Archive for March, 2017

On Shifted Legendre Polynomial Coefficients

March 27th, 2017 No comments

So here it is: the shifted Legendre polynomial coefficients are:

 a(k-1,n) = a_{k-1}^n = (-1)^k k \left( \prod_{m \neq 1} \frac{m-1}{m-n-1} \prod_{m \neq k} \frac{m-n-1}{m-k} \prod_{m \neq n} \frac{m-n-k}{m-n-1} \right)

with k \leq n and m = 1 \ldots n, so that

\tilde{P}_n(x) = \sum_{i = 0}^n a_i \cdot x^i

I'd love to hear your ideas of a proof.  Mine is a very particular thorny meticulous one. I may be off on the sign, but the absolute value is right. Let me know!