Asymptotic expansion of the modified lommel polynomials hn,ν(x) and their zeroszeros

The modified Lommel polynomials satisfy the second-order linear difference equation h(n +1),(v)(x) - 2(n + v) x h(n,v)(x)+ h(n-1,v)(x) = 0, n = 0, with initial values h(-1,v)(x) = 0 and h(0,v)(x) = 1, where x is a real variable and v is a fixed positive parameter. An asymptotic expansion, as n - infinity, is derived for these polynomials by using a turning-point theory for three-term recurrence relations developed by Wang and Wong Numer. Math. 91 (2002) and 94 (2003)]. The result holds uniformly in the infinite interval 0 = x infinity, containing the critical value x = 1/N, where N = n + v. Behavior of the zeros of these polynomials is also studied.