Explanation of the composition of two formal power series

Question

My university lecturer has given the following definition of the composition of two formal power series:

Let $F(X)=\sum_{n=0}^\infty a_nX^n$ with $a_0=0$ and $G(X)=\sum_{n=0}^\infty b_nX^n$.
We define $$G\circ F(X)=G(F(X))=\sum_{n=0}^\infty c_nX^n,$$ where the coefficients $c_n$ are as follows:

• There is a sequence $({d_k}^{(n)})_{k\geq0}$ such that $$\sum_{k=0}^\infty {d_k}^{(n)}X^k=\sum_{k=0}^n b_k(F(X))^k.$$
• Set $c_n={d_n}^{(n)}$.

My questions are:

• Firstly, does the notation ${d_k}^{(n)}$ refer to the $n^{th}$ derivative of $d_k$, or the $n^{th}$ power of $d_k$? (I think it refers to the latter; however, the use of notation seems ambiguous to me.)
• Secondly, how do we know that the sequence $({d_k}^{(n)})_{k\geq0}$ exists?
• Finally, why must we have $a_0=0$?

Answer 1

The composition is just the composition; that is, it's

$$G(F(x)) = \sum_{n=0}^{\infty} b_n F(x)^n.$$

In order for this to be well-defined as a formal power series we need $a_0 = 0$ so that $x^n | b_n F(x)^n$, which ensures that each term $b_n F(x)^n$ only contributes finitely many terms to the coefficient of $x^k$ in the composition. Otherwise, for example, $G(F(0))$ may be a sum of infinitely many terms. Said another way, the condition that $a_0 = 0$ guarantees that the above sum converges in the $x$-adic topology.

Answer 2

Firstly, does the notation ${d_k}^{(n)}$ refer to the $n^{th}$ derivative of $d_k$, or the $n^{th}$ power of $d_k$? (I think it refers to the latter; however, the use of notation seems ambiguous to me.)

It refers to none of those. The author just wants to put an $n$ in somewhere to indicate that for each $n$ one has a different series with coefficients $d_k$; since the subscript position was already taken, and unadorned superscript would have meant exponentiation, he went for a parenthesised superscript. Forgetting that that may sometimes means repeated derivative, so it was not really available either. In any case it would be clearer to write $d^{(n)}{}_k$ to indicate that is is coefficient $k$ of $d^{(n)}$. Personally I might have written $(d_n)_k$ to indicate that this is coefficient $k$ of the series $d_n$ that serves as auxiliary to define (the coefficient) $c_n$.

Secondly, how do we know that the sequence $({d_k}^{(n)})_{k\geq0}$ exists?

Because the right hand side is a finite sum of power series in $X$, and we are just taking the sequence of its coefficients.

Finally, why must we have $a_0=0$?

Because otherwise this whole set-up is pointless (though strictly speaking well defined). Note that the right hand side $\sum_{k=0}^n b_kF^k$ (writing $F(X)$ for the series $F$ is only confusing) is just the "infinite sum" of series $\sum_{k\geq0} b_k F^k$ truncated to its initial $n+1$ terms. In algebra infinite sums are suspect (not defined in general) but taking just $n+1$ terms (each of which are power series, and we know how to add two of those) is certainly a valid operation. But you will notice that only the coefficient $n$ of the resulting series is being used (to define $c_n$) the rest of this infinite series is just thrown out of the window, which is kind of a weird thing to do. But look what happens when $a=0$ ($a$ is the constant term of $F$). Then each power $F^k$ is divisible by $X^k$, so it is a power series that starts with zero terms, the first (possibly) nonzero coefficient being that of $X^k$. This means that the change from the series $d_n$ to $d_{n+1}$ will not change anything up to and including the coefficient of$~X^n$ that we used to define $c_n$; that coefficient will also be the coefficient of $X^n$ in the series $d_{n+1}$, and in all series formed after that. What is going on is really developing the infinite sum $\sum_{k\geq0} b_k F^k$, while "harvesting" its coefficients at the point where they become ripe, that is, when they are out of reach of any of the not yet contributed terms.

If one would do this with $a\neq0$, then there would be no point were the coefficients stabilise to a final value, and therefore no reasonable meaning to the infinite sum $\sum_{k\geq0} b_k F^k$. One may of course still pick of the coefficient of $X^n$ after $n+1$ terms have been contributed, but there would be no reason that is meaningful, and it would certainly not give a candidate for the value of $\sum_{k\geq0} b_k F^k$ (such a value is in fact not defined in this case).

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The whole story could be given more easily if one would consider infinite sums of power series in a more general setting. No meaning can be given to infinite sums in general, but as long as for each $k\geq0$ the coefficient of $X^k$ in the terms of the sum eventually becomes $0$, one can define the coefficient of $X^k$ in the sum to be that coefficient once all terms that could contribute to it have been included, and that gives a good definition of the infinite sum for this case. Such sums may be called convergent, and (unlike in analysis) there is a very simple condition for convergence of infinite sums: they converge if and only if their terms tend to $0$. Which means exactly that each coefficient eventually become $0$, as indicated above.

Once this is established, one can simply define for any series $G$, and for any $F$ without constant term: $$ G\circ F = \sum_{k\geq0}b_kF^k \qquad\text{where $G=\sum_ib_iX^i$.} $$