Geodesic
Sections of the generating series of a solution to a difference equation in a simplicial cone
The Bulletin of Irkutsk State University. Series Mathematics, Tome 42 (2022), pp. 75-89

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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 $\mathbb Z^n$ 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.
Keywords: generating series, difference equation, lattice cone, Stanley hierarchy
Mots-clés : section. 
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     title = {Sections of the generating series of a solution to a difference equation in a simplicial cone},
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A. P. Lyapin; T. Cuchta. Sections of the generating series of a solution to a difference equation in a simplicial cone. The Bulletin of Irkutsk State University. Series Mathematics, Tome 42 (2022), pp. 75-89. http://geodesic.mathdoc.fr/item/IIGUM_2022_42_a5/