Journal of the
Korean Mathematical Society JKMS

Let $k$ be an imaginary quadratic field, $\frak h$ the complex upper half plane, and let $\tau\in \frak h \cap k$, $q=e^{\pi i \tau}$. For $n,t\in \Bbb Z^+$ with $1\leq t\leq n-1$, set $n=\frak z \cdot 2^{l} \ (\frak z=2,3,5,7,9,13,15)$ with $l\geq0$ integer. Then we show that $q^{\frac{n}{ 12} - \frac{t}{ 2} +\frac{t^2}{ 2 n}} \prod_{m=1}^\infty (1-q^{n m-t })(1-q^{n m-(n -t )})$ are algebraic numbers (Theorem 3.1, Theorem 4.2).

Keywords: algebraic number, theta series, Rogers-Ramanujan identities

MSC numbers: 11Jxx, 11R04, 11F11