Applications to the Quadratic Forms and Product-to-sum Formulae

In this paper, we attempt to express two valuesG(q) :=∞∑n=1g(n)qn= 24q∞∏n=1(1 +qn)32(1−qn)16(1 +q2n)12,H(q) :=∞∑n=1h(n)qn= 212q3∞∏n=1(1 +q2n)4(1−q4n)16into sums of squares and obtaing(n) = 8∑(x1,x2,···,x11,x12)∈Z12x21+x22+···+x29+x210+2(x211+x212)=n(x210−2x212),h(n) = 1024∑(x1,x2,x3,x4)∈Z42(x21+x22+2x24)+x23=n(−1)x21+x22+1(x41−3x21x22)(x23−4x24)(see Theorem 1.1 and Theorem 1.2, respectively). Finally, we obtain some relationsφ1(q)according toξ1(q),φ(q), andψ(q).