In this paper, we present a comprehensive account of all Laguerre-type differential operators D that are symmetric with respect to a smooth, irreducible <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$2\times 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> weight W on the interval <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$(0, \infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. These operators are associated with monic orthogonal polynomials <InlineEquation ID="IEq5"> <EquationSource Format="TEX">${P_n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>P</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>, which satisfy the equation <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$DP_n = P_n\Delta _n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>D</mi> <msub> <mi>P</mi> <mi>n</mi> </msub> <mo>=</mo> <msub> <mi>P</mi> <mi>n</mi> </msub> <msub> <mi mathvariant="normal">Δ</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> for a certain lower triangular eigenvalue <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\Delta _n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Δ</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>. We introduce three distinct families of operators and weights, each characterized by explicit expressions depending on two or three parameters, along with a new expression based on a single parameter.

$2\times 2$ Laguerre-type differential operator with triangular eigenvalue

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2025年12月29日

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Yanina Gonzalez,
A. Victoria Torres

摘要

In this paper, we present a comprehensive account of all Laguerre-type differential operators D that are symmetric with respect to a smooth, irreducible $2\times 2$ $2 \times 2$ weight W on the interval $(0, \infty )$ $(0, \infty)$ . These operators are associated with monic orthogonal polynomials ${P_n}$ $P_{n}$ , which satisfy the equation $DP_n = P_n\Delta _n$ $D P_{n} = P_{n} Δ_{n}$ for a certain lower triangular eigenvalue $\Delta _n$ $Δ_{n}$ . We introduce three distinct families of operators and weights, each characterized by explicit expressions depending on two or three parameters, along with a new expression based on a single parameter.

\(2\times 2\) Laguerre-type differential operator with triangular eigenvalue

摘要