Journal of the
Korean Mathematical Society JKMS

Let $k$ be an imaginary quadratic field, $\frak h$ the complex upper half plane, and let $\tau\in \frak h \cap k$, $q=e^{\pi i \tau}$. In this article, we obtain algebraic numbers from the $130$ identities of Rogers-Ramanujan continued fractions investigated in [28] and [29] by using Berndt's idea ([3]). Using this, we get special transcendental numbers. For example, $\frac{q^{1/8}}{1} {{}\atop {+}} \frac{-q}{1+q}\ {{}\atop {+}} \frac{-q^2 }{1+q^2 } {{}\atop {+}} {{}\atop {\cdots}}$ ([1]) is transcendental.

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

MSC numbers: 11Jxx, 11R04, 11F11