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Heine Representations and Monotonicity Properties of Determinants and Pfaffians

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Abstract

We establish integral representations of Heine type for certain integrals of determinants. We use these representations to derive monotonicity properties of such determinants. New integral representations of Heine type for biorthogonal functions obtained from the general Gram–Schmidt orthonormalization process are given. We also establish the monotonicity of certain Pfaffians using the de Bruijn formula of Ishikawa and Zeng. The complete or absolute monotonicity of a number of Pfaffian functions is established.

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Acknowledgments

The authors would like to thank Jiang Zeng for his lecture on Hankel Pfaffians, which initiated this work, and for helpful discussions. We are also thankful to City University of Hong Kong where most of this work was done for its hospitality and support.

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Authors and Affiliations

  1. Department of Mathematics, University of Central Florida, Orlando, FL, 32816, USA

    Mourad E. H. Ismail

  2. Department of Mathematics, King Saud University, Riyadh, Saudi Arabia

    Mourad E. H. Ismail

  3. Department of Mathematics and Statistics, University of Houston-Downtown, Houston, TX, 77002, USA

    Plamen Simeonov

Authors
  1. Mourad E. H. Ismail
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  2. Plamen Simeonov
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Correspondence to Plamen Simeonov.

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Communicated by Edward B. Saff.

Research supported by King Saud University, Riyadh through Grant DSFP/MATH 01, by the DSFP program at King Saud University, and by he Research Grants Council of Hong Kong under contract CityU 1014111.

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Ismail, M.E.H., Simeonov, P. Heine Representations and Monotonicity Properties of Determinants and Pfaffians. Constr Approx 41, 231–249 (2015). https://doi.org/10.1007/s00365-014-9262-2

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  • DOI: https://doi.org/10.1007/s00365-014-9262-2

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