An expansion as a sum of squares of Jacobi polynomials \(P_n^{(\alpha , \beta )}(x)\) is used to prove that if \(0 \leq \lambda \leq \alpha + \beta\) and \(\beta \geq -1/2\), then \[ \sum_{k=1}^{n} \frac{(\lambda +1)_{n-k}}{(n-k)!} \frac{(\lambda +1)_k}{k!} \frac{P_k^{(\alpha ,\beta )}(x)}{P_k^{(\alpha ,\beta )}} = 0,\quad -1\leq x \leq \infty, \tag{\(*\)} \] 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 \(\alpha , \beta \geq -1/2\), then the Cesàro \((C,\alpha + \beta + 2)\) means of the Jacobi series of a nonnegative function are nonnegative. Also, it is shown that \[ \frac{d}{d\theta}\sum_{k=0}^{n} \frac{(\lambda +1)_{n-k}}{(n-k)!} \frac{(\lambda +1)_k}{k!} \frac{\sin (k+1)\theta}{(k+1)\sin (\theta /2)}<0,\quad 0 < \theta < \pi, \; \; 0 \leq \lambda \leq 1, \] which extends a recent result of Askey and Steinig.