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It is shown that when dependence on the second flow of the KP hierarchy isadded, the resulting semi-stationary wave function of certain points in GeorgeWilson's adelic Grassmannian are generating functions of the exceptionalHermite orthogonal polynomials. This surprising correspondence betweendifferent mathematical objects that were not previously known to be so closelyrelated is interesting in its own right, but also proves useful in two ways: itleads to new algorithms for effectively computing the associated differentialand difference operators and it also answers some open questions about them.

The Adelic Grassmannian and Exceptional Hermite Polynomials