Convergence properties of a quadrature formula of Clenshaw-Curtis type for the Gegenbauer weight function

A quadrature formula of Clenshaw-Curtis type for functions of the form (1-x(2))(lambda-1/2) f(x) over the interval [-1,1] exhibits a curious phenomenon when applied to certain analytic functions. As the number of points in the quadrature rule increases the error may sometimes decay to zero in two distinct stages rather than in one depending on the value of lambda. In this paper we shall derive explicit and asymptotic error formulae which describe this phenomenon.