# Holonomic functions and modular forms brought together by computer algebra

## Combinatorics/Partitions Seminar

## Meeting Details

For more information about this meeting, contact Kristin Berrigan, George Andrews.

**Speaker:** Peter Paule, Research Institute for Symbolic Computation (RISC) Johannes Kepler University Linz

**Abstract:** Holonomic functions and sequences satisfy linear differential
and difference equations, respectively, with polynomial coefficients.
It has been estimated that holonomic functions cover about 60 percent
of the functions contained in the 1964 "Handbook" by Abramowitz and Stegun.
A recent estimate says that holonomic sequences constitute about 20 percent
of Sloane's OEIS database. The study of these ubiquitous objects traces
back to the time of Gauss (at least).
Also tracing back to the time of Gauss (at least) are highly non-holonomic
objects: modular functions and modular forms with q-series representations
arising, for instance, as generating functions of partitions of various
kinds.
Using computer algebra, the talk connects these two different worlds.
Applications concern partition congruences, Frickeâ€“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 used. The results presented arose in joint work with
Silviu Radu (RISC).

## Room Reservation Information

**Room Number:** 114 McAllister

**Date:** 07/18/2023

**Time:** 11:00am - 12:00pm