## Arkiv för Matematik

• Ark. Mat.
• Volume 17, Number 1-2 (1979), 139-151.

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

#### Abstract

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.

#### Note

The first author was partially supported by the Danish Natural Science Research Council.

#### Article information

Source
Ark. Mat., Volume 17, Number 1-2 (1979), 139-151.

Dates
First available in Project Euclid: 31 January 2017

https://projecteuclid.org/euclid.afm/1485896579

Digital Object Identifier
doi:10.1007/BF02385463

Mathematical Reviews number (MathSciNet)
MR543509

Zentralblatt MATH identifier
0409.33009

Rights

#### Citation

Flensted-Jensen, Mogens; Koornwinder, Tom H. Jacobi functions: The addition formula and the positivity of the dual convolution structure. Ark. Mat. 17 (1979), no. 1-2, 139--151. doi:10.1007/BF02385463. https://projecteuclid.org/euclid.afm/1485896579

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