Algorithms for Computing with Hypergeometric-Type Sequences

Core Concepts

This paper presents algorithms for efficiently processing and analyzing hypergeometric-type sequences, including computing holonomic recurrence equations and performing Hadamard product operations.

Abstract

The paper introduces the concept of hypergeometric-type sequences, which are a class of sequences that can be expressed as linear combinations of interlaced hypergeometric terms. The author presents two key algorithms:

HolonomicRE: This algorithm computes holonomic recurrence equations for hypergeometric-type sequences. It first finds recurrence equations for the individual hypergeometric terms, then combines them using the P-recursive addition algorithm.

HTSproduct: This algorithm computes the Hadamard (element-wise) product of two hypergeometric-type sequences. It leverages the Chinese Remainder Theorem to efficiently perform the product operation.

The author also describes a Maple software package called HyperTypeSeq, which implements these algorithms and provides additional functionality for working with hypergeometric-type sequences, such as evaluating terms and converting expressions to hypergeometric-type normal form.
The paper includes several examples demonstrating the usage of the HyperTypeSeq package and the capabilities of the presented algorithms. It also discusses the challenges in defining canonical forms for hypergeometric-type sequences and the implications for recognizing equivalent formulas.

Stats

The paper does not contain any explicit numerical data or statistics. The focus is on the algorithmic aspects of working with hypergeometric-type sequences.

Quotes

The paper does not contain any notable quotes.

Key Insights Distilled From

Computing with Hypergeometric-Type Terms

by Bertrand Teg... at arxiv.org 04-17-2024

https://arxiv.org/pdf/2404.10143.pdf

Deeper Inquiries

What are the potential applications of hypergeometric-type sequences in fields beyond computer science, such as physics, biology, or finance

Hypergeometric-type sequences have various potential applications in fields beyond computer science. In physics, these sequences can be utilized to model complex physical systems, such as quantum mechanics, where hypergeometric functions often arise in the solutions of differential equations. In biology, hypergeometric-type sequences can be employed to analyze genetic sequences, protein structures, or population dynamics. For instance, in genetics, these sequences can help in understanding patterns of inheritance or gene expression. In finance, hypergeometric-type sequences can be used in modeling financial time series data, risk analysis, or option pricing models. By applying these sequences in different domains, researchers can gain insights into the underlying patterns and structures of the systems they are studying.

How can the algorithms presented in this paper be extended or adapted to handle more general classes of sequences or functions

The algorithms presented in the paper can be extended or adapted to handle more general classes of sequences or functions by incorporating additional mathematical techniques or data structures. For instance, to handle more complex sequences, the algorithms can be enhanced to work with multivariate sequences or functions. This extension would involve modifying the existing algorithms to accommodate multiple variables and dependencies. Additionally, the algorithms can be adapted to work with different types of functions, such as trigonometric, exponential, or logarithmic functions, by incorporating appropriate transformations or representations. By expanding the scope of the algorithms to handle a broader range of sequences and functions, researchers can apply them to a wider variety of problems in diverse fields.

What are the computational complexity and performance characteristics of the HolonomicRE and HTSproduct algorithms, and how do they compare to alternative approaches for similar tasks

The computational complexity and performance characteristics of the HolonomicRE and HTSproduct algorithms depend on various factors such as the size of the input sequences, the order of the recurrence equations, and the complexity of the hypergeometric-type terms involved. In general, the algorithms exhibit polynomial time complexity for typical cases, making them efficient for practical use. However, for extremely large sequences or highly complex functions, the algorithms may experience increased computational demands.
Comparing these algorithms to alternative approaches for similar tasks, the HolonomicRE algorithm is specifically designed to compute holonomic recurrence equations from hypergeometric-type terms efficiently. It leverages mathematical properties of hypergeometric sequences to provide accurate and concise results. On the other hand, the HTSproduct algorithm focuses on computing products of hypergeometric-type terms, utilizing techniques like the Chinese Remainder Theorem for efficient calculations. While alternative methods may exist, the specialized nature of these algorithms makes them well-suited for handling hypergeometric-type sequences with optimal performance and accuracy.