Creative Telescoping for Holonomic Functions

The aim of this article is twofold: on the one hand it is intended to serve as a gentle introduction to the topic of creative telescoping, from a practical point of view; for this purpose its application to several problems is exemplified. On the other hand, this chapter has the flavour of a survey article: the developments in this area during the last two decades are sketched and a selection of references is compiled in order to highlight the impact of creative telescoping in numerous contexts.

💡 Research Summary

The paper “Creative Telescoping for Holonomic Functions” serves both as a gentle tutorial and a concise survey of the developments in creative telescoping over the past two decades. It begins by explaining the basic idea: given a bivariate summand f(k, w) or integrand f(x, w) that satisfies a system of linear difference‑differential equations, one seeks an operator P that annihilates f. By writing P in the form P = T(w, ∂) + (S_k − 1)·C, where T is called the telescoper and C the certificate, the telescoping relation T·F = 0 (or an inhomogeneous version) is obtained for the sum or integral F(w) = ∑{k=a}^{b} f(k,w) or F(w) = ∫{a}^{b} f(x,w)dx. The telescoper thus yields a recurrence (or differential equation) for the whole sum or integral, which can be used for proving identities, deriving closed forms, or further symbolic manipulation.

The authors then place this method in a historical context. The term “creative telescoping” was coined by van Pooten in the 1970s, but the systematic algorithmic treatment began with Zeilberger’s 1990 “holonomic systems approach”. Zeilberger introduced a general elimination algorithm (later called the “slow algorithm”) and a fast algorithm for hypergeometric single sums. Parallel developments include the Almkvist–Zeilberger algorithm for hyperexponential integrals, the WZ theory for hypergeometric identities, and Takayama’s module‑based improvements. In the 1990s the focus was on hypergeometric summation, with extensions to q‑hypergeometric terms, multiple sums, and bounds on the order and degree of the output. Around the turn of the millennium, Chyzak and Salvy generalized Zeilberger’s method to arbitrary holonomic functions, leading to what is now known as Chyzak’s algorithm.

A substantial part of the paper is devoted to the algebraic foundations. The authors introduce Ore algebras O = F⟨∂₁,…,∂_ℓ⟩, where the non‑commutative multiplication obeys the familiar rules D·a = a·D + a′ (for derivations) and S·a = a(n+1)·S (for shifts). For a function f, the annihilating ideal Ann_O(f) = {P∈O | P·f = 0} is a left ideal; a left Gröbner basis of this ideal can be computed algorithmically. A function is called ∂‑finite (or D‑finite) if the quotient O/Ann_O(f) has finite dimension over the coefficient field; the dimension is the rank. The paper lists closure properties: sums, products, substitutions, and definite sums/integrals of ∂‑finite (hence holonomic) functions remain ∂‑finite, with explicit bounds on the resulting rank.

The core algorithmic section explains how to compute a telescoper. One first computes a Gröbner basis for Ann_O(f) and then searches for a relation of the form (2) in the paper. If the inhomogeneous part vanishes (e.g., the boundary terms are zero), the telescoper directly annihilates the sum/integral. Otherwise, the authors describe how to homogenize the relation by multiplying with an annihilator of the inhomogeneous term. The paper discusses several concrete algorithms: Zeilberger’s fast algorithm for hypergeometric terms, the Almkvist–Zeilberger algorithm for hyperexponential integrands, Takayama’s module‑based approach, Chyzak’s algorithm for general holonomic inputs, and more recent heuristic or complexity‑oriented methods (e.g., order‑degree trade‑offs, residue‑based telescopers for rational/algebraic functions, Hermite reduction for hyperexponential functions, and the Griffiths‑Dwork method for rational functions).

Implementation aspects are covered extensively. The authors list existing software: Maple’s SumTools