On the Zeros of Certain Jacobi Polynomial Sums

• Published in: SIAM journal on mathematical analysis 1977-04, Vol.8 (2), p.202-205
• Main Author: Bustoz, J.
• Format: Article
• Language: English
• Subjects: Inequality; Polynomials
• ISSN: 0036-1410; 1095-7154
• DOI: 10.1137/0508014

Let $P_k^{(\alpha ,\beta )} (x)$ be the Jacobi polynomial of degree $k$ with parameters $\alpha $, $\beta $. It is known that \[\sum\limits_{k = 0}^n {\frac{{P_k^{(\beta ,\beta )} (\cos \theta )}} {{P_k^{(\beta ,\beta )} (1)}}} \geqq 0\quad {\text{for }}\beta \geqq 0.\] It has been conjectured that \[\sum_{k = 0}^n {\frac{{P_k^{(\beta ,\beta )} (\cos \theta )}}{{P_k^{(\beta ,\beta )} (1)}}} z^k \ne 0\quad {\text{if }}| z | < 1,\quad \beta > 0.\] This conjecture has been verified for $\beta = 0$ and $\beta = \frac{1}{2}$. Here we prove the conjecture for $\beta = 1$ and $\beta = 2$ and give a more general inequality valid for $\beta = 0,\frac{1}{2},1,2$.