Holonomic functions and modular forms brought together by computer algebra
Sprache des Vortragstitels:
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, Fricke?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).