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

Let N(a,b,c,d;n) and t(a,b,c,d;n) denote the number of representations of n as ax2+by2+cz2+dw2 and the number of representations of n as ax(x+1)2+by(y+1)2+cz(z+1)2+dw(w+1)2, respectively, where a,b,c,d are positive integers, n is a nonnegative integer and x,y,z,w are integers. Sun established many relations between N(a,b,c,d;n) and t(a,b,c,d;n) and posed 23 conjectures. Yao proved five of them by using (p,k)-parametrization of theta functions. In this paper, we confirm four conjectures of Sun by employing theta function identities.