Geodesic
The Cauchy problem for multidimensional difference equations and the preservation of the hierarchy of generating functions of its solutions
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 11 (2018) no. 6, pp. 712-722.

Voir la notice de l'article provenant de la source Math-Net.Ru

We study the dependence of the properties of the generating function of the solution of the Cauchy problem on the properties of the generating function of the initial data for a difference equation with constant coefficients in a rational point cone. Conditions are found under which the generating functions of the solution remain in the same classes as the generating functions of the initial data.
Keywords: multidimensional difference equations, Cauchy problem, hierarchy of generating function
Mots-clés : Hadamard composition. 
@article{JSFU_2018_11_6_a6,
     author = {Evgeny K. Leinartas and Tatiana I. Yakovleva},
     title = {The {Cauchy} problem for multidimensional difference equations and the preservation of the hierarchy of generating functions of its solutions},
     journal = {\v{Z}urnal Sibirskogo federalʹnogo universiteta. Matematika i fizika},
     pages = {712--722},
     publisher = {mathdoc},
     volume = {11},
     number = {6},
     year = {2018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JSFU_2018_11_6_a6/}
}
TY  - JOUR
AU  - Evgeny K. Leinartas
AU  - Tatiana I. Yakovleva
TI  - The Cauchy problem for multidimensional difference equations and the preservation of the hierarchy of generating functions of its solutions
JO  - Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
PY  - 2018
SP  - 712
EP  - 722
VL  - 11
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/JSFU_2018_11_6_a6/
LA  - en
ID  - JSFU_2018_11_6_a6
ER  - 
%0 Journal Article
%A Evgeny K. Leinartas
%A Tatiana I. Yakovleva
%T The Cauchy problem for multidimensional difference equations and the preservation of the hierarchy of generating functions of its solutions
%J Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
%D 2018
%P 712-722
%V 11
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/JSFU_2018_11_6_a6/
%G en
%F JSFU_2018_11_6_a6
Evgeny K. Leinartas; Tatiana I. Yakovleva. The Cauchy problem for multidimensional difference equations and the preservation of the hierarchy of generating functions of its solutions. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 11 (2018) no. 6, pp. 712-722. http://geodesic.mathdoc.fr/item/JSFU_2018_11_6_a6/

[1] M. Bousquet-Mélou, M. Petkovšek, “Linear recurrences with constant coefficients: the multivariate case”, Discrete Mathematics, 225 (2000), 51–75 | DOI | MR | Zbl

[2] T.I. Yakovleva, “Well-posedness of the Cauchy problem for multidimensional difference equations in rational cones”, Siberian Mathematical Journal, 58:2 (2017), 363–372 | DOI | MR | Zbl

[3] R.P. Stanley, Enumerative Combinatorics, Mir, M., 1990 (Russian) | MR

[4] R.P. Stanley, Enumerative Combinatorics. Trees, generating functions, symmetric functions, Mir, M., 2009 (in Russian) | MR

[5] E.K. Leinartas, T.I. Nekrasova, “Constant coefficient linear difference equations on the rational cones of the integer lattice”, Siberian Mathematical Journal, 57:2 (2016), 98–112 | MR | Zbl

[6] L. Bieberbach, Analytic continuation, Nauka, M., 1967 (in Russian) | MR

[7] E.K. Leinartas, “Hadamard's theorem on the multiplication of singularities in $\mathbb C^n$”, Sib. Mat. Zh., 27:3 (1986), 209–212 (in Russian) | MR

[8] E.K. Leinartas, M.S. Rogozina, “Solvability of the Cauchy problem for a polynomial difference operator and monomial bases for the quotients of a polynomial ring”, Siberian Mathematical Journal, 56:1 (2015), 111–121 | DOI | MR | Zbl

[9] A.O. Gelfond, The calculus of finite differences, KomKniga, M., 2006 (in Russian) | MR

[10] O.A. Shishkina, “Bernoulli Polynomials in Several Variables and Summation of Monomials over Lattice Points of a Rational Parallelotope”, Izvestiya Irkutsk Gos. Univ., 16 (2016), 89–101 (in Russian) | Zbl

[11] O.A. Shishkina, “Multidimensional Analog of the Bernoulli polynomials and its Properties”, Journal of Siberian Federal University, Mathematics $\$ Physics, 9:3 (2016), 384–392 | DOI

[12] L. Lipshitz, “D-Finite power series”, Journal of Algebra, 122 (1989), 353–373 | DOI | MR | Zbl

[13] M.S. Apanovich, E.K. Leinartas, “Correctness of a Two-dimensional Cauchy Problem for a Polynomial Difference Operator with Constant Coefficients”, Journal of Siberian Federal University. Mathematics $\$ Physics, 10:2 (2017), 199–205 | DOI | MR

[14] R.W.K. Odoni, “On the norms of algebraic integers”, Matematica, 22 (1975), 71–80 | MR | Zbl

[15] D.Ẑ. Djokoviĉ, “A properties of the Taylor expansion of rational function in several variables”, J. of Math. Anal. and Appl., 66 (1978), 679–685 | DOI | MR | Zbl

[16] J.H. Poincare, New Methods of Celestial Mechanics, v. 1, Nauka, M., 1971 (in Russian) | MR

[17] A.K. Tsikh, “Conditions for absolute convergence of the Taylor coefficient series of a meromorphic function of two variables”, Math. Sb., 182:11 (1991), 1588–1612 | MR | Zbl

[18] B.V. Shabat, An Introduction to Complex Analysis, 1969 (in Russian) | MR

[19] J.W.S. Cassels, An Introduction to the Geometry of Numbers, Mir, M., 1965 (in Russian) | MR | Zbl

[20] D. Yu. Pochekutov, “Diagonals of the Laurent series of rational functions”, Sib. Math. J., 50 (2009), 1081–1090 | DOI | MR

[21] T.I. Nekrasova, “On the hierarchy of generating functions for solutions of multidimensional difference equations”, Izvestiya Irkutsk Gos. Univ., 9 (2014), 91–103 (in Russian)