Asymptotics of the Wilson polynomials

In this paper, we study the asymptotic behavior of the Wilson polynomials Wn (x; a,b,c,d) as their degree tends to infinity. These polynomials lie on the top level of the Askey scheme of hypergeometric orthogonal polynomials. Infinite asymptotic expansions are derived for these polynomials in various cases, for instance, (i) when the variable x is fixed and (ii) when the variable is rescaled as x = n²t  with t ≥ 0. Case (ii) has two subcases, namely, (a) zero-free zone (t > 1) and (b) oscillatory region (0 < t < 1). Corresponding results are also obtained in these cases (iii) when t lies in a neighborhood of the transition point t = 1, and (iv) when t is in the neighborhood of the transition point t = 0. The expansions in the last two cases hold uniformly in t. Case (iv) is also the only unsettled case in a sequence of works on the asymptotic analysis of linear difference equations.