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Shifted Darboux Transformations of the Generalized Jacobi Matrices, I

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Abstract

Let ℑ be a monic generalized Jacobi matrix, i.e., a three-diagonal block matrix of a special form. We find conditions for a monic generalized Jacobi matrix ℑ to admit a factorization ℑ = 𝔏𝔘 + αI with 𝔏 and 𝔘 being lower and upper triangular two-diagonal block matrices of special forms. In this case, the shifted parameterless Darboux transformation of ℑ defined by ℑ(p) = 𝔘𝔏 + αI is shown to be also a monic generalized Jacobi matrix. Analogs of the Christoffel formulas for polynomials of the first and second kinds corresponding to the Darboux transformation ℑ(p) are found.

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Authors and Affiliations

  1. Dragomanov National Pedagogical University, Kiev, Ukraine

    Ivan M. Kovalyov

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  1. Ivan M. Kovalyov
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Correspondence to Ivan M. Kovalyov.

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Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 15, No. 4, pp. 490–515 October–December, 2018.

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Kovalyov, I.M. Shifted Darboux Transformations of the Generalized Jacobi Matrices, I. J Math Sci 242, 393–412 (2019). https://doi.org/10.1007/s10958-019-04485-6

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  • DOI: https://doi.org/10.1007/s10958-019-04485-6

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