| Title:
|
Generalized Knopp identities for homogeneous Hardy sums and Cochrane-Hardy sums (English) |
| Author:
|
Liu, Huaning |
| Author:
|
Gao, Jing |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
62 |
| Issue:
|
4 |
| Year:
|
2012 |
| Pages:
|
1147-1159 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
Dedekind sum |
| Keyword:
|
Cochrane sum |
| Keyword:
|
Knopp identity |
| MSC:
|
11F20 |
| idZBL:
|
Zbl 1259.11044 |
| idMR:
|
MR3010262 |
| DOI:
|
10.1007/s10587-012-0069-7 |
| . |
| Date available:
|
2012-11-10T21:54:05Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143050 |
| . |
| Reference:
|
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