One way to define the Legendre polynomials is with the recurrence relation $$(n+1)P_{n+1} (x) = (2n+1)xP_{n} (x)-nP_{n-1} (x),$$ with $P_0(x)=1$ and $P_1(x)=x$. This standardization is normalized so that $P_n(1)=1$ for all $n$, and so that the polynomials are orthogonal on $[-1,1]$. I'm trying to form an iterative method to solve $Ax=b$, and one thing that came up is that I need the Legendre polynomials renormalized such that $P_n(0)=1$ instead of $P_n(1)=1$.
However I don't know how to do this. Simply dividing the polynomial by $P_n(0)$ for example is not very helpful, because half the degrees of the Legendre polynomial already have $P_n(0)=0$ and so I'd be dividing by zero.