Uniform asymptotic expansions of the Tricomi-Carlitz polynomials

Abstract:

The Tricomi-Carlitz polynomials satisfy the second-order linear difference equation \begin{equation*} (n+1)f_{n+1}^{(\alpha )}(x)-(n+\alpha ) x f_n^{(\alpha )}(x)+ f_{n-1}^{(\alpha )}(x)=0, \qquad \qquad n\geq 1, \end{equation*} with initial values $f_0^{(\alpha )}(x)=1$ and $f_1^{(\alpha )}(x)=\alpha x$, where $x$ is a real variable and $\alpha$ is a positive parameter. An asymptotic expansion is derived for these polynomials by using the 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 regions containing the critical values $x=\pm 2/\sqrt {\nu }$, where $\nu =n+2\alpha -1/2$.