<p>The sequences <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\{\delta _n^{(k)}\}_{n\in \mathbb {N}}, k=0,1,\ldots n,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mrow> <mo stretchy="false">{</mo> <msubsup> <mi>δ</mi> <mi>n</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </msub> <mo>,</mo> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mi>n</mi> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> associated to a Bochner differential operator are introduced as an effective tool to study this kind of operators. Some properties of this sequence are proven and used to deduce that a particular operator leads to solutions of a bispectral problem. In addition, the inverse problem is studied; that is, given a sequence <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\{\lambda _n\}_{n\in \mathbb {N}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">{</mo> <msub> <mi>λ</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> of complex numbers and a sequence <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\{P_n\}_{n\in \mathbb {N}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">{</mo> <msub> <mi>P</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> of polynomials with complex coefficients, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\deg {P_n}=n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>deg</mo> <msub> <mi>P</mi> <mi>n</mi> </msub> <mo>=</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation>, we find a necessary and sufficient condition for the existence of a Bochner differential operator that has those sequences as eigenvalues and eigenpolynomials, respectively. The mentioned condition also depends on <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\{\delta _n^{(k)}\}_{n\in \mathbb {N}}, k=0,1,\ldots n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mrow> <mo stretchy="false">{</mo> <msubsup> <mi>δ</mi> <mi>n</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </msub> <mo>,</mo> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation>.</p>

New tools for the study of Bochner differential operators

期刊论文
发表日期：

2026年2月9日

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L. M. Anguas,
D. Barrios Rolanía

摘要

The sequences $\{\delta _n^{(k)}\}_{n\in \mathbb {N}}, k=0,1,\ldots n,$ ${δ_{n}^{(k)}}_{n \in N}, k = 0, 1, \dots n,$ associated to a Bochner differential operator are introduced as an effective tool to study this kind of operators. Some properties of this sequence are proven and used to deduce that a particular operator leads to solutions of a bispectral problem. In addition, the inverse problem is studied; that is, given a sequence $\{\lambda _n\}_{n\in \mathbb {N}}$ ${λ_{n}}_{n \in N}$ of complex numbers and a sequence $\{P_n\}_{n\in \mathbb {N}}$ ${P_{n}}_{n \in N}$ of polynomials with complex coefficients, $\deg {P_n}=n$ $deg P_{n} = n$ , we find a necessary and sufficient condition for the existence of a Bochner differential operator that has those sequences as eigenvalues and eigenpolynomials, respectively. The mentioned condition also depends on $\{\delta _n^{(k)}\}_{n\in \mathbb {N}}, k=0,1,\ldots n$ ${δ_{n}^{(k)}}_{n \in N}, k = 0, 1, \dots n$ .