Sums of triangular numbers and sums of squares

Abstract

For non-negative integers a,b, and n, let N(a,b;n) be the number of representations of n as a sum of squares with coefficients 1 or 3 (a of ones and b of threes). Let N⁎(a,b;n) be the number of representations of n as a sum of odd squares with coefficients 1 or 3 (a of ones and b of threes). We have that N⁎(a,b;8n+a+3b) is the number of representations of n as a sum of triangular numbers with coefficients 1 or 3 (a of ones and b of threes). It is known that for a and b satisfying 1≤a+3b≤7, we haveN⁎(a,b;8n+a+3b)=22+(a4)+abN(a,b;8n+a+3b) and for a and b satisfying a+3b=8, we haveN⁎(a,b;8n+a+3b)=22+(a4)+ab(N(a,b;8n+a+3b)−N(a,b;(8n+a+3b)/4)). Such identities are not known for a+3b>8. In this paper, for general a and b with a+b even, we prove asymptotic equivalence of formulas similar to the above, as n→∞. One of our main results extends a theorem of Bateman, Datskovsky, and Knopp where the case b=0 and general a was considered. Our approach is different from Bateman-Datskovsky-Knopp's proof where the circle method and singular series were used. We achieve our results by explicitly computing the Eisenstein components of the generating functions of N⁎(a,b;8n+a+3b) and N(a,b;8n+a+3b). The method we use is robust and can be adapted in studying the asymptotics of other representation numbers with general coefficients.