The relations between N(a,b,c,d;n) and t(a,b,c,d;n) and (p,k)-parametrization of theta functions

In this paper, we confirm five conjectures on the relations between N(a,b,c,d;n) and t(a,b,c,d;n) given by Sun by utilizing (p,k)-parametrization of theta functions, where 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, with a,b,c,d∈N+, n∈N, and x,y,z,w∈Z.