Spectral properties of solutions of hypergeometric-type differential equations |
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Authors: | A Zarzo JS Dehesa |
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Institution: | a Departamento de Matemática Aplicada, Escuela Técnica Superior de Ingenieros Industriales, Universidad Politécnica de Madrid, C / José Gutiérrez Abascal 2, 28006 Madrid, Spain b Departamento de Física Moderna, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain |
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Abstract: | The second-order differential equation σ(x)y″ + τ(x)y′ + λy = 0 is usually called equation of hypergeometric type, provided that σ, τ are polynomials of degree not higher than two and one, respectively, and λ is a constant. Their solutions are commonly known as hypergeometric-type functions (HTFs). In this work, a study of the spectrum of zeros of those HTFs for which
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, and σ, τ are independent of ν, is done within the so-called semiclassical (or WKB) approximation. Specifically, the semiclassical or WKB density of zeros of the HTFs is obtained analytically in a closed way in terms of the coefficients of the differential equation that they satisfy. Applications to the Gaussian and confluent hypergeometric functions as well as to Hermite functions are shown. |
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Keywords: | Differential equations Zeros Special functions Semiclassical approximation |
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