PROOFS OF SOME CONJECTURES OF SUN ON THE RELATIONS BETWEEN T(a, b, c, d; n) AND N(a, b, c, d; n)

Let N(a, b, c, d; n) be the number of representations of n as ax(2) + by(2) + cz(2) + dw(2) and T (a, b, c, d; n) be the number of representations of n as alpha X( X + 1)/2 + b Y (Y + 1)/2 + cZ(Z + 1)/2 + dW(W + 1)/2, where a, b, c, d are positive integers, x, y, z, w are integers, and n, X, Y, Z, W are nonnegative integers. In this paper, by using theta function identities, we confirm Sun's conjectures that T (a, b, c, d; n) is a linear combination of N(a, b, c, d; m) and N(a, b, c, d; 4m), with m = 8n + a + b+ c+ d, for several values of a, b, c, d.