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«THE BULLETIN OF IRKUTSK STATE UNIVERSITY». SERIES «MATHEMATICS»
«IZVESTIYA IRKUTSKOGO GOSUDARSTVENNOGO UNIVERSITETA». SERIYA «MATEMATIKA»
ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2022. Vol 42

Sections of the Generating Series of a Solution to a Difference Equation in a Simplicial Cone

Author(s)
Alexander P. Lyapin1,2, Tom Cuchta2

1Siberian Federal University, Krasnoyarsk, Russian Federation

2Fairmont State University, Fairmont, West Virginia, USA

Abstract
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 Z𝑛 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.
About the Authors

Alexander P. Lyapin, Cand. Sci. (Phys.–Math.), Assoc. Prof., Siberian Federal University, Krasnoyarsk, 660041, Russian Federation, aplyapin@sfu-kras.ru

Tom Cuchta, PhD, Assist. Prof., Fairmont State University, Fairmont, WV, 26554, USA, tcuchta@fairmontstate.edu

For citation
Lyapin A. P., Cuchta T. Sections of the Generating Series of a Solution to a Difference Equation in a Simplicial Cone. The Bulletin of Irkutsk State University. Series Mathematics, 2022, vol. 42, pp. 75–89. https://doi.org/10.26516/1997-7670.2022.42.75
Keywords
generating series, difference equation, lattice cone, Stanley hierarchy, section
UDC
517.55+517.96
MSC
39A06, 32A10, 39A10, 39A14
DOI
https://doi.org/10.26516/1997-7670.2022.42.75
References
  1. Akhtamova S.S., Grishunov V.Y., Lyapin A.P., Tikhomirov S.A. On Sections Of Generating Series In Lattice Path Problems. Applied Mathematics & Physics, 2020,vol. 52, no. 2, pp. 146–151. https://doi.org/10.18413/2687-0959-2020-52-2-146-151
  2. Bostan A., Bousquet-M´elou M., Melczer S. Counting walks with large steps in an orthant. J. Eur. Math. Soc., 2021, vol. 23, pp. 2221–2297. https://doi.org/10.4171/JEMS/1053
  3. Bousquet-M´𝑒lou M., Petkovˇsek M. Linear recurrences with constant coefficients: the multivariate case. Discrete Mathematics, 2000, vol. 225, pp. 51–75. https://doi.org/10.1016/S0012-365X(00)00147-3
  4. Leinartas E.K. Multidimensional Hadamard composition and sums with linear constraints on the summation indices. Sib. Math. J., 1989, vol. 30, pp. 250–255. https://doi.org/10.1007/BF00971380
  5. Leinartas E.K., Lyapin A.P. On the Rationality of Multidimensional Recursive Series. Journal of Siberian Federal University. Mathematics & Physics, 2009, vol. 2,no. 4. pp. 449–455.
  6. Leinartas E.K., Nekrasova T.I. Constant Coefficient Linear Difference Equations On The Rational Cones Of The Integer Lattice. Siberian Math. J., 2016, vol. 57, no. 1, pp. 74–85. https://doi.org/10.1134/S0037446616010080
  7. Leinartas E.K., Yakovleva T.I. Generating Function of the Solution of a Difference Equation and the Newton Polyhedron of the Characteristic Polynomial. The Bulletin of Irkutsk State University. Series Mathematics, 2022, vol. 40, pp. 3–14. https://doi.org/10.26516/1997-7670.2022.40.3
  8. Leinartas E.K., Yakovleva T.I. The Cauchy Problem For Multidimensional Difference Equations And The Preservation Of The Hierarchy Of Generating Functions Of Its Solutions. Journal of Siberian Federal University. Mathematics & Physics, 2018, vol. 11, no. 6, pp. 712–722. https://doi.org/10.17516/1997-1397-2018-11-6-712-722
  9. Lipshitz L. D-Finite Power Series. J. of Algebra, 1989, vol. 122, pp. 353–373. https://doi.org/10.1016/0021-8693(89)90222-6
  10. Luz´on A., Mor´on M.A. Recurrence relations for polynomial sequences via Riordan matrices. Linear Algebra and its Applications, 2010, vol. 433, no. 7, pp. 1422–1446. https://doi.org/10.1016/j.laa.2010.05.021
  11. Lyapin A.P., Akhtamova S.S. Recurrence Relations For The Sections Of The Generating Series Of The Solution To The Multidimensional Difference Equation.Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2021, vol. 31, no. 3, pp. 414–423. https://doi.org/10.35634/vm210305
  12. Lyapin A.P., Chandragiri S. Generating Functions For Vector Partition Functions And A Basic Recurrence Relation. Journal of Difference Equations & Applications, 2019, vol. 25, no. 7, pp. 1052–1061. https://doi.org/10.1080/10236198.2019.1649396
  13. Lyapin A.P., Chandragiri S. The Cauchy Problem for Multidimensional Difference Equations in Lattice Cones. Journal of Siberian Federal University. Mathematics & Physics, 2020, vol. 13, no. 2, pp. 187–196. https://doi.org/10.17516/1997-1397-2020-13-2-187-196
  14. Pochekutov D.Yu. Analytic continuation of diagonals of Laurent series for rational functions. Journal of Siberian Federal University Mathematics & Physics, 2021, vol. 14, no. 3, pp. 360–368. https://doi.org/10.17516/1997-1397-2021-14-3-360-368
  15. Pochekutov D.Yu. Diagonals of the Laurent series of rational functions. Siberian Math. J., 2009, vol. 50, no. 6, pp. 1081–1091. https://doi.org/10.1007/s11202-009-0119-z
  16. Safonov K.V., Tsikh A.K. Singularities of the Grothendieck parametric residue and diagonals of a double power series. Izv. Vyssh. Uchebn. Zaved. Mat., 1984,vol. 4, pp. 57–58.
  17. Senashov A.V. A list of integral representations for diagonals of power series of rational functions. Journal of Siberian Federal University. Mathematics & Physics,2021, vol. 14, no. 5, pp. 624–631. https://doi.org/10.17516/1997-1397-2021-14-5-624-631
  18. Shabat B.V. Introduction to complex analysis. Part II. Functions of several variables. American Mathematical Society, 1992, 371 p.
  19. Stanley R.P. Differentiably Finite Power Series. Europ. J. Combinatorics, 1980, vol. 1, pp. 175–188. https://doi.org/10.1016/S0195-6698(80)80051-5
  20. Tsikh A. Multidimensional residues and their applications. Providence, American Mathematical Society, 1992, 188 p.

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