On the asymptotic behavior of Jacobi polynomials with first varying parameter

Abstract

We investigate the large n behavior of Jacobi polynomials with varying parameters Pn(an+α,bn+β)(1−2λ2) for a,b>−1 and λ∈(0,1). This is a well-studied topic in the literature but some of the published results appear to be discordant. To address this issue we provide an in-depth investigation of the case b=0, which is most relevant for our applications. Our approach is based on a new and surprisingly simple representation of Pn(an+α,β)(1−2λ2),a>−1 in terms of two integrals. The integrals’ asymptotic behavior is studied using standard tools of asymptotic analysis: one is a Laplace integral and the other is treated via the method of stationary phase. As a consequence we prove that if a∈(2λ1−λ,∞) then λanPn(an+α,β)(1−2λ2) shows exponential decay and we derive simple exponential upper bounds in this region. If a∈(−2λ1+λ,2λ1−λ) then the decay of λanPn(an+α,β)(1−2λ2) is O(n−1/2) and if a∈{−2λ1+λ,2λ1−λ} then λanPn(an+α,β)(1−2λ2) decays as O(n−1/3). A new phenomenon occurs in the parameter range a∈(−1,−2λ1+λ), where we find that the behavior depends on whether or not an+α is an integer: If a∈(−1,−2λ1+λ) and an+α is an integer then λanPn(an+α,β)(1−2λ2) decays exponentially. If a∈(−1,−2λ1+λ) and an+α is not an integer then λanPn(an+α,β)(1−2λ2) may increase exponentially depending on the proximity of the sequence (an+α)n to integers.