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Some generalizations of Knopp's identity*

Authors:

Huaning Liu

Affiliation:

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

Abstract:

For integers a, b and n > 0, define

$A_{Gamma } {left( {a,b,n} right)} = {sumlimits_{begin{array}{*{20}c} {{r = 1}} {{nnmid b}} end{array} }^n {';;{left( {{left( {frac{{ar}} {n}} right)}} right)}} }ln Gamma {left( {{left{ {frac{{bifmmodeexpandafterbarelseexpandafter=fi{r}}} {n}} right}}} right)}$

and

$B_{Gamma } {left( {a,b,n} right)} = {sumlimits_{begin{array}{*{20}c} {{r = 1}} {{nnmid b}} end{array} }^n {';;{left( {{left( {frac{{ar}} {n}} right)}} right)}} }frac{{{Gamma }ifmmode{'}else$'$fi{left( {{left{ {frac{{bifmmodeexpandafterbarelseexpandafter=fi{r}}} {n}} right}}} right)}}} {{Gamma {left( {{left{ {frac{{bifmmodeexpandafterbarelseexpandafter=fi{r}}} {n}} right}}} right)}}},$

where ${sumlimits{_r} ' }$ denotes the summation over all r such that (r, n) = 1, and $$
overline{r}
$$

is defined by the equation $$
roverline{r} equiv 1;bmod n
$$

. 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:

KeywordHeading" >: Dedekind sum Knopp's identity homogeneous

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