Hypercomplex systems originating from orthogonal polynomials
This is a mostly expository article discussing discrete hypergroups associated to families of orthogonal polynomials of one variable. Suppose \(\{P_ n(x):\) \(n=0,1,2,...\}\) is a suitably normalized family of such polynomials associated to the positive measure \(d\sigma\) (usually supported in a bounded interval), so that \(P_ n\) is of degree n, \(n=0,1,2,... \). The hypergroup is determined by the structural constants \(\gamma\) (m,n,k), where \[ P_ m(x)P_ n(x)=\sum^{m+n}_{k=| m- n|}\gamma (m,n,k)P_ k(x),\quad x\in {\mathbb{R}}. \] The relation between the \(\gamma\) (m,n,k) and the coefficients of the three-term recurrence for \(\{P_ n(x)\}\) is investigated. Some interesting results are possible even if \(\gamma\) (m,n,k) satisfies a boundedness rather than a positivity condition. This includes the situation of \(d\sigma (t)=h(t)(1-t)^{\alpha}(1+t)^{\beta}dt\) on \(-1<t<1\), where \(h(t)>0\) and satisfies certain smoothness and boundedness conditions (note that \(h\equiv 1\) is the case of Jacobi polynomials).
scientific article; zbMATH DE number 4040528
scientific article