Iterated integrals of Jacobi polynomials
The \(m\)th Jacobi polynomial with parameters \(\alpha,\beta\in\mathbb{R}\) is defined as \[P_{n}^{(\alpha, \beta)}(z)=\sum_{k=0}^{n}\binom{n+\alpha}{n-k}\binom{n+\beta}{k}\binom{2n+\alpha+\beta}{n}^{-1}(z-1)^{k}(z+1)^{n-k}.\] For a fixed \(m \in \mathbb{Z}_{+},\) let \(\mathcal{P}_{n, m}^{(\alpha, \beta)}\) be the monic polynomial of degree \(n+m\) given by \[\mathscr{P}_{n, m}^{(\alpha, \beta)}=P_{n+m}^{(\alpha-m, \beta-m)}.\] This polynomial is called the \(m\)th fundamental iterated integral of \(\frac{(n+m)!}{n!}P_n^{(\alpha, \beta)}\). There is renewed interest in the location of zeros of these polynomials, because this knowledge can be applied to the numerical solution of differential equations. Motivated by these applications, the authors give several theorems about the location if zeros of the \(\mathscr{P}_{n, m}^{(\alpha, \beta)}\) polynomials. Among many other nice results, they prove that the union of the zero sets of all these polynomials can be limited by a compact set of the complex plane.
scientific article; zbMATH DE number 7192870
scientific article