Discrete Generating Series and Linear Difference Equations
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866909597789323264 |
|---|---|
| author | Alekseev, Vitaly Cuchta, Tom Lyapin, Alexander |
| author_facet | Alekseev, Vitaly Cuchta, Tom Lyapin, Alexander |
| contents | We define discrete generating series for arbitrary functions \( f \colon \mathbb{Z}^n \rightarrow \mathbb{C} \) and derive functional relations that these series satisfy. For linear difference equations with constant coefficients, we establish explicit functional equations linking the generating series to the initial data, and for equations with polynomial coefficients, we introduce an analogue of Stanley's \( D \)-finiteness criterion, proving that a discrete generating series is \( D \)-finite if and only if the corresponding sequence is polynomially recursive. The framework is further generalized to multidimensional settings, where we investigate the interplay between discrete generating series and solutions to Cauchy problems for difference equations. Key structural properties are uncovered through the introduction of polynomial shift operators and projection techniques. The theory is illustrated with concrete examples, including the Tribonacci recurrence and Schröder's second problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_21722 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Discrete Generating Series and Linear Difference Equations Alekseev, Vitaly Cuchta, Tom Lyapin, Alexander Classical Analysis and ODEs Dynamical Systems 05A15, 39A05, 39A06 We define discrete generating series for arbitrary functions \( f \colon \mathbb{Z}^n \rightarrow \mathbb{C} \) and derive functional relations that these series satisfy. For linear difference equations with constant coefficients, we establish explicit functional equations linking the generating series to the initial data, and for equations with polynomial coefficients, we introduce an analogue of Stanley's \( D \)-finiteness criterion, proving that a discrete generating series is \( D \)-finite if and only if the corresponding sequence is polynomially recursive. The framework is further generalized to multidimensional settings, where we investigate the interplay between discrete generating series and solutions to Cauchy problems for difference equations. Key structural properties are uncovered through the introduction of polynomial shift operators and projection techniques. The theory is illustrated with concrete examples, including the Tribonacci recurrence and Schröder's second problem. |
| title | Discrete Generating Series and Linear Difference Equations |
| topic | Classical Analysis and ODEs Dynamical Systems 05A15, 39A05, 39A06 |
| url | https://arxiv.org/abs/2504.21722 |