From continuous to discrete: weak limit of normalized Askey–Wilson measure

In this paper, we consider the weak limit of the normalized measure for Askey–Wilson polynomials when the parameter q approaches \(-1\) $-1$ from the right. We use two different methods to prove that the weak limit is a discrete measure with two mass points that are symmetric about the origin. The weights on these two mass points are, however, not always the same. We also calculate the weak limit of the q-ultraspherical measure when q approaches a complex root of unity.