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Register a new userOn the convergence of generalized power series satisfying an algebraic ODE
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We propose a sufficient condition of the convergence of a generalized power series formally satisfying an algebraic (polynomial) ordinary differential equation. The proof is based on the majorant method.
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We propose a sufficient condition of the convergence of a Dulac series formally satisfying an algebraic ordinary differential equation (ODE). Such formal solutions of algebraic ODEs appear rather often, in particular, the third, fifth, and sixth Painleve equations possess formal Dulac series solutions, whose convergence follows from the proposed sufficient condition.
A sufficient condition of the convergence of an exotic formal series (a kind of power series with complex exponents) solution to an ODE of a general form is proposed.
A sufficient condition for the convergence of a generalized formal power series solution to an algebraic $q$-difference equation is provided. The main result leans on a geometric property related to the semi-group of (complex) power exponents of such a series. This property corresponds to the situation in which the small divisors phenomenon does not arise. Some examples illustrating the cases where the obtained sufficient condition can be or cannot be applied are also depicted.
We propose a sufficient condition of the convergence of a complex power type formal series of the form $varphi=sum_{k=1}^{infty}alpha_k(x^{{rm i}gamma}),x^k$, where $alpha_k$ are functions meromorphic at the origin and $gammain{mathbb R}setminus{0}$, that satisfies an analytic ordinary differential equation (ODE) of a general type. An example of a such type formal solution of the third Painleve equation is presented and the proposed sufficient condition is applied to check its convergence.
We propose an analytic proof of the Malgrange-Sibuya theorem concerning a sufficient condition of the convergence of a formal power series satisfying an ordinary differential equation. The proof is based on the majorant method and allows to estimate the radius of convergence of such a series.
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