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On a conjectured inequality of Gautschi and Leopardi for Jacobi polynomials

机译:关于Jacobi多项式的Gautschi和Leopardi的猜想不等式

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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)} ? 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 $(alpha, ,beta)in big{(frac{1}{2},frac{1}{2}),,(frac{1}{2},-frac{1}{2}),,(-frac{1}{2},frac{1}{2})big}$ .
机译:受到球形表面上积极的培养公式公式的启发,Gautschi和Leopardi推测不等式$ frac {P_ {n} ^ {(alpha,beta)}(cosfrac {theta} {n})} {P_ {n} ^ {(alpha,beta)}(1)} ? 1和n≥1,θ∈(0,π),其中$ P_ {n} ^ {(alpha,beta)}(x)$是阶数n和参数(α,β)的Jacobi多项式。我们在特殊情况下解决这个猜想,其中$ {alpha,,beta)在大{(frac {1} {2},frac {1} {2}),,(frac {1} {2},-frac { 1} {2}),((-frac {1} {2},frac {1} {2})big} $。

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