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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2025-01-01
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| author | Roberto S. Costas-Santos |
| author_facet | Roberto S. Costas-Santos |
| author_sort | Roberto S. Costas-Santos |
| collection | DOAJ |
| description | 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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| institution | Kabale University |
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| language | English |
| publishDate | 2025-01-01 |
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| spelling | doaj-art-e0f7c32f2ce54a5aa2fefd79e1f2311b2025-01-24T13:40:00ZengMDPI AGMathematics2227-73902025-01-0113227710.3390/math13020277The Laguerre Constellation of Classical Orthogonal PolynomialsRoberto S. Costas-Santos0Department of Quantitative Methods, Universidad Loyola Andalucía, 41704 Sevilla, SpainA 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.https://www.mdpi.com/2227-7390/13/2/277recurrence relationcharacterization theoremclassical orthogonal polynomialsLaguerre constellation |
| spellingShingle | Roberto S. Costas-Santos The Laguerre Constellation of Classical Orthogonal Polynomials Mathematics recurrence relation characterization theorem classical orthogonal polynomials Laguerre constellation |
| title | The Laguerre Constellation of Classical Orthogonal Polynomials |
| title_full | The Laguerre Constellation of Classical Orthogonal Polynomials |
| title_fullStr | The Laguerre Constellation of Classical Orthogonal Polynomials |
| title_full_unstemmed | The Laguerre Constellation of Classical Orthogonal Polynomials |
| title_short | The Laguerre Constellation of Classical Orthogonal Polynomials |
| title_sort | laguerre constellation of classical orthogonal polynomials |
| topic | recurrence relation characterization theorem classical orthogonal polynomials Laguerre constellation |
| url | https://www.mdpi.com/2227-7390/13/2/277 |
| work_keys_str_mv | AT robertoscostassantos thelaguerreconstellationofclassicalorthogonalpolynomials AT robertoscostassantos laguerreconstellationofclassicalorthogonalpolynomials |