Explicit evaluation of triple convolution sums of the divisor functions

In this paper, we use the theory of modular forms and give a general method to obtain the convolution sums $$W_{d_1,d_2,d_3}^{r_1,r_2,r_3}(n)=\sum _{\genfrac{}{}{0ex}{}{l_1,l_2,l_3\in \mathbb{N}}{d_1{l}_1+d_2{l}_2+d_3{l}_3=n}}{\sigma }_{r_1}(l_1){\sigma }_{r_2}(l_2){\sigma }_{r_3}(l_3),$$ for odd integers $r_1,r_2,r_3\ge 1,$ and $d_1,d_2,d_3,n\in \mathbb{N}$, where ${\sigma }_r(n)$ is the sum of the $r$th powers of the positive divisors of $n$. We consider four cases, namely (i) $r_1=r_2=r_3=1$, (ii) $r_1=r_2=1$; $r_3\ge 3$ (iii) $r_1=1$; $r_2,r_3\ge 3$ and (iv) $r_1,r_2,r_3\ge 3$, and give explicit expressions for the respective convolution sums. We provide several examples of these convolution sums in each case and further use these formulas to obtain explicit formulas for the number of representations of a positive integer $n$ by certain positive definite quadratic forms. The existing formulas for $W_{1,1,1}(n)$ (in [20]), $W_{1,1,2}(n),W_{1,2,2}(n),W_{1,2,4}(n)$ (in [7]), $W_{1,1,1}^{1,3,3}(n),W_{1,1,3}^{1,3,3}(n),W_{1,3,3}^{1,3,3}(n),W_{3,1,1}^{1,3,3}(n),W_{3,3,1}^{1,3,3}(n)$ (in [35]), $W_{d_1,d_2,d_3}(n)$, $\mathrm{\text{lcm}}(d_1,d_2,d_3)\le 6$ (in [30]) and $\mathrm{\text{lcm}}(d_1,d_2,d_3)=7,8,9$ (in [31]), which were all obtained by using the theory of quasimodular forms, follow from our method.