Convergence of generalized power series solutions of functional equations

R. R. Gontsov^ab, I. V. Goryuchkina<sup>c

a</sup> HSE University
^b Kharkevich Institute for Information Transmission Problems of the Russian Academy of Sciences
^c Keldysh Institute of Applied Mathematics of Russian Academy of Sciences

Abstract: We consider questions relating to the convergence of generalized power series (with complex-valued exponents) that formally satisfy some analytic functional equations: a differential equation, a $q$-difference one, or a Mahler equation. We present new results, as well as generalizations of some of our previous results, thus summing up our investigations of the subject. We also present a selection of results on the existence and uniqueness of local holomorphic solutions of such equations and review some classical results on the convergence of Taylor power series that solve them formally.
Bibliography: 54 titles.

Keywords: analytic functional equations, ordinary differential equation, $q$-difference equation, Mahler equation, generalized power series, formal solution, convergence.

UDC: 517.925+517.96

MSC: 30B10, 39B32

Received: 24.12.2024

DOI: 10.4213/rm10235

 English version: Russian Mathematical Surveys, 2025, 80:3, 367–425

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