Creative Telescoping (2505.05345v1)

Abstract: These notes on creative telescoping are based on a series of lectures at the Institut Henri Poincare in November and December 2023.

Overview of Creative Telescoping

Creative telescoping is a symbolic computation technique employed in the field of summation and integration problems for discrete and continuous functions. It is particularly relevant when handling functions described by operators such as recurrence, differential, or difference operators. The technique is pivotal for deriving relations, known as telescopers, that can transform complex summation or integration into manageable recurrence or differential equations. This paper elucidates the foundational methods and recent advancements in creative telescoping, grounded in the context of D-finite functions and hypergeometric terms.

Key Concepts and Algorithms

• D-Finite Functions: These functions are characterized by satisfying linear differential equations with polynomial coefficients. The class encompasses many elementary functions and special functions, backed by robust closure properties under operations like addition, multiplication, and integration. The paper highlights the equivalence between D-finiteness and holonomic ideals in differential settings.
• Symbolic Summation and Integration: For hypergeometric terms, Gosper's algorithm serves as a core tool for indefinite summation, while Zeilberger's algorithm addresses definite summation. In the case of D-finite functions, Abramov--van Hoeij's algorithm provides a solution for symbolic integration, guiding the discovery of primitive elements. Chyzak's algorithm advances the computation of telescopers in the context of bivariate D-finite functions, facilitating effective integration.
• Reduction-Based Telescoping: This technique leverages the notion of simplification through algebraic structures, focusing on efficient computations of telescopers by identifying linear dependencies among derivatives of the integrand or summand with respect to a free variable. The recent development in handling multiple variables and function classes enhances its applicability in broader contexts.

Applications and Examples

The practical utility of creative telescoping spans diverse applications in computer algebra, numerical analysis, and even real-world scientific problems. A notable example is within electromagnetic simulations, where difference-differential relations for basis functions are efficiently derived using holonomic techniques. Another compelling application lies in proving identities and evaluating special function expressions appearing in mathematical literature, such as those tabulated in Gradshteyn and Ryzhik.

Advanced applications include the holonomic ansatz for symbolic determinant evaluations, which has been instrumental in tackling conjectures from areas such as quantum physics models. The technique's adaptability is showcased in tackling nested sums and complex integral evaluations by creatively expanding their formal representations.

Conclusion and Future Directions

While the paper reflects on the historical development and core principles of creative telescoping, it nudges towards future challenges in extending these algorithmic frameworks. The fusion of approaches based on operator techniques and differential/difference fields holds promise for a unified theory, proliferating the computational prospects for even more generalized function classes. The ongoing enhancements in reduction-based methods signal continuous growth in the efficiency and applicability of creative telescoping, paving the way for more groundbreaking solutions in automated symbolic computation.