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On the inverse problem of the product of a form by a monomial: the case n =4. Part I

2009; Taylor & Francis; Volume: 21; Issue: 1 Linguagem: Inglês

10.1080/10652460903016117

1476-8291

P. Maroni, I. Nicolau,

Quantum Mechanics and Non-Hermitian Physics

Abstract A form (linear functional) u is called regular if there exists a unique sequence of monic polynomials {P n } n≥0, deg P n =n, which is orthogonal with respect to u. On certain regularity conditions, the product of a regular form by a polynomial is still a regular form. In this paper, we consider the particular inverse problem: given a regular form v, find all the regular forms u that satisfy the equation x 4 u=−λ v, λ∈ℂ \ {0}. We give the second-order recurrence relation of the orthogonal polynomial sequence with respect to u. An example is studied. Keywords: orthogonal polynomialssemi-classical polynomialsintegral representations AMS Subject Classification : 42C0533C45 Acknowledgements Part of this work was performed while the first author was in residence at the Departamento de Matemática Aplicada da Faculdade de Ciências da Universidade do Porto. He is grateful for the support of the Fundação para a Ciência e Tecnologia, Praxis XXI.