Sections of the Generating Series of a Solution to a Difference Equation in a Simplicial Cone
We consider a multidimensional difference equation in a simplicial lattice cone with coefficients from a field of characteristic zero and sections of a generating series of a solution to the Cauchy problem for such equations. We use properties of the shift and projection operators on the integer lat...
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Format: | Article |
Language: | English |
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Irkutsk State University
2022-12-01
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Series: | Известия Иркутского государственного университета: Серия "Математика" |
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Online Access: | https://mathizv.isu.ru/en/article/file?id=1432 |
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author | A.P. Lyapin T. Cuchta |
author_facet | A.P. Lyapin T. Cuchta |
author_sort | A.P. Lyapin |
collection | DOAJ |
description | We consider a multidimensional difference equation in a simplicial lattice cone with coefficients from a field of characteristic zero and sections of a generating series of a solution to the Cauchy problem for such equations. We use properties of the shift and projection operators on the integer lattice $Z^{n}$ to find a recurrence relation (difference equation with polynomial coefficients) for the section of the generating series. This formula allows us to find a generating series of a solution to the Cauchy problem in the lattice cone through a generating series of its initial data and a right-side function of the difference equation. We derived an integral representation for sections of the holomorphic function, whose coefficients satisfy the difference equation with complex coefficients. Finally, we propose a system of differential equations for sections that represent D-finite functions of two complex variables. |
first_indexed | 2024-04-11T06:32:58Z |
format | Article |
id | doaj.art-b62b7504e3cf411caad55001a8f62c89 |
institution | Directory Open Access Journal |
issn | 1997-7670 2541-8785 |
language | English |
last_indexed | 2024-04-11T06:32:58Z |
publishDate | 2022-12-01 |
publisher | Irkutsk State University |
record_format | Article |
series | Известия Иркутского государственного университета: Серия "Математика" |
spelling | doaj.art-b62b7504e3cf411caad55001a8f62c892022-12-22T04:39:58ZengIrkutsk State UniversityИзвестия Иркутского государственного университета: Серия "Математика"1997-76702541-87852022-12-014217589https://doi.org/10.26516/1997-7670.2022.42.75Sections of the Generating Series of a Solution to a Difference Equation in a Simplicial ConeA.P. LyapinT. CuchtaWe consider a multidimensional difference equation in a simplicial lattice cone with coefficients from a field of characteristic zero and sections of a generating series of a solution to the Cauchy problem for such equations. We use properties of the shift and projection operators on the integer lattice $Z^{n}$ to find a recurrence relation (difference equation with polynomial coefficients) for the section of the generating series. This formula allows us to find a generating series of a solution to the Cauchy problem in the lattice cone through a generating series of its initial data and a right-side function of the difference equation. We derived an integral representation for sections of the holomorphic function, whose coefficients satisfy the difference equation with complex coefficients. Finally, we propose a system of differential equations for sections that represent D-finite functions of two complex variables.https://mathizv.isu.ru/en/article/file?id=1432generating seriesdifference equationlattice conestanley hierarchysection |
spellingShingle | A.P. Lyapin T. Cuchta Sections of the Generating Series of a Solution to a Difference Equation in a Simplicial Cone Известия Иркутского государственного университета: Серия "Математика" generating series difference equation lattice cone stanley hierarchy section |
title | Sections of the Generating Series of a Solution to a Difference Equation in a Simplicial Cone |
title_full | Sections of the Generating Series of a Solution to a Difference Equation in a Simplicial Cone |
title_fullStr | Sections of the Generating Series of a Solution to a Difference Equation in a Simplicial Cone |
title_full_unstemmed | Sections of the Generating Series of a Solution to a Difference Equation in a Simplicial Cone |
title_short | Sections of the Generating Series of a Solution to a Difference Equation in a Simplicial Cone |
title_sort | sections of the generating series of a solution to a difference equation in a simplicial cone |
topic | generating series difference equation lattice cone stanley hierarchy section |
url | https://mathizv.isu.ru/en/article/file?id=1432 |
work_keys_str_mv | AT aplyapin sectionsofthegeneratingseriesofasolutiontoadifferenceequationinasimplicialcone AT tcuchta sectionsofthegeneratingseriesofasolutiontoadifferenceequationinasimplicialcone |