D -Finite Discrete Generating Series and Their Sections

Svetlana S. Akhtamova, Vitaly S. Alekseev and Alexander P. Lyapin ()
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Svetlana S. Akhtamova: Lesosibirskij Pedagogical Institute–Branch of Siberian Federal University, 662544 Lesosibirsk, Russia
Vitaly S. Alekseev: School of Mathematics and Computer Science, Siberian Federal University, 660041 Krasnoyarsk, Russia
Alexander P. Lyapin: School of Mathematics and Computer Science, Siberian Federal University, 660041 Krasnoyarsk, Russia

Mathematics, 2025, vol. 13, issue 20, 1-13

Abstract: This paper investigates D-finite discrete generating series and their sections. The concept of D-finiteness is extended to multidimensional discrete generating series and its equivalence to p-recursive sequences is rigorously established. It is further shown that sections of the D-finite series preserve D-finiteness, and that their generating functions satisfy systems of linear difference equations with polynomial coefficients. In the two-dimensional case, explicit difference relations are derived that connect section values with boundary data, while in higher dimensions general constructive methods are developed for obtaining such relations, including cases with variable coefficients. Several worked examples illustrate how the theory applies to solving difference equations and analyzing multidimensional recurrent sequences. The results provide a unified framework linking generating functions and recurrence relations, with applications in combinatorics, number theory, symbolic summation, and the theory of discrete recursive filters in signal processing.

Keywords: D -finite series; p -recursive sequences; difference equations; series sections; multidimensional series; recurrence relations (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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