A COMBINATORIAL PROOF OF THE FARKAS-KRA THETA-FUNCTION IDENTITIES ANDTHEIR GENERALIZATIONS

Recently, Farkas acid Kra found some cubic theta function identities f rom their work on automorphic forms. Shortly thereafter, Farkas and Ro peliovich were able to generalize these to pth order theta function id entities using the theory of elliptic functions. We give short, elemen tary proofs of the cubic identities. We show that the pth order identi ties follow from more general relations between the coefficients of ce rtain theta functions. Our proof is combinatorial and utilizes certain orthogonal transformations. (C) 1995 Academic Press, Inc.