Generating Function of Sequence by Index

Theorem

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

Then:

$z \map {G'} z$ is the generating function for the sequence $\sequence {n a_n}$

where $\map {G'} z$ is the derivative of $\map G z$ with respect to $z$.

Proof

\(\ds \map {G'} z\)

\(=\)

\(\ds \sum_{k \mathop \ge 0} \paren {k + 1} a_{k + 1} z^k\)

Derivative of Generating Function

\(\ds \leadsto \ \ \)

\(\ds z \map {G'} z\)

\(=\)

\(\ds \sum_{k \mathop \ge 0} \paren {k + 1} a_{k + 1} z^{k + 1}\)

\(\ds \)

\(=\)

\(\ds \sum_{k + 1 \mathop \ge 0} k a_k z^k\)

Translation of Index Variable of Summation

\(\ds \)

\(=\)

\(\ds \sum_{k \mathop \ge 0} k a_k z^k\)

as the term vanishes when $k = 0$

The result follows by definition of generating function.

$\blacksquare$

Sources

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