Expansion of $(a_{{0}}+a_{{1}}x+a_{{2}}{x}^{2}+a_{{3}}{x}^{3}+\cdots)^n$

Question

I am looking for a way to obtain the coefficient $c_k$ of $x^k$ in the expansion of $(a_{{0}}+a_{{1}}x+a_{{2}}{x}^{2}+a_{{3}}{x}^{3}+\cdots)^n$. I know it can be done by the multinomial theorem, but I am looking for an alternative expression.

It is easy to show by induction that the coefficient $c_k$ of $x^k$ is given by $c_k=\frac {\sum _{i=1}^{k} \left( in-k+i \right) a_{{i}}c_{{k-i}}}{ka_{{0}}}$.

However I wonder whether there exists a way to express $c_k$ in a closed form in the sense of not necessitating to calculate all of the preceding coefficients. I was hoping that someone here knew an answer to this.

Thank you very much in advance for any help.

Answer 1

Besides the representations via multinomial coefficients there are no other explicit expressions of the coefficients $[x^k]$ of $x^k$ of the $n$-the power of a general power series

\begin{align*} \left(\sum_{j=0}^{\infty}a_jx^j\right)^n \end{align*}

Your recurrence formula of $c_k$ is nice. It is also stated in the section Power series raised to powers of formal power series. Note, that in order to fully specify this recurrence, the initial condition $c_0=a^n$ is also stated as well as the restriction that the factors in the denominator have to be invertible.

Answer 2

Yes. For two powerseries, their product coefficient is readily defined in terms of their coefficients. Now iterate the calculation.