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Czechoslovak Mathematical Journal, Vol. 62, No. 4, pp. 1147-1159, 2012
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Generalized Knopp identities for homogeneous Hardy sums and Cochrane-Hardy sums

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Huaning Liu, Jing Gao

* Huaning Liu*, Department of Mathematics, Northwest University, Xi'an 710069, Shaanxi, P. R. China, e-mail: ` hnliumath@hotmail.com`; * Jing Gao*, School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, Shaanxi, P. R. China, e-mail: ` jgao@mail.xjtu.edu.cn`

**Abstract:** 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]-\frac12, & \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}da+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}da+r_1q,\frac{n}db+r_2q,dq\Big)=n\sigma(n)s(a,b,q),

where $\sigma(n)=\sum\nolimits_{d\mid n}d$. 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.

**Keywords:** Dedekind sum, Cochrane sum, Knopp identity

**Classification (MSC 2010):** 11F20

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