On Solutions of the q-Hypergeometric Equation with q N = 1

We consider the q-hypergeometric equation with q ^N = 1 and , , . We solve this equation on the space of functions given by a power series multiplied by a power of the logarithmic function. We prove that the subspace of solutions is two-dimensional over the field of quasi-constants. We get a basis for this space explicitly. In terms of this basis, we represent the q-hypergeometric function of the Barnes type constructed by Nishizawa and Ueno. Then we see that this function has logarithmic singularity at the origin. This is a difference between the q-hypergeometric functions with 0 < |q| < 1 and at |q| = 1.