Zeros of Jacobi polynomials $P_{n}^{(\alpha ,\beta)}$, − 2 < α, β < − 1

Abstract

The sequence of Jacobi polynomials $\{P_{n}^{(\alpha ,\beta )}\}_{n = 0}^{\infty }$ is orthogonal on (− 1,1) with respect to the weight function (1 − x)^α(1 + x)^β provided α > − 1,β > − 1. When the parameters α and β lie in the narrow range − 2 < α, β < − 1, the sequence of Jacobi polynomials $\{P_{n}^{(\alpha ,\beta )}\}_{n = 0}^{\infty }$ is quasi-orthogonal of order 2 with respect to the weight function (1 − x)^α+ 1(1 + x)^β+ 1 and each polynomial of degree n,n ≥ 2, in such a Jacobi sequence has n real zeros. We show that any sequence of Jacobi polynomials $\{P_{n}^{(\alpha ,\beta )}\}_{n = 0}^{\infty }$ with − 2 < α, β < − 1, cannot be orthogonal with respect to any positive measure by proving that the zeros of Pn− 1(α,β) do not interlace with the zeros of Pn(α,β) for any $n \in \mathbb {N},$n ≥ 2, and any α,β lying in the range − 2 < α, β < − 1. We also investigate interlacing properties satisfied by the zeros of equal degree Jacobi polynomials Pn(α,β),Pn(α,β+ 1) and Pn(α+ 1,β+ 1) where − 2 < α, β < − 1. Upper and lower bounds for the extreme zeros of quasi-orthogonal order 2 Jacobi polynomials Pn(α,β) with − 2 < α, β < − 1 are derived.