One major theme of this paper concerns the expansion of modular forms and functions in terms of fractional (Puiseux) series. This theme is connected with another major theme, holonomic functions and sequences. With particular attention to algorithmic aspects, we study various connections between these two worlds. Applications concern partition congruences, Fricke–Klein relations, irrationality proofs a la Beukers, or approximations to pi studied by Ramanujan and the Borweins. As a major ingredient to a “first guess, then prove” strategy, a new algorithm for proving differential equations for modular forms is introduced.

中文翻译：

---

模函数和形式的完整关系：首先猜测，然后证明

本文的一个主要主题是关于分数（Puiseux）级数的模块化形式和功能的扩展。这个主题与另一个主要主题完整的功能和序列有关。我们特别关注算法方面，研究这两个世界之间的各种联系。应用涉及分区同余、Fricke-Klein 关系、非理性证明啦啦Beukers 或 Ramanujan 和 Borweins 研究的 pi 近似值。作为“先猜测后证明”策略的主要成分，引入了一种用于证明模形式微分方程的新算法。