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Hermite–Sobolev Polynomials with Symmetric Purely Imaginary Mass Points and Their Laguerre–Sobolev Reduction
1
Departamento de Matemáticas, Universidad Carlos III de Madrid, Av. de la Universidad, 30, 28911 Leganés, Madrid, Spain
2
Instituto de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Santo Domingo, Av. Alma Mater, Santo Domingo 10105, Dominican Republic
3
Instituto de Matemáticas e Instituto de Física, Facultad de Ciencias, Universidad Autónoma de Santo Domingo, Av. Alma Mater, Santo Domingo 10105, Dominican Republic
*
Author to whom correspondence should be addressed.
Axioms 2026, 15(9), 700; https://doi.org/10.3390/axioms15090700 (registering DOI)
Submission received: 21 August 2026
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Revised: 14 September 2026
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Accepted: 16 September 2026
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Published: 19 September 2026
Abstract
We study monic Hermite–Sobolev polynomials for the Gaussian weight with discrete Sobolev masses placed at conjugate purely imaginary points . 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 , 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 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.
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MDPI and ACS Style
Pijeira-Cabrera, H.; Féliz-Sánchez, C.; Toribio-Milane, J.; Gómez-Hernández, J.
Hermite–Sobolev Polynomials with Symmetric Purely Imaginary Mass Points and Their Laguerre–Sobolev Reduction. Axioms 2026, 15, 700.
https://doi.org/10.3390/axioms15090700
AMA Style
Pijeira-Cabrera H, Féliz-Sánchez C, Toribio-Milane J, Gómez-Hernández J.
Hermite–Sobolev Polynomials with Symmetric Purely Imaginary Mass Points and Their Laguerre–Sobolev Reduction. Axioms. 2026; 15(9):700.
https://doi.org/10.3390/axioms15090700
Chicago/Turabian Style
Pijeira-Cabrera, Héctor, Carlos Féliz-Sánchez, Juan Toribio-Milane, and José Gómez-Hernández.
2026. "Hermite–Sobolev Polynomials with Symmetric Purely Imaginary Mass Points and Their Laguerre–Sobolev Reduction" Axioms 15, no. 9: 700.
https://doi.org/10.3390/axioms15090700
APA Style
Pijeira-Cabrera, H., Féliz-Sánchez, C., Toribio-Milane, J., & Gómez-Hernández, J.
(2026). Hermite–Sobolev Polynomials with Symmetric Purely Imaginary Mass Points and Their Laguerre–Sobolev Reduction. Axioms, 15(9), 700.
https://doi.org/10.3390/axioms15090700
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