On the zeros of Jacobi polynomials $P_{n}^{(\alpha _{n},\beta _{n})}(x)$

Abstract:

If ${r_n}$ and ${s_n}$ denote, respectively, the smallest and largest zeros of the Jacobi polynomial $P_n^{({\alpha _n},{\beta _n})}$, where ${\alpha _n} > 1$, ${\beta _n} - 1$, and if ${\lim _{n \to \infty }} {\alpha _n}/(2n + {\alpha _n} + {\beta _n} + 1) = a$ and if ${\lim _{n \to \infty }}{\beta _n}/(2n + {\alpha _n} + {\beta _n} + 1) = b$, then the numbers ${r_{a,b}}$ and ${s_{a,b}}$ are determined where \[ \lim \limits _{n \to \infty } {r_{n }} = {r_{a,b}},\lim \limits _{n \to \infty } {s_{n }} = {s_{a,b}}\] . Furthermore, the zeros of $\{ P_n^{({\alpha _n},{\beta _n})}(x)\} _{n = 0}^\infty$ are dense in $[{r_{a,b}},{s_{a,b}}]$.