Uniform asymptotic expansions of the Tricomi-Carlitz polynomials

The Tricomi-Carlitz polynomials satisfy the second-order linear difference equation

$$(n+1)f_{n+1}^{(\alpha)}(x)-(n+\alpha) x f_n^{(\alpha)}(x)+ f_{n-1}^{(\alpha)}(x)=0, \qquad\qquad n\geq1,$$

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 $x=\pm2/\sqrt{\nu}$, where $\nu=n+2\alpha-1/2$.