# Coefficient of a polynomial defined by Legendre polynomial

1. Sep 14, 2015

### duc

1. The problem statement, all variables and given/known data
The polynomial of order $(l-1)$ denoted $W_{l-1}(x)$ is defined by
$W_{l-1}(x) = \sum_{m=1}^{l} \frac{1}{m} P_{m-1}(x) P_{l-m}(x)$ where $P_m(x)$ is the Legendre polynomial of first kind. In addition, one can also write
$W_{l-1}(x) = \sum_{n=0}^{l-1} a_n \cdot x^n$

Find the coefficient $a_n$ in terms of $n$ and $l$.

2. The attempt at a solution
I think the binomial form of $P_m(x)$ would help
$P_m(x) = 2^m \cdot \sum_{k=0}^{m} C^{k}_{m} C^{\frac{m+k-1}{2}}_{m} x^k$, with $C^{k}_{m} = \frac{m!}{k!(m-k)!}$. The next thing is to know "how to count" the number of terms in both expressions of $W_{l-1}(x)$. This is where I stuck at.

Last edited: Sep 14, 2015
2. Sep 14, 2015

### duc

I've found the solution which is of the following form

$a_n = \sum_{m=1}^{l} \frac{1}{m} \sum_{i=0}^{n} a_i^{(m-1)} a_{n-i}^{(l-m)}$

where a_i^(m-1) is the coefficient corresponding to the power $x^i$ of the polynomial $P_{m-1}(x)$ (the same convention for $P_{l-m}(x)$).