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Positivity of more Jacobi polynomial sums. (English) Zbl 0847.33006

Positivity of integrals of Bessel functions and sums of Jacobi polynomials has been a surprisingly fruitful topic to consider. One problem is to find the values of \((\alpha, \beta)\) for which \(\sum^n_{k=0} P_k^{(\alpha, \beta)} (x)/ P_k^{(\beta, \alpha)} (1)\geq 0\), \(-1\leq x\leq 1\). For \(\alpha\geq 1/2\) this problem has been completely solved by Gasper, who showed that this inequality holds for \(\beta\geq -1/2\). For \(-1< \alpha< 1/2\) good but not best possible results are known. Earlier, the authors of this paper completely solved this problem for \(\alpha= \beta\). Here they solve this problem for \(\alpha=- 1/2\). In both cases the inequalities hold if a corresponding inequality holds for an integral of a Bessel function, and this holds when the area under the first arch is at least as great as the area above the second arch.
The Jacobi polynomial problem is translated into an alternating sum of even ultraspherical polynomials by a quadratic transformation. The Dirichlet-Mehler integral and sharp inequalities gives a proof for polynomials of degre at least 24. Some numerical work was needed. For polynomials of degree at most 22, an argument using Sturm sequences is outlined. These results are sharp in a number of different ways, which explains why the proof is so complicated.
Reviewer: R.Askey (Madison)

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)

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References:

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