(untitled) (European Journal of Pure and Applied Mathematics, 2015)
We evaluate the convolution sums âˆ‘_l+135m=nÏƒ(l)Ïƒ(m),Â Â âˆ‘_3l+45m=nÏƒ(l)Ïƒ(m), Â âˆ‘_5l+27m=nÏƒ(l)Ïƒ(m),âˆ‘_9l+15m=nÏƒ(l)Ïƒ(m),Â âˆ‘_l+45m=nÏƒ(l)Ïƒ(m),âˆ‘_5l+9m=nÏƒ(l)Ïƒ(m), and âˆ‘_3l+15m=nÏƒ(l)Ïƒ(m) for all nÎµN using the theory of quasimodular forms and use some of these convolution sums to determine the number of representations of a positive integer n by some direct sum of the forms xâ‚Â²+xâ‚xâ‚‚+34xâ‚‚Â²,5xâ‚Â²+5xâ‚xâ‚‚+8xâ‚‚Â²,4xâ‚Â²Â±3xâ‚xâ‚‚+9xâ‚‚Â², 2xâ‚Â²Â±xâ‚xâ‚‚+17xâ‚‚Â² of discriminant -135