Generalizations of some conjectures of Sun on the relations between N(a, b, c, d; n) and T(a, b, c, d; n)

Let T(a, b, c, d; n) denote the number of representations of n as \(a\frac{x(x+1)}{2}+ b\frac{y(y+1)}{2}+c\frac{z(z+1) }{2}+d\frac{w(w+1)}{2}\), where \(a,\ b,\ c,\ d\) are positive integers and \( n,\ x,\ y,\ z,\ w\) are arbitrary non-negative integers, and let N(a, b, c, d; n) denote the number of representations of n as \(ax^2+by^2+cz^2+dw^2\), where this time x, y, z and w are integers. Recently, Sun proved relations between N(a, b, c, d; n) and T(a, b, c, d; n) and posed 23 conjectures. In this paper, we not only confirm five conjectures due to Sun, but also generalize four of them.