[bib]

@**techreport**{RISC6108,

author = {Peter Paule and Cristian-Silviu Radu},

title = {{An algorithm to prove holonomic differential equations for modular forms}},

language = {english},

abstract = {Express a modular form $g$ of positive weight locally in terms of a modular function $h$ as$y(h)$, say. Then $y(h)$ as a function in $h$ satisfiesa holonomic differential equation; i.e., one which islinear with coefficients being polynomials in $h$.This fact traces back to Gau{ss} and has beenpopularized prominently by Zagier. Using holonomicprocedures, computationally it is often straightforwardto derive such differential equations as conjectures.In the spirit of the ``first guess, then prove'' paradigm,we present a new algorithm to prove such conjectures.},

number = {20-05},

year = {2020},

month = {May},

note = {A final version appeared in: Transcendence in Algebra, Combinatorics, Geometry and Number Theory. TRANS 2029, eds.: A. Bostan and K. Raschel, Springer, 2021},

keywords = {modular functions, modular forms, holonomic differential equations, holonomic functions and sequences, Fricke-Klein relations, $q$-series, partition congruences},

sponsor = {FWF SFB F50},

length = {48},

license = {CC BY 4.0 International},

type = {RISC Report Series},

institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},

address = {Altenberger Straße 69, 4040 Linz, Austria},

issn = {2791-4267 (online)}

}