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Some generalizations of Knopp's identity*

Authors:

Huaning Liu

Institution:

(1) Department of Mathematics, Northwest University, Xi'an, Shaanxi, P.R. CHINA

Abstract:

For integers a, b and n > 0, define

$A_{\Gamma } {\left( {a,b,n} \right)} = {\sum\limits_{\begin{array}{*{20}c} {{r = 1}} \\ {{n\nmid b}} \\ \end{array} }^n {'{\left( {{\left( {\frac{{ar}} {n}} \right)}} \right)}} }\ln \Gamma {\left( {{\left\{ {\frac{{b\ifmmode\expandafter\bar\else\expandafter\=\fi{r}}} {n}} \right\}}} \right)}$

and

$B_{\Gamma } {\left( {a,b,n} \right)} = {\sum\limits_{\begin{array}{*{20}c} {{r = 1}} \\ {{n\nmid b}} \\ \end{array} }^n {'{\left( {{\left( {\frac{{ar}} {n}} \right)}} \right)}} }\frac{{{\Gamma }\ifmmode{'}\else$'$\fi{\left( {{\left\{ {\frac{{b\ifmmode\expandafter\bar\else\expandafter\=\fi{r}}} {n}} \right\}}} \right)}}} {{\Gamma {\left( {{\left\{ {\frac{{b\ifmmode\expandafter\bar\else\expandafter\=\fi{r}}} {n}} \right\}}} \right)}}},$

where ${\sum\limits{_r} ' }$ denotes the summation over all r such that (r, n) = 1, and $\overline{r}$ is defined by the equation $r\overline{r} \equiv 1\bmod n$ . The two sums are analogous to the homogeneous Dedekind sum S(a,b, n). The functional equations for A _Γ and B _Γ are established. Furthermore, Knopp's identity on Dedekind sum is extended. *This work is supported by the N.S.F. (10271093, 60472068) of P.R. China.

Keywords:

:" target="_blank">: Dedekind sum Knopp's identity homogeneous

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