Rogers-Ramanujan Functions, Modular Functions, and Computer Algebra ∗ Dedicated to the symbolic summation pioneer Sergei Abramov who concretely passed milestone 70 Peter Paule and Silviu Radu Abstract Many generating functions for partitions of numbers are strongly related to mod- ular functions. This article introduces such connections using the Rogers-Ramanujan func- tions as key players. After exemplifying basic notions of partition theory and modular func- tions in tutorial manner, relations of modular functions to q-holonomic functions and se- quences are discussed. Special emphasis is put on supplementing the ideas presented with concrete computer algebra. Despite intended as a tutorial, owing to the algorithmic focus the presentation might contain aspects of interest also to the expert. One major application concerns an algorithmic derivation of Felix Klein’s classical icosahedral equation. 1 Introduction The main source of inspiration for this article was the truly wonderful paper [14] by William Duke. When reading Duke’s masterly exposition, the first named author started to think of writing kind of a supplement which relates the beautiful ingredients of Duke’s story to com- puter algebra. After starting, the necessity to connect to readers with diverse backgrounds soon became clear. As a consequence, this tutorial grew longer than originally intended. As a compensation for its length, we hope some readers will find it useful to find various things presented together at one place the first time. Owing to the algorithmic focus, some aspects might have a new appeal also to the expert. Starting with partition generating functions and using the Omega package, in Section 2 the key players of this article are introduced, the Rogers-Ramanujan functions F (1) and F (q). To prove non-holonomicity, in Section 3 the series presentations of F (1) and F (q) are converted into infinite products. Viewing things analytically, the Dedekind eta function, also defined via an infinite product on the upper half complex plane H, is of fundamental importance, in particular, owing to its modular transformation properties. Peter Paule Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Altenbergerstraße 69, 4040, Linz, Austria, e-mail: ppaule@risc.jku.at Silviu Radu Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Altenbergerstraße 69, 4040, Linz, Austria, e-mail: sradu@risc.jku.at ∗ Both authors were supported by grant SFB F50-06 of the Austrian Science Fund (FWF) 1