scholarly journals Elementary proof of the transformation formula for Lambert series involving generalized Dedekind sums

1982 ◽  
Vol 15 (1) ◽  
pp. 14-24 ◽  
Author(s):  
Tom M. Apostol
Keyword(s):  
Elementary Proof ◽  
Transformation Formula ◽  
Dedekind Sums ◽  
Lambert Series

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.)

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.

Dedekind sums and the transformation formula in function fields

2016 ◽  
Vol 12 (08) ◽  
pp. 2061-2072 ◽  
Author(s):  
Yoshinori Hamahata
Keyword(s):  
Function Fields ◽  
Transformation Formula ◽  
Dedekind Sums ◽  
Dedekind Sum ◽  
Sum Formula

Dedekind used the classical Dedekind sum [Formula: see text] to describe the transformation of [Formula: see text] under the substitution [Formula: see text]. In this paper, we use the Dedekind sum [Formula: see text] in function fields to describe the transformation of a certain series under the substitution [Formula: see text].

scholarly journals Franel integrals of order four

1996 ◽  
Vol 60 (2) ◽  
pp. 192-203 ◽  
Author(s):  
Richard J. McIntosh
Keyword(s):  
Lattice Point ◽  
Elementary Proof ◽  
Unit Interval ◽  
Dedekind Sums ◽  
Dedekind Sum ◽  
Elementary Method ◽  
The Third ◽  
Reciprocity Law ◽  
Order Four

AbstractLet ((x)) =x−⌊x⌋−1/2 be the swatooth function. Ifa, b, cand e are positive integeral, then the integral or ((ax)) ((bx)) ((cx)) ((ex)) over the unit interval involves Apolstol's generalized Dedekind sums. By expressing this integral as a lattice-point sum we obtain an elementary method for its evaluation. We also give an elementary proof of the reciprocity law for the third generalized Dedekind sum.

scholarly journals An identity involving Nörlund polynomials

1991 ◽  
Vol 43 (2) ◽  
pp. 307-315
Author(s):  
A.E. Özlük ◽  
C. Snyder
Keyword(s):  
Lattice Points ◽  
Transformation Formula ◽  
Lambert Series

We prove an identity involving Nörlund polynomials, the proof of which is elementary and involves the enumeration of lattice points. The identity is slightly stronger than an identity of Carlitz which he obtained by using Apostol's transformation formula for Lambert series.

An alternative transformation formula for the Dedekind η-function via the Chinese Remainder Theorem

2016 ◽  
Vol 12 (02) ◽  
pp. 513-526 ◽  
Author(s):  
Heng Huat Chan ◽  
Teoh Guan Chua
Keyword(s):  
Chinese Remainder Theorem ◽  
Transformation Formula ◽  
Dedekind Sums ◽  
Reciprocity Law

In this article, we will give a new proof of the reciprocity law for Dedekind sums, as well as a proof of the transformation formula for the Dedekind [Formula: see text]-function using the Chinese Remainder Theorem.

scholarly journals A new transformation formula involving derived WP-bailey pair and its applications

Filomat ◽  
2020 ◽  
Vol 34 (13) ◽  
pp. 4245-4252
Author(s):  
Zhizheng Zhang ◽  
Jing Gu ◽  
Hanfei Song
Keyword(s):  
Transformation Formula ◽  
Lambert Series ◽  
Summation Formulas ◽  
Transformation Formulas

The main purpose of this paper is to obtain a new transformation formula involving the derived WP-Bailey pair. As applications, by using two 10?9 summation formulas, some transformation formulas in terms of generalized Lambert series are obtained.

Sign in / Sign up

Export Citation Format

Share Document