Open AccessHolonomic relations for modular functions and forms: First guess, then prove
Author(s) -
Peter Paule,
Cristian-Silviu Radu
Publication year - 2020
Publication title -
international journal of number theory
Language(s) - English
Resource type - Journals
eISSN - 1793-0421
pISSN - 1793-7310
DOI - 10.1142/s1793042120400278
Subject(s) - mathematics , ramanujan's sum , mathematical proof , holonomic , modular form , modular design , irrationality , theme (computing) , algebra over a field , pure mathematics , rationality , computer science , artificial intelligence , epistemology , geometry , philosophy , operating system
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.