A Combinatorial Proof of the Farkas-Kra Theta Function Identities and Their Generalizations

Recently, Farkas acid Kra found some cubic theta function identities from their work on automorphic forms. Shortly thereafter, Farkas and Ropeliovich were able to generalize these to pth order theta function identities using the theory of elliptic functions. We give short, elementary proofs of the cubic identities. We show that the pth order identities follow from more general relations between the coefficients of certain theta functions. Our proof is combinatorial and utilizes certain orthogonal transformations.