On the infinite products derived from theta series II

Abstract

Let k be an imaginary quadratic field, the complex upper half plane, and let tau is an element of h boolean AND k, q = e(pi it). For n, t is an element of Z(+) with 1 <= t <= n-1, set n = z.2(t) (z = 2, 3, 5, 7, 9, 13, 15) with l >= 0 integer. Then we show that q (n/12 - t/2 + t2/2n) Pi(infinity)(m=1) (1 - q(nm-t)) (1 - q(nm-(n-t))) are algebraic numbers.