How to find a set of polynomials that are orthonormal with respect to this inner product?

Question

How to find a set of polynomials $P_j(x)$ of degree $j$, for $j=0,1,2$, that are orthonormal with respect to the inner product $$\langle f,g\rangle =\int_0^{+\infty}e^{-t}f(t)\overline{g}(t)\,dt\,?$$

Answer 1

You should use the Gram–Schmidt process: Take any three linearly independent polynomials, for example $1,x,x^2$. Now apply the process to this set. $$\|1\|^2=\langle1,1\rangle=\int_0^\alpha e^{-t}dt=1-e^{-\alpha}\hspace{5pt}\Rightarrow\hspace{5pt} p_0(x)=\frac{1}{\|1\|}=\frac{1}{\sqrt{1-e^{-\alpha}}}$$ $$proj_{p_0}(v_1)=\langle v_1,p_0 \rangle p_0=p_0\cdot\langle x,p_0 \rangle=\frac{1}{1-e^{-\alpha}}\int_0^\alpha e^{-t}tdt$$ and then $u_1=v_1-proj_{p_0}(v_1)$ and $p_1=\frac{u_1}{\|u_1\|}$ and so on.

Answer 2

If $\alpha=\infty$ you will obtain the Laguerre-polynomials.