can Legendre polynomials take on different forms?

My prof. assigned some homework that has us compute legendre polynomials, but I'm getting polynomials that are different from ones that I reference with on like Wolfram and Wikipedia. I think the inner product given to work with is the issue: the integrand of factors is f(x)*g(x), where neither one of them is conjugated.

I think typically one of those factors is conjugated.

The set of polynomials I got, after using Gram-Schmidt is certainly an orthogonal set (I checked this). My question is: is this set also referred to as "legendre" polynomials, even though they're are different from the ones I am finding on reference websites?

Is the only requirement that these polynomials be mutually orthogonal?

Also, what about the magnitude? We were not asked to normalize any of these polynomials. Is a set of legendre polynomials necessarily orthonormal, too?

Thanks,

$$[(1-x^2) P_n'(x)]' = -n(n+1)P_n(x)$$
with $P_n(1) = 1, n = 0, 1, 2, \ldots$.