Asymptotic expansion of the modified Lommel polynomials $h_{n,\nu }(x)$ and their zeros

Abstract:

The modified Lommel polynomials satisfy the second-order linear difference equation \begin{equation*} h_{n+1,\nu }(x)-2(n+\nu ) x h_{n,\nu }(x)+ h_{n-1,\nu }(x)=0, \qquad n\geq 0, \end{equation*} with initial values $h_{-1,\nu }(x)=0$ and $h_{0,\nu }(x)=1$, where $x$ is a real variable and $\nu$ is a fixed positive parameter. An asymptotic expansion, as $n \to \infty$, 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 $0\leq x<\infty$, containing the critical value $x=1/{N}$, where $N=n+\nu$. Behavior of the zeros of these polynomials is also studied.