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Orthogonal polynomials relative to a generalized Marchenko–Pastur probability measure

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Abstract

The Marchenko–Pastur probability measure, of interest in the asymptotic theory of random matrices, is generalized in what appears to be a natural way. The orthogonal polynomials and their three-term recurrence relation for this generalized Marchenko–Pastur measure are obtained in explicit form, analytically as well as symbolically using Mathematica. Special cases involve Chebyshev polynomials of all four kinds. Supporting Matlab software is provided.

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Notes

  1. In the formula for \({\beta _{0}^{J}}\) at the bottom of Table 1.1 of [5], the denominator should read Γ(α + β + 2) instead of Γ(α + β + 1).

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Acknowledgements

The authors are indebted to Walter Van Assche for helpful remarks and for the references [3, 12, 14].

Funding

The work of the second author was supported in part by the Serbian Academy of Sciences and Arts (Φ-96).

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Authors and Affiliations

  1. Department of Computer Science, Purdue University, West Lafayette, IN, 47907-2107, USA

    Walter Gautschi

  2. Serbian Academy of Sciences and Arts, Kneza Mihaila 35, Belgrade, 11000, Serbia

    Gradimir V. Milovanović

  3. Faculty of Sciences and Mathematics, University of Niš, Niš, 18000, Serbia

    Gradimir V. Milovanović

Authors
  1. Walter Gautschi
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  2. Gradimir V. Milovanović
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Correspondence to Walter Gautschi.

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Gautschi, W., Milovanović, G.V. Orthogonal polynomials relative to a generalized Marchenko–Pastur probability measure. Numer Algor 88, 1233–1249 (2021). https://doi.org/10.1007/s11075-021-01073-1

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  • DOI: https://doi.org/10.1007/s11075-021-01073-1

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