Positive Sums of the Classical Orthogonal Polynomials

An expansion as a sum of squares of Jacobi polynomials $P_n^{(\alpha ,\beta )} (x)$ is used to prove that if $0 \leqq \lambda \leqq \alpha + \beta $ and $\beta \geqq - {1 / 2}$, then \[( * )\qquad \sum_{k = 0}^n {\frac{{(\lambda + 1)_{n - k} }}{{(n - k)!}}} \frac{{(\lambda + 1)_k }}{{k!}}\frac{{P_k^{(\alpha ,\beta )} }}(x){{P_k^{(\beta ,\alpha )}(i) }} \geqq 0,\quad - 1 \leqq x < \infty ,\] and the only cases of equality occur when $x = - 1$ for n odd and when $\lambda = 0$, $\alpha = - \beta = {1 / 2}$. Additional conditions are given under which $( * )$ holds and some special uses, limit cases, and important applications are pointed out. In particular, the case $\lambda = \alpha + \beta $ of $( * )$ is used to prove that if $\alpha $, $\beta \geqq - {1 / 2}$ then the Cesaro $(C,\alpha + \beta + 2)$ means of the Jacobi series of a nonnegative function are nonnegative. Also, it is shown that \[\frac{d}{{d\theta }}\sum\limits_{k = 0}^n {\frac{{(\lambda + 1)_{n - k} }}{{(n - k)!}}} \frac{{(\lambda + 1)_k }}...