 Composition coefficients of power series

ID: composition-coefficients-of-power-series

f (z) = \sum_{m \geq 0} f_{m} z^{m}

and

g (x) = \sum_{j \geq 1} g_{j} x^{j}

x^{n}

receives contributions only from

m \leq n

, since

g

has zero constant term. For

m \geq 1

, expand the coefficient as

[x^{n}] g (x)^{m} = \sum_{j_{1} + \dots + j_{m} = n j_{1}, \dots, j_{m} \geq 1} g_{j_{1}} \dots g_{j_{m}} .

The

m = 0

term is one for

n = 0

and zero otherwise. The first four coefficients of the function composition are therefore

f_{0}

f_{1} g_{1}

f_{1} g_{2} + f_{2} g_{1}^{2}

, and

f_{1} g_{3} + 2 f_{2} g_{1} g_{2} + f_{3} g_{1}^{3}

. For convergent power series, the same calculation holds wherever

g (x)

f

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