@**incollection**{RISC6395,author = {Peter Paule and Cristian-Silviu Radu},

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

booktitle = {{Transcendence in Algebra, Combinatorics, Geometry and Number Theory. TRANS 2019.}},

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 $h$ satisfies
a holonomic differential equation; i.e., one which is
linear with coefficients being polynomials in $h$.
This fact traces back to Gau{ss} and has been
popularized prominently by Zagier. Using holonomic
procedures, computationally it is often straightforward
to 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.},

series = {Springer Proceedings in Mathematics & Statistics},

volume = {373},

pages = {367--420},

publisher = {Springer, Cham},

isbn_issn = {978-3-030-84303-8},

year = {2021},

editor = {Bostan A. and Raschel K. },

refereed = {yes},

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

sponsor = {FWF SFB F50},

length = {54},

url = {https://doi.org/10.1007/978-3-030-84304-5_16}

}