An inequality for Jacobi polynomials of form P_n^(α_n,β_n)(x)

2017/04/21 by He, Zhulin, Ouyang, Yuyuan
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1704.06381

Abstract

We prove an inequality for Jacobi polynomials that Δ_n(x):=P_n^(α_n,β_n)(x)P_n^(α_n+1,β_n+1)(x)- P_n-1^(α_n,β_n)(x)P_n+1^(α_n+1,β_n+1)(x)≤ 0, ∀ x≥ 1, where α_n=an and β_n=bn for some a,b≥ 0. The above inequality has a similar taste as the Tuŕan type inequalities, but with α_n and β_n that depends linearly on n.

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