Use Impulse Response Sequences in the Construction of Number Sequence Identities

TL;DR

This paper introduces impulse response sequences for linear recurrence relations, deriving their properties and identities, and applies them to analyze Stirling numbers, the Wythoff array, and the Boustrophedon transform.

Contribution

It defines impulse response sequences for linear recurrences, establishes their identities, and demonstrates their applications to various combinatorial structures.

Findings

Derived generating functions and explicit expressions for impulse response sequences.

Established identities including nonlinear expressions for these sequences.

Applied impulse response sequences to analyze Stirling numbers, Wythoff array, and Boustrophedon transform.

Abstract

We define impulse response sequence in the set of all linear recurring sequences satisfying a linear recurrence relation of order $r$. The generating function and expression of the impulse response sequence are presented. Some identities of impulse response sequences including a type of nonlinear expressions are established. The interrelationship between the impulse response sequence and other linear recurring sequences in the same set is given, which is used to transfer the identities of impulse response sequences to those of the linear recurring sequences in the same set. Some applications of impulse response sequences to the structure of Stirling numbers of the second order, the Wythoff array, and the Boustrophedon transform are studied.