Evaluation of Some Convolution Sums by Quasimodular Forms (European Journal of Pure and Applied Mathematics, 2015)
We evaluate the convolution sums âˆ‘_{l+135m=n}Ïƒ(l)Ïƒ(m),Â Â âˆ‘_{3l+45m=n}Ïƒ(l)Ïƒ(m), Â âˆ‘_{5l+27m=n}Ïƒ(l)Ïƒ(m),âˆ‘_{9l+15m=n}Ïƒ(l)Ïƒ(m),Â âˆ‘_{l+45m=n}Ïƒ(l)Ïƒ(m),âˆ‘_{5l+9m=n}Ïƒ(l)Ïƒ(m), and âˆ‘_{3l+15m=n}Ïƒ(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