Find the first $3$ orthogonal polynomials of Laguerre (in relation to $\langle f,g\rangle = \int_0^{\infty} f(x)g(x)e^{-x}\ dx$)
I don't know which definition I should use for the Laguerre polynomials and if my definition generate orthogonal polynomials.
I've found this recursive definition:
$$L_0 = 1, L_1 = 1-x\\ $$
$$ L_{k+1}(x)=\frac{(2k+1-x)L_{k}(x)-kL_{k-1}(x)}{k+1}$$
So I can easily get the third one. However I don't know if these polynomials are orthogonal because the wikipedia article mentions that the generalized Laguerre polynomials are orthogonal and I don't think these are generalized. Also, I didn't use the dot product from the exercise. So I guess my way is not what my teacher intended for me to do.