Author’s abstract: In 1980, Richard Askey conjectured several \(q\)- integral extensions of Selberg’s multidimensional beta integral formula. We prove the last (and most general) of these. The proof utilizes our multivariate extension of a \(q\)-integral of Andrews and Askey along with a \(q\)-extension of an innovative method of Greg Anderson. The remainder of the paper is devoted to an exposition of Anderson’s method as applied to Selberg’s gamma integrals. Bombieri proved the Mehta-Dyson gamma integral formula by taking a limiting case of Selberg’s beta integral formula, thus exploiting the extra parameters in the beta integral. Richard Askey stressed the importance of finding direct evaluations of such limiting cases without introduction of extra parameters. This can be accomplished via Anderson’s method, and we give detailed proofs.
For the entire collection see [Zbl 0798.00010].