[Paule]

### An algorithm to prove holonomic differential equations for modular forms

#### Peter Paule, Cristian-Silviu Radu

Submitted to the RISC Report Series. May 2020.@

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.},

year = {2020},

month = {May},

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},

type = {RISC Report Series},

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

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

**techreport**{RISC6109,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.},

year = {2020},

month = {May},

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},

type = {RISC Report Series},

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

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}