Discrete Generating Series and Linear Difference Equations

TL;DR

The paper develops a theory of discrete generating series for functions on integer lattices and links them to linear difference equations with constant or polynomial coefficients. By introducing polynomial shift operators and projection techniques, it derives functional equations that encode the Cauchy problem in both one- and multi-dimensional settings, and extends the Stanley $D$-finite framework to discrete generating series, proving that $D$-finiteness is equivalent to polynomial recursiveness. The multidimensional generalization uses componentwise forward differences and projection operators to formulate and solve Cauchy problems for constant- and polynomial-coefficient difference equations. Key examples, including Tribonacci and Schröder's second problem, illustrate the method and the structure of the resulting equations. The framework promises applications in combinatorics, lattice-path analysis, and digital signal processing, with potential extensions to time scales and broader multidimensional recurrences.

Abstract

We define discrete generating series for arbitrary functions \( f \colon \mathbb{Z}^n \rightarrow \mathbb{C} \) and derive functional relations that these series satisfy. For linear difference equations with constant coefficients, we establish explicit functional equations linking the generating series to the initial data, and for equations with polynomial coefficients, we introduce an analogue of Stanley's \( D \)-finiteness criterion, proving that a discrete generating series is \( D \)-finite if and only if the corresponding sequence is polynomially recursive. The framework is further generalized to multidimensional settings, where we investigate the interplay between discrete generating series and solutions to Cauchy problems for difference equations. Key structural properties are uncovered through the introduction of polynomial shift operators and projection techniques. The theory is illustrated with concrete examples, including the Tribonacci recurrence and Schröder's second problem.

Figures (1)

• Figure 1: Illustration of the sets $x \geqslant m$, $x \leqslant m$, and $x\not\geqslant m$.

Theorems & Definitions (23)

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