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# Positive Jacobi Polynomial Sums, II

American Journal of Mathematics
Vol. 98, No. 3 (Autumn, 1976), pp. 709-737
DOI: 10.2307/2373813
Stable URL: http://www.jstor.org/stable/2373813
Page Count: 29
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## Abstract

Among the positive polynomial sums of Jacobi polynomials, there are two which have been very useful, the Cesàro means of the formal reproducing kernel and the sum (*) ∑n k = 0 Pk (α,β)(x)/Pk (β,α)(1), which was first considered by Fejér when α = 1/2, β = ± 1/2 and when α = β = 0. A conjecture is given which connects these two sets of inequalities and this conjecture is proven for many values of (α, β). In particular, it is shown that if β ⩾ 0, then the sum (*) is nonnegative for -1 ⩽ x ⩽ 1 if and only if α + β ⩾ -2. It is also shown that $\sum_{k=0}^n \frac{(\lambda+1)_{n-k}{(n-k)!}\frac{(\lambda+1}_k}{k!} \frac{\sin(k+1)^\theta}{k+1} > 0, 0 < \theta < \pi, -1 < \lambda \leqslant 1$, and that for real α the function (1 - r)-|α|[ 1 ± r + (1 - 2xr + r2)1/2 ]α is absolutely monotonic for -1 ⩽ x ⩽, i.e., it has nonnegative power series coefficients when it is expanded in a power series in r. Limiting cases involving Laguerre and Hermite polynomials are also considered.

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