I am aware of some famous families of polynomials, for example Bessel, Jacobi, Chebyshev, Hermite and so on. They have one thing in common : they are all ON bases $\{P_0,P_1,\cdots\}$ with respect to their own specific integral inner product $\langle \cdot,\cdot \rangle_g$. In other words, they have their own characteristic $g$ function, so that:
$$\langle P_n,P_m \rangle_g =\int_{-\infty}^\infty P_n(t)\cdot P_m(t)\cdot g(t) dt\\[0.75cm]\text{so that}\\[0.5cm]\langle P_n,P_m\rangle _g = \delta_{n-m}$$
Now to my question : given some known $g$, what methods exist to numerically calculate new such families of polynomials $\{P(k)\}$.