973 ◽  
Vol 50 ◽  
pp. 21-47 ◽  
Author(s):  
Larry Joel Goldstein
Keyword(s):  
Fuchsian Group ◽  
Dedekind Sums ◽  
Dedekind Eta Function ◽  
Limit Formula ◽  
Group I

The well-known first limit formula of Kronecker asserts thatwhere z = x + iy is contained in the complex upper halfplane H, C = the Euler-Mascheroni constant, and η(z) is the Dedekind eta-function defined by

scholarly journals On a Reciprocity Law for Finite Multiple Zeta Values

2010 ◽  
Vol 2010 ◽  
pp. 1-13
Author(s):  
Markus Kuba ◽  
Helmut Prodinger
Keyword(s):  
Reciprocity Relation ◽  
Weighted Sums ◽  
Combinatorial Proof ◽  
Multiple Zeta Values ◽  
Reciprocity Law ◽  
Multiple Zeta ◽  
Reciprocity Relations ◽  
Zeta Values ◽  
Identity Based ◽  
Partial Fraction Decomposition

It was shown by Kirschenhofer and Prodinger (1998) and Kuba et al. (2008) that harmonic numbers satisfy certain reciprocity relations, which are in particular useful for the analysis of the quickselect algorithm. The aim of this work is to show that a reciprocity relation from Kirschenhofer and Prodinger (1998) and Kuba et al. (2008) can be generalized to finite variants of multiple zeta values, involving a finite variant of the shuffle identity for multiple zeta values. We present the generalized reciprocity relation and furthermore a combinatorial proof of the shuffle identity based on partial fraction decomposition. We also present an extension of the reciprocity relation to weighted sums.

Dedekind sums and Hecke operators

1980 ◽  
Vol 88 (1) ◽  
pp. 11-14 ◽  
Author(s):  
L. Alayne Parson
Keyword(s):  
Elementary Proof ◽  
Hecke Operators ◽  
Transformation Formula ◽  
Dedekind Sums ◽  
Lambert Series ◽  
Dedekind Eta Function ◽  
Dedekind Sum ◽  
Modular Transformation ◽  
Special Case

By considering the action of the Hecke operators on the logarithm of the Dedekind eta function together with the modular transformation formula for this function, Knopp (8) proved an extension of an identity of Dedekind for the classical Dedekind sums first mentioned by H. Petersson. By looking at the action of the Hecke operators on certain Lambert series studied by Apostol(l) together with the transformation formulae for these series, Parson and Rosen (9) established an analogous identity for a type of generalized Dedekind sum. A special case of this identity was initially proved by Carlitz(6). In this note an elementary proof of these identities is given. The Hecke operators are applied directly to the Dedekind sums without invoking the transformation formulae for the logarithm of the eta function or for the Lambert series. (Recently, L. Goldberg has given another elementary proof of Knopp's identity.)

On generic complexity of the subset sum problem in monoids and groups of integer matrix of order two

2020 ◽  
Vol 25 (4) ◽  
pp. 10-15
Author(s):  
Alexander Nikolaevich Rybalov
Keyword(s):  
Linear Group ◽  
Modular Group ◽  
Special Linear Group ◽  
Second Order ◽  
Integer Matrix ◽  
Subset Sum ◽  
Algorithmic Problems ◽  
Almost All ◽  
Subset Sum Problems ◽  
Generic Complexity

Generic-case approach to algorithmic problems was suggested by A. Miasnikov, I. Kapovich, P. Schupp and V. Shpilrain in 2003. This approach studies behavior of an algo-rithm on typical (almost all) inputs and ignores the rest of inputs. In this paper, we prove that the subset sum problems for the monoid of integer positive unimodular matrices of the second order, the special linear group of the second order, and the modular group are generically solvable in polynomial time.

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