English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

It is known that all $\tau$ functions of the Painlev\'{e} equations satisfythe fourth-order quadratic differential equation. Among them, for the III, V,and VI equations, it is possible to express the formal series solutionsexplicitly by using combinatorics. In this paper, we show the convergence ofthe formal series, including the solutions of more general equations. And bythe absolute convergence of $\tau $ series, the convergence of the conformalblock function ($c=1$) also follows since it is a partial sum of the $\tau $series. We also characterized the form of a homogeneous quadratic equation witha series solution similar to the tau functions of the Painlev\'{e} equations.The Painlev\'{e} equations are classified into six types, and it is known thatthey can be obtained by sequentially degenerating from type VI to type I.

On Formal Series Solutions To 4th-order Quadratic Homogeneous Differential Equations And Their Convergence