Continuation of Holomorphic Solutions of Linear Di#erence Equations with Polynomial Coe#cients Summary: Continuation of Holomorphic Solutions of Linear Di#erence Equations with Polynomial Coe#cients S. A. Abramov and M. A. Barkatou Computing Center of RAS, Limoges University We consider a linear di#erence operator L = a d (z)E d + ˇ ˇ ˇ + a 1 (z)E + a 0 (z), (1) and the corresponding linear di#erence equation Ly = 0. Here the coe#cients a i (z) are polynomial over C and E is the ``shift operator'' acting on functions of the complex variable z as Ey(z) = y(z + 1). The di#erence operator L is an element of the non­commutative ring C[z, E]. We shall call a d the leading coef­ ficient and a 0 the trailing coe#cient of L and we shall suppose that a 0 a d #= 0. We shall call the singularities of both L and Ly = 0 the zeros of the polyno­ mials a 0 (z) and a d (z - d). A point p # C will be said to be congruent to the singularities of L if a 0 (p +#) = 0 or a d (p -#) = 0 for some non­negative integer #. Equation Ly = 0 can be used as a tool to define a sequence or a function. If we know the value y(z) at every point z of a given strip # # Re z < # + d we can find the value of y(z) in the strip # - 1 # Re z < # , and hence in the strip # - 2 # Re z < # - 1, and so on. We can continue y(z) indefinitely to the left except at the points that are congruent to the zeros of the trailing coe#cient