Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I have $n$ power series. How can I find the power series of the product of these $n$ series? If there are two series $(a_m)$ and $(b_m)$ then the product series $(c_m)$ is given by the Cauchy product,

$$c_m = \sum_{k=0}^m a_k b_{m-k}$$

How does this generalize to more series?

share|cite|improve this question
Please do a little research before asking a broad non specific question such as the one above. have you tried any multiplication? can you add some more detail please. – Arjang Jun 15 '11 at 9:11
The question needed a little cleaning up, but it was a long way from a 'broad non specific question' and it shows clear evidence of some research. I've edited to make it more obvious what's being asked. In general I think the first response to a question like this should be to edit for clarity rather than downvote. – Chris Taylor Jun 15 '11 at 9:20
@Chris, It is your cleaned upversion that deservesthe the +1 not the original question. – Arjang Jun 15 '11 at 10:00
@Arjang: I found the OP's original question to be clear and specific. – Pete L. Clark Jun 15 '11 at 12:21
@Pete @Chris, cool then my bad. I should have commented that I could'nt make sense of it and nothing towards the OP. – Arjang Jun 15 '11 at 21:24

$$\left(\sum_{n=0}^\infty a_{1,n} x^n\right)\left(\sum_{n=0}^\infty a_{2,n} x^n\right)\dots\left(\sum_{n=0}^\infty a_{l,n} x^n\right) = \sum_{n=0}^\infty \sum_{k_1 + k_2 + \dots + k_l = n } a_{1,k_1}a_{2,k_2}\dots a_{l,k_l} x^n $$

share|cite|improve this answer

Presumably you are looking for an expression for the coefficients of $x^i$ in the product

$$\left(\sum_j a_j x^j \right) \left(\sum_k b_k x^k \right) \left(\sum_l c_l x^l \right) \ldots \left(\sum_m n_m x^m \right)$$

Generalising the Cauchy product, this is

$$\sum_i \left(\sum_{j=0}^{i} \sum_{k=0}^{i-j} \sum_{l=0}^{i-j-k} \cdots a_j b_k c_l \ldots n_{i-j-k-l-\cdots} \right) x^i$$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.