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Jacobi functions: The addition formula and the positivity of the dual convolution structure

Mogens Flensted-Jensen Matematisk Institut, Universitetsparken 5, Copenhagen, Denmark Tom H. Koornwinder Mathematisch Centrum, 2e Boerhaavestraat 49, Amsterdam, The Netherlands

TBD mathscidoc:1701.332502

Arkiv for Matematik, 17, (1), 139-151, 1978.7
We prove an addition formula for Jacobi functions $$\varphi _\lambda ^{\left( {\alpha ,\beta } \right)} \left( {\alpha \geqq \beta \geqq - \tfrac{1}{2}} \right)$$ analogous to the known addition formula for Jacobi polynomials. We exploit the positivity of the coefficients in the addition formula by giving the following application. We prove that the product of two Jacobi functions of the same argument has a nonnegative Fourier-Jacobi transform. This implies that the convolution structure associated to the inverse Fourier-Jacobi transform is positive.
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@inproceedings{mogens1978jacobi,
  title={Jacobi functions: The addition formula and the positivity of the dual convolution structure},
  author={Mogens Flensted-Jensen, and Tom H. Koornwinder},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203505600923311},
  booktitle={Arkiv for Matematik},
  volume={17},
  number={1},
  pages={139-151},
  year={1978},
}
Mogens Flensted-Jensen, and Tom H. Koornwinder. Jacobi functions: The addition formula and the positivity of the dual convolution structure. 1978. Vol. 17. In Arkiv for Matematik. pp.139-151. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203505600923311.
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