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Borel summability of divergent solutions for singular first order linear partial differential equations with polynomial coefficients. (English) Zbl 1036.35051
Singular first order linear partial differential equation with polynomial coefficients $(a+bx+c(x,y))yu_x(x,y)+(d+e(x,y))y^2u_y(x,y)+u(x,y)=f(x,y),\tag{*}$ where $$x,y\in\mathbb C$$, $$a,b$$ and $$d$$ are complex constants, $$c(x,y)$$ and $$e(x,y)$$ are polynomials of at least first degree with respect to $$y$$, and $$f(x,y)$$ is a holomorphic function at the origin is considered. The Borel summability of the formal solutions to (*) is studied: $$u(x,y)$$ is a Borel summable in $$\theta\in\mathbb R$$ if there exists a holomorphic function $$U(x,y)$$ such that there exists some positive constants $$C$$ and $$K$$ for which $$\max_{| x| \leq r}| U(x,y)-\sum_{k=0}^{n-1}u_k(x)y^k| \leq CK^nn!| y| ^n$$ for $$y\in\{y: | y-Te^{i\theta}| <T\}$$. The conditions under which the formal solutions to (*) are Borel summable are given by analytic continuation properties and growth estimates for the coefficients of (*).

##### MSC:
 35C20 Asymptotic expansions of solutions to PDEs 35C10 Series solutions to PDEs 35F05 Linear first-order PDEs