On a conjectured inequality of Gautschi and Leopardi for Jacobi polynomials期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

On a conjectured inequality of Gautschi and Leopardi for Jacobi polynomials

Authors:	Stamatis Koumandos

Affiliation:	(1) Department of Mathematics and Statistics, University of Cyprus, P. O. Box 20537, 1678 Nicosia, Cyprus

Abstract:	Motivated by work on positive cubature formulae over the spherical surface, Gautschi and Leopardi conjectured that the inequality $frac{P_{n}^{(alpha,beta)}(cosfrac{theta}{n})}{P_{n}^{(alpha,beta)}(1)}<frac{P_{n+1}^{(alpha,beta)}(cosfrac{theta}{n+1})}{P_{n+1}^{(alpha,beta)}(1)}$ holds for α,β > − 1 and n ≥ 1, θ ∈ (0, π), where $P_{n}^{(alpha,beta)}(x)$ are the Jacobi polynomials of degree n and parameters (α, β). We settle this conjecture in the special cases where $(alpha, ,beta)in big{(frac{1}{2},frac{1}{2}),,(frac{1}{2},-frac{1}{2}),,(-frac{1}{2},frac{1}{2})big}$ .

Keywords:	Jacobi polynomials Inequalities Trigonometric functions
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