This paper is taken up for the following difference equation problem (P e); $$\begin{gathered} (L_\varepsilon y)_k \equiv \varepsilon y(k + 1) + a(k,{\mathbf{ }}\varepsilon )y(k) + b(k,\varepsilon )y(k - 1) = f(k,\varepsilon ){\mathbf{ }}(1 \leqslant k \leqslant N - 1), \hfill \\ B_1 y \equiv - (0) + c_1 y(1) = a,{\mathbf{ }}B_2 y \equiv - c_2 y(N - 1) + y(N) = \beta \hfill \\ \end{gathered} $$ where e is a small parameter, c1, c2, α, β constants and a(k,e), b(k,e), f(k,e) (1⩽k⩽N) functions of k and e. Firstly, the case with constant coefficients is considered. Secondly, a general method based on extended transformation is given to handle (P e) where the coefficients may be variable and uniform asymptotic expansions are obtained. Finally, a numerical example is provided to illustrate the proposed method.