A computer cannot store certain mathematical objects like the entire set of real numbers, because the set of real numbers is infinitely large. Some special functions, for example, various transcendental functions, cannot be stored using algebraic means. We need an efficient method of storing functions which occupies finite data. Instead of storing special functions themselves, we may be able to just store the differential equation that it is a solution of. F is a holonomic system of d-variable functions if we have d (essentially independent) mixed homogeneous linear (partial) difference-differential equations with polynomial coefficients in all variables. The solutions of holonomic systems are called holonomic functions. These are very useful in computer algebra.
A holonomic function can be represented by its normal form, the lowest order holonomic equation that it is a solution of. Using this representation, computers can verify/disprove many identities involving special functions in finite time. If two holonomic functions are equivalent, a computer can verify they correspond to the same normal form and satisfy the same initial conditions.
Computer-aided verification of identities using holonomic systems has been extensively used to help mathematicians and scientists! In fact the HolonomicSystems package in Mathematica helped solve an open problem in enumerative combinatorics, the proof of Ira Gessel’s conjecture on the number of lattice paths within a step set.
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