The determinants and Pfaffians appearing in the enumeration of plane partitions and its applications on mathematical physics

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In the enumeration problems of plane partitions under several symmetries and their generalizations we obtain several interesting determinants and Pfaffians as their generating functions. We have been interested in the evaluation of these determinants and Pfaffians and have been intensively studied those evaluation problems. Several studies have been hinted close relations between the evaluation of Hankel determinants and so-called Hankel Pfaffians and the evaluation of the determinants and Pfaffians appearing in the enumeration problems. In our recent study we find a method to evaluate Hankel Pfaffians by the Selberg integrals through de Bruijn's formula. In fact it is well-known that the Selberg integrals and its extensions are closely related to the orthogonal polynomials and Dunkl operators and its evaluation can be obtained by multivariate generalizations of the classical orthogonal polynomials. Further we can relate the Hankel determinants and Pfaffians with the determinants and Pfaffians appearing in our enumeration problems by manipulations of the generating functions of the entries of determinants(Pfaffians). Especially the associated Jacobi polynomials play an important role in the enumeration of the plane partitions because its distribution measure is related to Gauss's hypergeometric series_3F2. We consider Hankel determinants or Pfaffians whose entries are obtained from moments of a class of orthogonal polynomials. The moment generating function related to our enumeration problems seem to be related to the associated Jacobi polynomials by considering the continued fraction expansion of the moment generating function related to our enumeration problems.