Dimitrov, Dimitar K.; Merlo, Clinton A. Two new conjectures concerning positive Jacobi polynomials sums. (English) Zbl 0967.33004 Rev. Colomb. Mat. 32, No. 2, 133-142 (1998). 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.) Keywords:Jacobi polynomials; positive sums; Bessel functions; discriminant of a polynomial PDFBibTeX XMLCite \textit{D. K. Dimitrov} and \textit{C. A. Merlo}, Rev. Colomb. Mat. 32, No. 2, 133--142 (1998; Zbl 0967.33004) Full Text: EuDML