  For eleven examples of one-dimensional quantum mechanics with shape-invariant
potentials, the Darboux-Crum transformations in terms of multiple pseudo
virtual state wavefunctions are shown to be equivalent to Krein-Adler
transformations deleting multiple eigenstates with shifted parameters. These
are based upon infinitely many polynomial Wronskian identities of classical
orthogonal polynomials, i.e. the Hermite, Laguerre and Jacobi polynomials,
which constitute the main part of the eigenfunctions of various quantum
mechanical systems with shape-invariant potentials.
