Special Seminar by Associate Professor Dr. Alexander Lyapin, Department of Computational and Information Technology, Siberian Federal University.
Title: Multidimensional difference equations with constant coefficients, generating functions and applications in enumerative combinatorial analysis
Date: Thursday, May 26, 2022 Time: 9.00 – 10.00 AM.
Online: WebEx Meeting Link: https://mahidol.webex.com/mahidol/j.php?MTID=mdadf7fb1d2f4264426294d40c9852a3e
One-dimensional difference equations are well-studied and have various combinatorics and applied mathematics applications. However, the multidimensional case is more difficult and is a subject of study (M. Bouquet-Melou, M. Petkovsek, S. Abramov, R. Stanley, E. Leinartas, etc.).  Generating functions of solutions to difference equations can be found in terms of coefficients and initial data of the equations. In the one-dimensional case, the corresponding Cauchy problem has finite initial data, but in the multidimensional case, the initial data of the Cauchy problem is infinite.  R. Stanley considered a hierarchy of generating functions: rational, algebraic, and D-finite. In 1722 Moivre proved that generating function for one-dimensional difference equations is a rational function. In the multidimensional case, a type of generating function depends on the generating function of initial data. We have found closed formulas of the generating function of a solution to the multidimensional difference equation with constant coefficients and applied them to problems of enumerative combinatorial analysis, one of them is the integer lattice paths problem (Dyck, Motzkin, and Schroeder paths).
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