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The Mehler Formula for the Generalized Clifford–Hermite Polynomials

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Abstract

The Mehler formula for the Hermite polynomials allows for an integral representation of the one–dimensional Fractional Fourier transform. In this paper, we introduce a multi–dimensional Fractional Fourier transform in the framework of Clifford analysis. By showing that it coincides with the classical tensorial approach we are able to prove Mehler’s formula for the generalized Clifford–Hermite polynomials of Clifford analysis.

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Authors and Affiliations

  1. Clifford Research Group, Ghent University, Department of Mathematical Analysis, Faculty of Engineering, Galglaan 2, B-9000, Gent, Belgium

    F. Brackx & N. de Schepper

  2. University of Macau, Department of Mathematics, Faculty of Science and Technology, 3001, Macau

    K. I. Kou

  3. Clifford Research Group, Ghent University, Department of Mathematical Analysis, Faculty of Sciences, Galglaan 2, B-9000, Gent, Belgium

    F. Sommen

Authors
  1. F. Brackx
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  2. N. de Schepper
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  3. K. I. Kou
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  4. F. Sommen
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Corresponding author

Correspondence to N. de Schepper.

Additional information

The work is supported by Research Grant of the University of Macau No. RG021/03–045/QT/FST

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Brackx, F., de Schepper, N., Kou, K.I. et al. The Mehler Formula for the Generalized Clifford–Hermite Polynomials. Acta Math Sinica 23, 697–704 (2007). https://doi.org/10.1007/s10114-005-0754-7

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  • DOI: https://doi.org/10.1007/s10114-005-0754-7

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