Iterated Integrals of Jacobi Polynomials

Abstract

Let $P_n^{(\alpha ,\beta )}$ be the n-th monic Jacobi polynomial with $\alpha ,\beta >-1$. Given m numbers $\omega _1, \ldots , \omega _m \in \mathbb {C} \setminus [-1,1]$, let $\varOmega _m=( \omega _1, \ldots , \omega _m)$ and $\mathscr {P}^{(\alpha ,\beta )}_{n,m,\varOmega _m}$ be the m-th iterated integral of $\frac{(n+m)!}{n!}\;P^{(\alpha ,\beta )}_{n}$ normalized by the conditions

$$\begin{aligned} \frac{\mathrm{d}^k\,\mathscr {P}^{(\alpha ,\beta )}_{n,m,\varOmega _m}}{\mathrm{d}z^k}(\omega _{m-k})=0, \; \text { for } \; k=0,1,\ldots , m-1. \end{aligned}$$

The main purpose of the paper is to study the algebraic and asymptotic properties of the sequence of monic polynomials $\{ \mathscr {P}_{n,m,\varOmega _m}^{(\alpha , \beta )}\}_n$. In particular, we obtain the relative asymptotic for the ratio of the sequences $\{ \mathscr {P}^{(\alpha ,\beta )}_{n,m,\varOmega _m}\}_n$ and $\{P_n^{(\alpha ,\beta )}\}_n$. We prove that the zeros of these polynomials accumulate on a suitable ellipse.