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We obtain large $n$ asymptotics of $n \times n$ Hankel determinants whoseweight has a one-cut regular potential and Fisher-Hartwig singularities. Werestrict our attention to the case where the associated equilibrium measurepossesses either one soft edge and one hard edge (Laguerre-type) or two hardedges (Jacobi-type). We also present some applications in the theory of randommatrices. In particular, we can deduce from our results asymptotics forpartition functions with singularities, central limit theorems, correlations ofthe characteristic polynomials, and gap probabilities for (piecewise constant)thinned Laguerre and Jacobi-type ensembles. Finally, we mention some links withthe topics of rigidity and Gaussian multiplicative chaos.

Asymptotics of Hankel determinants with a Laguerre-type or Jacobi-type potential and Fisher-Hartwig singularities