On the number of representations of integers by sums of mixed numbers (Q2805963)

Denote by \(N(a_1,a_2,a_3;b_1,b_2;c_1,c_2,c_3,c_4;n)\) the number of representations of \(n\) in the form NEWLINE\[NEWLINE n=\sum_{i=1}^{a_1} (w_i^2+w_im_i+m_i^2)+2\sum_{i=1}^{a_2} (u_i^2+u_iv_i+v_i^2)+4 \sum_{i=1}^{a_3} (r_i^2+r_is_i+s_i^2)+NEWLINE\]NEWLINE NEWLINE\[NEWLINE+\sum_{i=1}^{b_1} p_i^2+3\sum_{i=1}^{b_2} q_i^2+\sum_{i=1}^{c_1} \frac{t_i^2+t_i}{2}+\sum_{i=1}^{c_2} (x_i^2+x_i)+3\sum_{i=1}^{c_3} \frac{y_i^2+y_i}{2}+3\sum_{i=1}^{c_4} (z_i^2+z_i).NEWLINE\]NEWLINE Some explicit formulae for \(N(a_1,a_2,a_3;b_1,b_2;c_1,c_2,c_3,c_4;n)\) with some special values of \(a_i\), \(b_i\), and \(c_i\) are established. These formulae represent \(N(a_1,\dots,c_4;n)\) as linear combinations of \(\sigma_3(n/d)\), where \(d=1,2,3,4,6,12\), and \(\sigma_3(n)=\sum_{d\mid n} d^3\). The proofs are based on some theta function identities.