Selberg integrals and Catalan-Pfaffian Hankel determinants

In our previous works "Pfaffian decomposition and a Pfaffian analogue of q-Catalan Hankel determinants" (by M.Ishikawa, H. Tagawa and J. Zeng, J. Combin. Theory Ser. A, 120, 2013, 1263-1284) we have proposed several ways to evaluate certain Catalan-Hankel Pffafians and also formulated several conjectures. In this work we propose a new approach to compute these Catalan-Hankel Pffafians using Selberg's integral as well as their q-analogues. In particular, this approach permits us to settle most of the conjectures in our previous paper. R´ esum´ e. Dans nos travaux pr´´ "Pfaffian decomposition and a Pfaffian analogue of q-Catalan Hankel determi- nants" (by M.Ishikawa, H. Tagawa and J. Zeng, em J. Combin. Theory Ser. A, 120, 2013, 1263-1284) nous avons propos´ e plusieurs mpourcertains Catalan-Pffafian dde Hankel et avons aussi formul´ e plusieurs conjectures. Dans ce travail nous proposons une nouvelle approche pour calculer ces Catalan-Pffafian de- terminants de Hankel en utilisant l'intde Selberg ainsi que leurs q-analogues. En particulier, cette approche nous permet de confirmer la plus part de nos conjectures pr´ ec´ edentes.