Table 1 Identities of convolution sums

\(\sum_{k=1}^{n_{1} -1}\sigma _{1}(k)\sigma _{1}(n_{1} -k)\)

(1)

\(\sum_{k=1}^{n_{1} -1} \bar{\sigma }(k)\bar{\sigma }(n_{1} -k) \)

Theorem 4

\(\sum_{\substack{a_{1}+a_{2}+a_{3}=n_{2} \\a_{1},a_{2},a_{3} \ge 1}}\sigma (a_{1})\sigma (a_{2})\sigma (a_{3})\)

[11], [20, p. 148]

\(\sum_{\substack{a_{1}+a_{2}+a_{3}=n_{2} \\a_{1},a_{2},a_{3} \ge 1}}\bar{\sigma }(a_{1})\bar{\sigma }(a_{2})\bar{\sigma }(a_{3}) \)

Theorem 5

\(\sum_{\substack{a_{1}+a_{2}+a_{3}+a_{4}=n_{3} \\ a_{1},a_{2},a_{3},a_{4} \ge 1}}\sigma (a_{1})\sigma (a_{2})\sigma (a_{3})\sigma (a_{4})\)

[11], [20, p. 158]

\(\sum_{\substack{a_{1}+a_{2}+a_{3}+a_{4}=n_{3} \\ a_{1},a_{2},a_{3},a_{4} \ge 1}}\bar{\sigma }(a_{1})\bar{\sigma }(a_{2})\bar{\sigma }(a_{3})\bar{\sigma }(a_{4})\)

Theorem 6