Hypergeometric-Type Sequences: A Comprehensive Approach to Representing Discrete Functions

Core Concepts

Hypergeometric-type sequences are a generalization of hypergeometric sequences, allowing for linear combinations of interlaced hypergeometric terms. They form a subring of the ring of holonomic sequences and can be used to represent a wide range of discrete functions, including those involving trigonometric functions with linear arguments.

Abstract

The paper introduces the concept of hypergeometric-type sequences, which are linear combinations of interlaced hypergeometric sequences. It proves that these sequences form a subring of the ring of holonomic sequences and discusses their properties, including their connection to hypergeometric-type power series.
The key highlights and insights are:

Hypergeometric-type sequences are defined as linear combinations of interlaced hypergeometric sequences, where the interlacement is controlled by m-fold indicator sequences.
The set of hypergeometric-type sequences forms a ring, and every hypergeometric-type sequence is P-recursive (satisfies a holonomic recurrence equation).
There is a one-to-one correspondence between proper hypergeometric-type power series and hypergeometric-type sequences.
The paper presents an algorithmic approach to finding normal forms of hypergeometric-type terms, which involves:

Finding a holonomic recurrence equation satisfied by the given expression
Determining a basis of m-fold hypergeometric term solutions of the recurrence equation
Using the initial values to deduce a hypergeometric-type normal form

The authors also provide a Maple implementation of the algorithm, which can automatically generate closed-form expressions for sequences involving trigonometric functions with linear arguments in the index.

Stats

sn = (3/2)^n - (5)^(n/2) χ{n≡0 mod 2} + 2^(n/3) χ{n≡0 mod 3}
an = (1/2) - ((-1)^n)/2 χ{n≡0 mod 2}
an = (31/3) - χ{n≡0 mod 2}
sin^2(nπ/4) = (1/2)(1 - (-1)^(n/2) χ{n≡0 mod 2})

Quotes

"Hypergeometric-type sequences are a generalization of hypergeometric sequences, allowing for linear combinations of interlaced hypergeometric terms."
"Every hypergeometric-type sequence is P-recursive (satisfies a holonomic recurrence equation)."
"There is a one-to-one correspondence between proper hypergeometric-type power series and hypergeometric-type sequences."

Key Insights Distilled From

Hypergeometric-Type Sequences

by Bertrand Teg... at arxiv.org 04-22-2024

https://arxiv.org/pdf/2401.00256.pdf

Deeper Inquiries

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

Hypergeometric-type sequences have various potential applications in fields beyond computer science. In physics, these sequences can be used to model and analyze physical phenomena that exhibit periodic or oscillatory behavior. For example, in quantum mechanics, hypergeometric-type sequences can be utilized to describe the energy levels of certain systems or the behavior of wave functions. In statistical mechanics, these sequences can help in analyzing the distribution of particles in a system or the fluctuations in a physical quantity over time.
In biology, hypergeometric-type sequences can be applied to study biological rhythms, such as circadian rhythms, which exhibit periodic behavior. These sequences can also be used to model genetic patterns, protein interactions, or population dynamics. For instance, in genetics, hypergeometric-type sequences can be employed to analyze the inheritance patterns of certain traits or the expression levels of genes over time.
Overall, the versatility and mathematical properties of hypergeometric-type sequences make them valuable tools for modeling and analyzing various phenomena in physics and biology.

How could the concept of hypergeometric-type sequences be extended to handle more complex mathematical structures, such as multivariate sequences or sequences over non-commutative rings?

To extend the concept of hypergeometric-type sequences to handle more complex mathematical structures, such as multivariate sequences or sequences over non-commutative rings, several modifications and adaptations can be made:

Multivariate Sequences: For multivariate sequences, each term in the sequence would depend on multiple variables instead of just one. The hypergeometric-type sequences can be extended to include terms that are functions of several variables, leading to a more intricate and versatile framework for modeling multidimensional phenomena.

Sequences over Non-Commutative Rings: When dealing with sequences over non-commutative rings, the algebraic properties of the ring need to be considered. The concept of hypergeometric-type sequences can be generalized to accommodate non-commutative operations, allowing for the study of sequences with non-trivial algebraic structures.

Generalized Hypergeometric-Type Sequences: Introducing generalized hypergeometric-type sequences that can handle multiple variables and non-commutative operations would provide a more comprehensive framework for addressing complex mathematical structures. These generalized sequences could capture a wider range of mathematical phenomena and offer more flexibility in modeling diverse systems.

By extending the concept of hypergeometric-type sequences in these ways, researchers can explore and analyze a broader array of mathematical structures and systems, leading to new insights and applications in various fields of mathematics and beyond.

Are there any connections between hypergeometric-type sequences and other areas of mathematics, such as number theory or algebraic geometry, that could lead to further insights or applications?

Hypergeometric-type sequences have connections to various areas of mathematics, including number theory and algebraic geometry, which can lead to further insights and applications:

Number Theory: In number theory, hypergeometric-type sequences can be used to study properties of integers, such as divisibility patterns, prime factorization, or congruences. These sequences can provide a framework for analyzing arithmetic functions, generating functions, and Diophantine equations, leading to new results and conjectures in number theory.

Algebraic Geometry: In algebraic geometry, hypergeometric-type sequences can be related to algebraic varieties, curves, and surfaces. By considering the solutions of hypergeometric-type equations in the context of algebraic geometry, researchers can explore the geometric properties of these sequences and their connections to geometric objects. This can lead to insights into the intersection of algebraic geometry and combinatorics.

Modular Forms: Hypergeometric-type sequences can also be linked to modular forms, which are important objects in number theory and algebraic geometry. By studying the modular properties of hypergeometric-type sequences, researchers can uncover new relationships between these sequences and modular forms, providing deeper insights into the underlying structures and symmetries.

Overall, the connections between hypergeometric-type sequences and other areas of mathematics offer a rich source of exploration and research, with the potential to uncover new connections, theorems, and applications across different mathematical disciplines.