ON THE INFINITE PRODUCTS DERIVED FROM THETA SERIES II
ON THE INFINITE PRODUCTS DERIVED FROM THETA SERIES II
Let k be an imaginary quadratic field, <TEX>${\eta}$</TEX> the complex upper half plane, and let <TEX>${\tau}{\in}{\eta}{\cap}k,\;q=e^{{\pi}{i}{\tau}}$</TEX>. For n, t <TEX>${\in}{\mathbb{Z}}^+$</TEX> with <TEX>$1{\leq}t{\leq}n-1$</TEX>, set n=<TEX>${\delta}{\cdot}2^{\iota}$</TEX>(<TEX>${\delta}$</TEX>=2, 3, 5, 7, 9, 13, 15) with <TEX>${\iota}{\geq}0$</TEX> integer. Then we show that <TEX>$q{\frac}{n}{12}-{\frac}{t}{2}+{\frac}{t^2}{2n}{\prod}_{m=1}^{\infty}(1-q^{nm-t})(1-q^{{nm}-(n-t)})$</TEX> are algebraic numbers.