PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 144, Number 2, February 2016, Pages 535–545 http://dx.doi.org/10.1090/proc/12689 Article electronically published on October 1, 2015 ASYMPTOTIC BEHAVIOUR OF JACOBI POLYNOMIALS AND THEIR ZEROS DIMITAR K. DIMITROV AND ELIEL J. C. DOS SANTOS (Communicated by Walter Van Assche) Abstract. We obtain the explicit form of the expansion of the Jacobi poly- nomial P (α,β) n (1 - 2x/β) in terms of the negative powers of β. It is known that the constant term in the expansion coincides with the Laguerre polyno- mial L (α) n (x). Therefore, the result in the present paper provides the higher terms of the asymptotic expansion as β →∞. The corresponding asymptotic relation between the zeros of Jacobi and Laguerre polynomials is also derived. 1. Introduction and statement of the main result Denote by P (α,β) n (x), C (λ) n (x), L (α) n (x), H n (x) the classical Jacobi, Gegenbauer (ultraspherical), Laguerre and Hermite orthogonal polynomials. We adopt the stan- dard normalizations as in [6, 7]. For a fixed n ∈ N, denote by x nk (α, β), u nk (λ), ℓ n,k (α) and h nk , k =1,...,n, the corresponding zeros of the above polynomials, all arranged in increasing order with respect to k. The following limit relations are well known (see [6, p. 57]): lim β→∞ P (α,β) n (1 − 2x β )= L (α) n (x) (1) and lim λ→∞ λ −n/2 C (λ) n ( x √ λ )= H n (x) n! . (2) These imply the following asymptotic formulae for the corresponding zeros of the classical orthogonal polynomials: lim β→∞ β(1 − x n,k (α, β)) 2 = ℓ n,n+1−k (α), k =1,...,n, (3) and lim λ→∞ √ λu nk (λ)= h nk , k =1,...,n. (4) It is clear that (2) and (4) are rather straightforward consequences of (1) and (3). On the other hand, Elbert and Laforgia [4] obtained nice extensions of (2) and (4), Received by the editors May 11, 2014 and, in revised form, October 10, 2014 and November 4, 2014. 2010 Mathematics Subject Classification. Primary 26C10, 33C45. Key words and phrases. Jacobi polynomials, Laguerre polynomials, zeros, asymptotics. The authors’ research was supported by the Brazilian foundations CNPq under Grant 307183/2013–0 and FAPESP under Grant 2009/13832–9. c 2015 American Mathematical Society 535 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use