The Laguerre Constellation of Classical Orthogonal Polynomials
A linear functional <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">u</mi></semantics></math></inline-formula> is classical if there exist polynomials <...
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-01-01
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| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/13/2/277 |
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| Summary: | A linear functional <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">u</mi></semantics></math></inline-formula> is classical if there exist polynomials <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϕ</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ψ</mi></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo form="prefix">deg</mo><mi>ϕ</mi><mo>≤</mo><mn>2</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo form="prefix">deg</mo><mi>ψ</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula> such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">D</mi><mfenced separators="" open="(" close=")"><mi>ϕ</mi><mo>(</mo><mi>x</mi><mo>)</mo><mi mathvariant="bold">u</mi></mfenced><mo>=</mo><mi>ψ</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi mathvariant="bold">u</mi></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">D</mi></semantics></math></inline-formula> is a certain differential, or difference, operator. The polynomials orthogonal with respect to the linear functional <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">u</mi></semantics></math></inline-formula> are called classical orthogonal polynomials. In the theory of orthogonal polynomials, a correct characterization of the classical families is of great interest. In this work, on the one hand, we present the Laguerre constellation, which is formed by all the classical families for which <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo form="prefix">deg</mo><mi>ϕ</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>, obtaining for all of them new algebraic identities such as structure formulas and orthogonality properties, as well as new Rodrigues formulas; on the other hand, we present a theorem that characterizes the classical families belonging to the Laguerre constellation. |
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| ISSN: | 2227-7390 |