The multi-dimensional beta integral evaluation given by Selberg has been shown to be equivalent to, and thus imply, a root system conjecture made by I. G. Macdonald [SIAM J. Math. Anal. 13, 988-1007 (1982; Zbl 0498.17006)]. R. Askey [SIAM J. Math. Anal. 11, 938-951 (1980; Zbl 0458.33002)] conjectured a q-analog of the Selberg integral, motivated at least in part by the existence of q-analogs by Andrews, Macdonald, and Morris of the Macdonald conjecture.
The author proves the Askey conjecture, following the outline of Selberg’s original proof, with some clever analytical arguments at points where Selberg’s arguments from symmetry break down. He also applies the evaluation of the Askey integral to proving a conjecture by Morris for the root system \(A_ n\).