Generalized Knopp identities for homogeneous Hardy sums and Cochrane-Hardy sums

Title:

Generalized Knopp identities for homogeneous Hardy sums and Cochrane-Hardy sums

Creator:

Liu, Huaning and Galo, Jing

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:0e3aafda-1989-4147-9a7a-f596207c37b5
uuid:0e3aafda-1989-4147-9a7a-f596207c37b5

Subject:

Dedekind sum, Cochrane sum, and Knopp identity

Type:

model:article and TEXT

Format:

bez média and svazek

Description:

Let $q$, $h$, $a$, $b$ be integers with $q>0$. The classical and the homogeneous Dedekind sums are defined by $$ s(h,q)=\sum _{j=1}^q\Big (\Big (\frac {j}{q}\Big )\Big )\Big (\Big (\frac {hj}{q}\Big )\Big ),\quad s(a,b,q)=\sum _{j=1}^q\Big (\Big (\frac {aj}{q}\Big )\Big )\Big (\Big (\frac {bj}{q}\Big )\Big ), $$ respectively, where $$ ((x))= \begin {cases} x-[x]-\frac {1}{2}, & \text {if $x$ is not an integer};\\ 0, & \text {if $x$ is an integer}. \end {cases} $$ The Knopp identities for the classical and the homogeneous Dedekind sum were the following: $$ \gathered \sum _{d\mid n}\sum _{r=1}^d s\Big (\frac {n}{d}a+rq,dq\Big )=\sigma (n)s(a,q),\\ \sum _{d\mid n}\sum _{r_1=1}^d\sum _{r_2=1}^d s\Big (\frac {n}{d}a+r_1q,\frac {n}{d}b+r_2q,dq\Big )=n\sigma (n)s(a,b,q), \endgathered $$ where $\sigma (n)=\sum \nolimits _{d\mid n}d$. \endgraf In this paper generalized homogeneous Hardy sums and Cochrane-Hardy sums are defined, and their arithmetic properties are studied. Generalized Knopp identities for homogeneous Hardy sums and Cochrane-Hardy sums are given.

Language:

English

Rights:

http://creativecommons.org/publicdomain/mark/1.0/
policy:public

Coverage:

1147-1159

Source:

Czechoslovak Mathematical Journal | 2012 Volume:62 | Number:4

Harvested from:

CDK

Metadata only:

false