On the number of representations of a positive integer as a sum of two binary quadratic forms

Let N denote the set of positive integers and Z the set of all integers. Let N-0 = N boolean OR{0}. Let a(1)x(2) + b(1)xy + c(1)y(2) and a(2)z(2) + b(2)zt + c(2)t(2) be two positive-definite, integral, binary quadratic forms. The number of representations of n is an element of N-0 as a sum of these two binary quadratic forms is N(a(1), b(1), c(1), a(2), b(2), c(2); n) := card{(x, y, z, t) is an element of Z(4) vertical bar n = a(1)x(2) + b(1)xy + c(1)y(2) + a(2)z(2) + b(2)zt + c(2)t(2)}. When (b1, b2) not equal (0, 0) we prove under certain conditions on a(1), b(1), c(1), a(2), b(2) and c(2) that N(a(1), b(1), c(1), a(2), b(2), c(2); n) can be expressed as a finite linear combination of quantities of the type N(a, 0, b, c, 0, d; n) with a, b, c and d positive integers. Thus, when the quantities N(a, 0, b, c, 0, d; n) are known, we can determine N(a(1), b(1), c(1), a(2), b(2), c(2); n). This determination is carried out explicitly for a number of quaternary quadratic forms a(1)x(2) + b(1)xy + c(1)y(2) + a(2)z(2) + b(2)zt + c(2)t(2). For example, in Theorem 1.2 we show for n is an element of N that N(3, 2, 3, 3, 2, 3; n) = {0 if n equivalent to 1 (mod 4), 2 sigma(N) if n equivalent to 3 (mod 4), 0 if n equivalent to 2 (mod 8), 4 sigma(N) if n equivalent to 4,6 (mod 8), 8 sigma(N) if n equivalent to 8 (mod 16), 24 sigma(N) if n equivalent to 0 (mod 16), where N is the largest odd integer dividing n and s(N) = Sigma(d is an element of Nd vertical bar N) d.