Holonomic relations for modular functions and forms: first guess, then prove. (English) Zbl 07924591
Andrews, George E. (ed.) et al., Analytic and combinatorial number theory: the legacy of Ramanujan. Contributions in honor of Bruce C. Berndt. Selected papers based on the presentations at the conference, Champaign, IL, USA, June 6–9, 2019. Singapore: World Scientific. Monogr. Number Theory 12, 535-581 (2024).
Summary: 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.
For the entire collection see [Zbl 07852528].
For the entire collection see [Zbl 07852528].
MSC:
| 05A30 | \(q\)-calculus and related topics |
| 11F03 | Modular and automorphic functions |
| 68W30 | Symbolic computation and algebraic computation |
| 11F33 | Congruences for modular and \(p\)-adic modular forms |
| 11P83 | Partitions; congruences and congruential restrictions |
