IRREGULAR DISTRIBUTION OF (N-BETA), N=1,2,3, ... QUADRATURE OF SINGULAR INTEGRANDS, AND CURIOUS BASIC HYPERGEOMETRIC-SERIES

Let beta-epsilon (0,1) be irrational and let {n-beta} denote the fractional part of n-beta, n greater-than-or-equal-to 1. The uniform distribution of {b-beta}, n greater-than-or-equal-to 1, implies that [GRAPHICS] for each bounded and Riemann integrable g. Hardy and Littlewood proved that this relation persists when g has integrable singularities at 0 and 1, under suitable conditions on g and beta. We show that by choosing suitable beta, and g with an arbitrarily weak singularity at a suitable interior point alpha-epsilon(0, 1), one can ensure that [GRAPHICS] On the other hand, if the singularity lies at 0, then at least [GRAPHICS] The motivation for these results lies in determination of the radius of convergence of the q or basic hypergeometric series [GRAPHICS] the solution of the functional equation [GRAPHICS] Especially for \A\ = \q\ = 1, these power series are of interest in Pade approximation. Although the radius of convergence is 1 for "most" A and q, we show that f may be a transcendental entire function for suitable \A\ = \q\ = 1.