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Two new conjectures concerning positive Jacobi polynomials sums. (English) Zbl 0967.33004

Summary: A refinement of a conjecture of Gasper concerning the values of \((\alpha,\beta), -1/2< \beta<0\), \(-1<\alpha +\beta<0\), for which the inequalities \[ \sum^n_{k=0} P_k^{(\alpha, \beta)}(x)/P_k^{(\beta, \alpha)}(1) \geq 0,\quad -1\leq x\leq 1,\;n=1,2,\dots \] hold, is stated. An algorithm for checking the new conjecture using the package Mathematica is provided. Numerical results in support of the conjecture are given and a possible approach to its proof is sketched.

MSC:

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