Recurrence relations for the sections of the generating series of the solution to the multidimensional difference equation

Abstract

In this paper, we study the sections of the generating series for solutions to a linear multidimensional difference equation with constant coefficients and find recurrent relations for these sections. As a consequence, a multidimensional analogue of Moivre's theorem on the rationality of sections of the generating series depending on the form of the initial data of the Cauchy problem for a multidimensional difference equation is proved. For problems on the number of paths on an integer lattice, it is shown that the sections of their generating series represent the well-known sequences of polynomials (Fibonacci, Pell, etc.) with a suitable choice of steps.

Remarks on the Coefficients of Inverse Cyclotomic Polynomials

“…In future works, we aim to study the computational complexity of Formula ( 19) and to express the results in matrix language as has been calculated for polynomial sequences [23]. We also plan to explore practical applications, such as, for example, regarding difference equations [24].…”

On Higher-Order Generalized Fibonacci Hybrinomials: New Properties, Recurrence Relations and Matrix Representations

“…The Fibonacci sequence has many applications in various fields, including mathematics, physics and engineering. This has made it a subject of interest to many researchers [1,2]. The study of Fibonacci and Lucas numbers has been the subject of extensive research by mathematicians in the literature.…”