Vanishing coefficients in three families of products of theta functions

Recently, vanishing coefficients with arithmetic progressions in various products of theta functions were subsequently considered by the author, Baruah and Kaur, Xia and the author, Mc Laughlin, and Vandna and Kaur. In this paper, we further investigate this phenomenon in three families of products of theta functions, and prove that some arithmetic progressions on vanishing coefficients could be enjoyed by a family of products of theta functions, which significantly extend the previous results. For instance, one result proved in the present paper is that if the sequence {gamma(j,k,r,s,t) (n)}(n >= n0) is defined by Sigma(infinity)(n=n0) gamma(j,k,r,s,t)(n)q(n) = (-q(j), -q(r-j); q(r))(infinity)(s)(q(k), q(2r-k); q(2r))(infinity)(t), then gamma(k,5l-k,5l,1,3)(5n + 3k) = 0 holds for all n, where l >= 1, 1 <= k <= 5l and gcd(k, 5) = 1. Finally, we present two related conjectures.