The t-coefficient method III: A general series expansion for the product of theta functions with different bases and its applications

By means of Jacobi's triple product identity and the t-coefficient method, we establish a general series expansion formula for the product of arbitrary two theta functions with bases p and q:θ(az;p)θ(bzt;q)=∑n=−∞∞τp(n)(az)nθ((−1/a)tbpt(t+1)/2−tn;pt2q), from which some new results on products of arbitrary finitely many theta functions and theta identities associated with Ramanujan's circular summation can be derived, among them the most interesting ones include∑k=0mn−1∏i=1nθ(−ayiωk;q)=mnΓ(0;y1,y2,⋯,yn)×θ(−(y1y2⋯yn)mamnqnm22−nm2;qnm2), where ω is a primitive mn-th root of unity andΓ(s;y1,y2,⋯,yn)=∑i1+i2+⋯+in=sq12∑j=1nij2∏j=1nyjij. Furthermore, for y1y2y3=−q and ω=exp⁡(πi/3),Γ3(0;y1,y2,y3)+q−3/2Γ3(1;y1,y2,y3)=1(q;q)∞3∏i=02∏j=13θ(−yjω2i;q). The former contains Chan and Liu's circular summation (cf. Chan and Liu (2010) [12]) as a special case y1y2⋯yn=1. The latter generalizes both Borweins and Garvan's well-known cubic theta function identity and Schultz's bivariate-generalization (cf. Borwein et al. (1994) [9] and Schultz (2013) [28]), in which y1=y2=y3=ωq1/3 and y1=y3/z1,y2=y3/z2,y33=−qz1z2 respectively.