Hermite–Sobolev Polynomials with Symmetric Purely Imaginary Mass Points and Their Laguerre–Sobolev Reduction

We study monic Hermite–Sobolev polynomials for the Gaussian weight with discrete Sobolev masses placed at conjugate purely imaginary points ±ick. The paired geometry preserves reflection symmetry and exact parity while allowing genuinely nonreal zeros. The main contribution is an exact quadratic Hermite–Laguerre reduction: under t=z2, the even and odd subsequences become Laguerre–Sobolev families, and derivative masses induce explicit positive semidefinite matrices that may contain off-diagonal couplings between derivative orders. We also obtain a minimal annihilating polynomial, exact quasi-orthogonality, a finite-term recurrence, and, under strict positivity of the Sobolev weights, a degree-independent horizontal strip containing all zeros. In the case of positive zero-order masses, the reduction is scalar and transfers Laguerre–Sobolev relative asymptotics, implying that, for all sufficiently large degrees, exactly 2N nonreal zeros occur and are attracted to the prescribed mass points. For completeness, Hermitian kernel connection formulas yield rational ladder operators and a second-order differential equation.