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

@article{Liu2012,
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 \[ ((x))= \{\left\lbrace \begin\{array\}\{ll\} x-[x]-\frac\{1\}\{2\}, & \text\{if $x$ is not an integer\};\\ 0, & \text\{if $x$ is an integer\}. \end\{array\}\right.\} \] The Knopp identities for the classical and the homogeneous Dedekind sum were the following: \[ \begin\{array\}\{c\}\sum \_\{d\mid n\}\sum \_\{r=1\}^d s\Big (\frac\{n\}\{d\}a+rq,dq\Big )=\sigma (n)s(a,q),\\ \sum \_\{d\mid n\}\sum \_\{r\_1=1\}^d\sum \_\{r\_2=1\}^d s\Big (\frac\{n\}\{d\}a+r\_1q,\frac\{n\}\{d\}b+r\_2q,dq\Big )=n\sigma (n)s(a,b,q), \end\{array\}\] 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.},
author = {Liu, Huaning, Gao, Jing},
journal = {Czechoslovak Mathematical Journal},
keywords = {Dedekind sum; Cochrane sum; Knopp identity; Dedekind sum; Cochrane sum; Knopp identity},
language = {eng},
number = {4},
pages = {1147-1159},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Generalized Knopp identities for homogeneous Hardy sums and Cochrane-Hardy sums},
url = {http://eudml.org/doc/246962},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Liu, Huaning
AU - Gao, Jing
TI - Generalized Knopp identities for homogeneous Hardy sums and Cochrane-Hardy sums
JO - Czechoslovak Mathematical Journal
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 4
SP - 1147
EP - 1159
AB - 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 \[ ((x))= {\left\lbrace \begin{array}{ll} x-[x]-\frac{1}{2}, & \text{if $x$ is not an integer};\\ 0, & \text{if $x$ is an integer}. \end{array}\right.} \] The Knopp identities for the classical and the homogeneous Dedekind sum were the following: \[ \begin{array}{c}\sum _{d\mid n}\sum _{r=1}^d s\Big (\frac{n}{d}a+rq,dq\Big )=\sigma (n)s(a,q),\\ \sum _{d\mid n}\sum _{r_1=1}^d\sum _{r_2=1}^d s\Big (\frac{n}{d}a+r_1q,\frac{n}{d}b+r_2q,dq\Big )=n\sigma (n)s(a,b,q), \end{array}\] 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.
LA - eng
KW - Dedekind sum; Cochrane sum; Knopp identity; Dedekind sum; Cochrane sum; Knopp identity
UR - http://eudml.org/doc/246962
ER -