Use Impulse Response Sequences in the Construction of Number Sequence Identities

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.

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