Zeros of Jacobi Polynomials With Varying Non-Classical Parameters

Asymptotic distributions of the zeros of a family of hypergeometric polynomials

“…It is interesting to note in conclusion that one of our results (Theorem 4 above) proves a special case of the conjecture made by Martínez-Finkelshtein et al [19]. Moreover, in [16,17,19,20], the asymptotic distribution of the zeros of the Jacobi polynomials P (α n ,β n ) n (z) was investigated when Finally, by applying (3.1) and Corollary 2, we can deduce Corollary 3.…”

Uniform asymptotics for Jacobi polynomials with varying large negative parameters— a Riemann-Hilbert approach

“…Both of them have been established recently for the polynomials under consideration; see [9] and [11]. However, for convenience of the reader, we will include in this paper brief sketches of the arguments used.…”

“…Furthermore, The results in Proposition 2 are essentially given in [11,Theorem 1]. However, since not many details of proof are provided there, and since some of the formulas to be used in the following "sketch of proof" will become useful later in our discussion, we decide to include an outline of the argument.…”

“…When α, β < −1, these polynomials are of course no longer orthogonal with respect to the weight function w(z) on the interval (−1, 1). Interest seems to be growing in recent years to study the zeros of the polynomials in (1.1), when α and β are negative and tend to −∞ as n → ∞; see [6], [11] and [14]. In fact, Temme [14, p. 461] has expressed that new research is needed in this area of asymptotics.…”

“…In fact, Temme [14, p. 461] has expressed that new research is needed in this area of asymptotics. Motivated by the work in [9] and [11], let us restrict our attention to the case (1.2) α = α n = −An + a and β = β n = −Bn + b,…”