In the previous paper (J. Combin. Theory Ser. A, 120, 2013, 1263–1284) H.
Tagawa and the two authors proposed an algebraic method to compute certain
Pfaffians whose form resemble to Hankel determinants associated with moment
sequences of the classical orthogonal polynomials. At the end of the paper they
offered several conjectures. In this work we employ a completely different
method to evaluate this type of Pfaffians. The idea is to apply certain de
Bruijn type formula and convert the evaluation of the Pfaffians to the certain
Selberg type integrals. This approach works not only for Pfaffians but also for
hyperpfaffians. Hence it enables us to establish much more generalized
identities than those conjectured in the previous paper. We also investigate
some Pfaffians related to classical $q$-orthogonal polynomials.
