- #1
duc
- 9
- 0
Homework Statement
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: