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.

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