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 Formula: see text for odd integers Formula: see text and Formula: see text, where Formula: see text is the sum of the Formula: see textth powers of the positive divisors of Formula: see text. We consider four cases, namely (i) Formula: see text, (ii) Formula: see text; Formula: see text (iii) Formula: see text; Formula: see text and (iv) Formula: see text, 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 Formula: see text by certain positive definite quadratic forms. The existing formulas for Formula: see text (in 20), Formula: see text (in 7), Formula: see text (in 35), Formula: see text, Formula: see text (in 30) and Formula: see text (in 31), which were all obtained by using the theory of quasimodular forms, follow from our method.