Table 5 Some convolution formulas

${\sum }_{k=1}^{N-1}{\sigma }_1^{\ast }(k){\sigma }_5^{\ast }(N-k)$

$\frac{1}{408}\{12{\sigma }_7^{\ast }(N)-17N{\sigma }_5^{\ast }(N)+5b(N)\}$

${\sum }_{k=1}^{N-1}{\sigma }_3^{\ast }(k){\sigma }_3^{\ast }(N-k)$

$\frac{1}{136}\{{\sigma }_7^{\ast }(N)-b(N)\}$

${\sum }_{k=1}^{N-1}{\sigma }_1^{\ast }(k){\sigma }_7^{\ast }(N-k)$

$\frac{1}{992}\{17{\sigma }_9^{\ast }(N)-31N{\sigma }_7^{\ast }(N)+448l(N)+14c(N)\}$

${\sum }_{k=1}^{N-1}{\sigma }_3^{\ast }(k){\sigma }_5^{\ast }(N-k)$

$\frac{1}{496}\{{\sigma }_9^{\ast }(N)-32l(N)-c(N)\}$

${\sum }_{k=1}^{N-1}{\sigma }_5^{\ast }(k){\sigma }_5^{\ast }(N-k)$

$\frac{1}{2,764}\{{\sigma }_{11}^{\ast }(N)-\tau (N)+692\tau (\frac{N}{2})\}$

${\sum }_{k=1}^{N-1}{\sigma }_1^{\ast }(k){\sigma }_9^{\ast }(N-k)$

$\frac{1}{27,640}\{310{\sigma }_{11}^{\ast }(N)-691N{\sigma }_9^{\ast }(N)+381\tau (N)+109,488\tau (\frac{N}{2})\}$

${\sum }_{k=1}^{N-1}{\sigma }_3^{\ast }(k){\sigma }_7^{\ast }(N-k)$

$\frac{1}{22,112}\{17{\sigma }_{11}^{\ast }(N)-17\tau (N)-13,112\tau (\frac{N}{2})\}$