Uniform asymptotic expansions of the Tricomi-Carlitz polynomials

The Tricomi-Carlitz polynomials satisfy the second-order linear difference equation ( n + 1 ) f n + 1 ( α ) ( x ) − ( n + α ) x f n ( α ) ( x ) + f n − 1 ( α ) ( x ) = 0 , n ≥ 1 , \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 ( α ) ( x ) = 1 f_0^{(\alpha )}(x)=1 and f 1 ( α ) ( x ) = α x f_1^{(\alpha )}(x)=\alpha x , where x x is a real variable and α \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 = ± 2 / ν x=\pm 2/\sqrt {\nu } , where ν = n + 2 α − 1 / 2 \nu =n+2\alpha -1/2 .