Askey–Gasper inequality

Remarks

The denominators are positive: $P_k^{(\beta, \alpha)}(1) = \frac{(\beta + 1)_k}{k!} > 0$ for $\beta \ge 0$. For $\beta = 0$ all denominators equal $1$, and the inequality says $\sum_{k=0}^{n} P_k^{(\alpha, 0)}(x) \ge 0$ on $[-1, 1]$. Check for $n = 1$ and $\alpha = \beta = 0$: the sum is $1 + x$, which is nonnegative on $[-1, 1]$ and vanishes at $x = -1$. Askey and Gasper proved it in 1976. A short proof by Ekhad (1993) treats the case $\beta = 0$ in its ${}_3F_2$ form. An identity found with the Wilf–Zeilberger method writes this ${}_3F_2$ as a positive combination of other ${}_3F_2$ series. These are nonnegative by Clausen's formula, which expresses them as squares of ${}_2F_1$ series.

Going deeper

For a student

Sums of the form $\sum_k c_k P_k^{(\alpha, \beta)}(x)$ are partial sums of Jacobi expansions, and their positivity is a question of positive summability kernels. The case $\beta = 0$ can be rewritten as the positivity of a terminating ${}_3F_2$ hypergeometric series on $0 \le t < 1$. De Branges needed this ${}_3F_2$ form for integer values of $\alpha$ to finish his proof.