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Transseries for a class of nonlinear difference equations. (English) Zbl 1001.39002

The nonlinear analytic difference equations studied in this article have the form \[ y(x+1)=\Lambda (x)y(x)+g(x,y)\qquad \tag{1} \] where \(g(x,y)\) is an analytic \(C^n\)-valued function of \((x,y)\) in a neighborhood of \((\infty ,0)\) and \[ \Lambda (x):=\text{diag}(e^{-\mu _1}(1+x^{-1})^{a_1},..., e^{-\mu _n}(1+x^{-1})^{a_n}) \] is assumed to satisfy certain conditions such that (1) has a formal solution \(\hat{y}_0\in x^{-2}C^n[[x^{-1}]]\). A formal transformation \[ y=\sum_{k\in N^n}z^k\hat{y}_k(x) \] where \(\hat{y}_k(x)\) are formal series in \(C^n[[x^{-1}]]\) and \(z^k:=z_1^{k_1}...z_n^{k_n}\), is used in order to reduce (1) to a normal form \(z(x+1)=\Lambda _1(x)z(x)\). From the solutions of the equation in normal form one obtains the formal exponential series solutions, transseries, of the difference equation \[ \hat{y}(x)=\sum_{k\in N^n}C^k(x)e^{-k\cdot \mu x}x^{k \cdot a} \hat{y}_k(x) \] where \(\mu , a\in C^n\) and \(k\cdot \mu :=k_1\mu _1+...+k_n\mu _n\). Some of their properties are investigated. The treatment is based on the use of Laplace and Borel transforms for such equations.

MSC:

39A10 Additive difference equations
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