Trigonometric convolution structures on \(\mathbb{Z}\) derived from Jacobi polynomials (Q1917952)

In the present paper, the author studies trigonometric polynomials \(\psi^{(\alpha,\beta)}_n\), which are orthogonal on the unit circle with respect to \(d\mu^{(\alpha,\beta)}(e^{it})=(1-\cos t)^\alpha\) \((1+\cos\beta)^\beta\sin t dt\), and have as real part the Jacobi polynomials \(R^{(\alpha,\beta)}_n\). By a result of Gasper the coefficients in the linearization of the products \(R^{(\alpha,\beta)}_n R^{(\alpha,\beta)}_m\) are nonnegative provided the parameters \(\alpha\), \(\beta\) satisfy certain conditions. As a consequence of the result of Gasper, the system \(R^{(\alpha,\beta)}_n\), \(n\in\mathbb{N}_0\), induces a hypergroup structure on \(\mathbb{N}_0\). In this article, the author shows a similar result for the trigonometric polynomials \(\psi^{(\alpha,\beta)}_n\), \(n\in\mathbb{Z}\). In fact \[ \psi^{(\alpha,\beta)}_n(z)\psi^{(\alpha,\beta)}_m(z)= \sum_{k\in I(n,m)}h^{(\alpha,\beta)}(n,m,k) \psi^{(\alpha,\beta)}_k(z), \] where \(I(n,m)=\{-|n|-|m|,\dots,-||n|-|m||\}\cup \{||n|-|m||,\dots,|n|+|m|\}\) and where -- which is the most important -- there exists a constant \(C=C_{\alpha,\beta}\) such that for all \(n,m\in \mathbb{Z}\), \[ \sum_{k\in I(n,m)} |h^{(\alpha,\beta)}(n,m,k)|\leq C. \] This boundedness condition guarantees a signed hypergroup structure on \(\mathbb{Z}\), which leads to a convolution algebra of measures on \(\mathbb{Z}\).