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<p>General Geronimus transformations, defined by regular matrix polynomials that are neither required to be monic nor restricted by the rank of their leading coefficients, are applied through both right and left multiplication to a rectangular matrix of measures associated with mixed multiple orthogonal polynomials. These transformations produce Christoffel-type formulas that establish relationships between the perturbed and original polynomials. Moreover, it is proven that the existence of Geronimus-perturbed orthogonality is equivalent to the non-cancellation of certain <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1036_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">\(\tau \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>τ</mi> </math></EquationSource> </InlineEquation>-determinants. The effect of these transformations on the Markov–Stieltjes matrix functions is also determined. As a case study, we examine the Jacobi–Piñeiro orthogonal polynomials with three weights.</p>

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General geronimus perturbations for mixed multiple orthogonal polynomials

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年4月5日

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  • Manuel Mañas,
  • Miguel Rojas

摘要

General Geronimus transformations, defined by regular matrix polynomials that are neither required to be monic nor restricted by the rank of their leading coefficients, are applied through both right and left multiplication to a rectangular matrix of measures associated with mixed multiple orthogonal polynomials. These transformations produce Christoffel-type formulas that establish relationships between the perturbed and original polynomials. Moreover, it is proven that the existence of Geronimus-perturbed orthogonality is equivalent to the non-cancellation of certain \(\tau \) τ -determinants. The effect of these transformations on the Markov–Stieltjes matrix functions is also determined. As a case study, we examine the Jacobi–Piñeiro orthogonal polynomials with three weights.