Mehler Formula for Wronskians of Hermite Polynomials

We prove that the bilinear generating function for Wronskians of Hermite polynomials can be expressed as the classical Mehler kernel multiplied by a polynomial, thereby extending the result of Pupasov-Maksimov [Ann. Physics 363 (2015), 122-135, arXiv:1502.01778] for exceptional Hermite polynomials. We establish several properties of the polynomials appearing in this extended version of the Mehler formula and present four conjectures about them.