Relations among representations of integers by certain quadratic forms (Q2088983)

In this work, the authors consider the representation of integers by quadratic forms. For this purpose, they set three quadratic forms, namely, \[ c_{1}(x_{1}^{2}+x_{1}x_{2}+x_{2}^{2})+c_{2}(x_{3}^{2}+x_{3}x_{4}+x_{4}^{2})+\cdots+c_{k}(x_{2k-1}^{2}+x_{2k-1}x_{2k}+x_{2k}^{2}) \] \[ c_{1}x_{1}^{2}+c_{2}x_{2}^{2}+\cdots+c_{k}x_{k}^{2} \] and \[ c_{1}x_{1}(\frac{x_{1}+1}{2})+c_{2}x_{2}(\frac{x_{2}+1}{2})+\cdots+c_{k}x_{k}(\frac{x_{k}+1}{2}), \] where \(c_{i}\)'s and \(x_{i}\) 's are integers. For a positive integer \(n\), let \[ N(c_{1},c_{2},\cdots,c_{k};n),r(c_{1},c_{2},\cdots,c_{k};n) \ \ and \ \ t(c_{1},c_{2},\cdots,c_{k};n) \] denote the count of the representations of \(n\) by quadratic forms above. They derive some formulas on \(N(c_{1},c_{2},\cdots,c_{k};n),r(c_{1},c_{2},\cdots,c_{k};n)\) and \(t(c_{1},c_{2},\cdots,c_{k};n)\) for \(1<2k\leq 8\) by using Ramanujan's theta functions. Further they also deduce some new explicit formulas on \(r(c_{1},c_{2},\cdots,c_{k};n)\) for \(c_{i}\in \{1,2,3,6,8,9,16,18,24,48\}.\)