Generating Function by Power of Parameter

Theorem

Let $\map G z$ be the generating function for the sequence $\sequence {a_n}$.

Let $m \in \Z_{\ge 0}$ be a non-negative integer.

Then $z^m \map G z$ is the generating function for the sequence $\sequence {a_{n - m} }$.

$\ds z^m \map G z$

$=$

$\ds z^m \sum_{n \mathop \ge 0} a_n z^n$

Definition of Generating Function

$\ds $

$=$

$\ds \sum_{n \mathop \ge 0} a_n z^{n + m}$

$\ds $

$=$

$\ds \sum_{n + m \mathop \ge 0} a_{n - m} z^n$

$\ds $

$=$

$\ds \sum_{n \mathop \ge m} a_{n - m} z^n$

By letting $a_n = 0$ for all $n < 0$:

$z^m \map G z = \ds \sum_{n \mathop \ge 0} a_{n - m} z^n$

Hence the result.

$\blacksquare$

1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.9$: Generating Functions: $(3)$