Journal of the Mathematical Society of Japan

A functional equation with Borel summable solutions and irregular singular solutions


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A Functional equation $\sum_{i=1}^{m}a_{i}(z)u(\varphi_{i}(z))=f(z)$ is considered. First we show the existence of solutions of formal power series. Second we study the homogeneous equation $(f(z)\equiv 0)$ and construct formal solutions containing exponential factors. Finally it is shown that there exists a genuine solution in a sector whose asymptotic expansion is a formal solution, by using the theory of Borel summability of formal power series. The equation has similar properties to those of irregular singular type in the theory of ordinary differential equations.

Article information

J. Math. Soc. Japan, Volume 70, Number 2 (2018), 711-731.

First available in Project Euclid: 18 April 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30D05: Functional equations in the complex domain, iteration and composition of analytic functions [See also 34Mxx, 37Fxx, 39-XX]
Secondary: 39B32: Equations for complex functions [See also 30D05] 44A10: Laplace transform

Borel summable irregular singular functional equation


ŌUCHI, Sunao. A functional equation with Borel summable solutions and irregular singular solutions. J. Math. Soc. Japan 70 (2018), no. 2, 711--731. doi:10.2969/jmsj/07027491.

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  • W. Balser, From Divergent Power Series to Analytic Functions, Lecture Notes in Math., 1582, Springer, 1994.
  • W. Balser, Formal power series and linear systems of meromorphic ordinary differential equations, Universitext, Springer, 2000.
  • W. Balser, B. L. J. Braaksma, J.-P. Ramis and Y. Sibuya, Multisummabilty of formal power series solutions of linear ordinary differential equations, Asymptotic Analysis, 5 (1991), 27–45.
  • S. Ōuchi, Multisummability of formal solutions of some linear partial differential equations, J. Differential Equations, 185 (2002), 513–549.
  • S. Ōuchi, On some functional equation with Borel summable solutions, Funkcial. Ekvac., 58 (2015), 223–251.
  • W. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Dover Publications, New York, 1987.