 Suppose that "y"0 = 1, "y"1, ... is a sequence of polynomials where "y""n" has degree "n".If this is a sequence of orthogonal polynomials for some positive weight function then it satisfies a 3-term recurrence relation.Favard's theorem is roughly a converse of this, and states that if these polynomials satisfy a 3-term recurrence relation of the form
for some numbers "c""n" and "d""n",
then the polynomials "y""n" form an orthogonal sequence for some linear functional Λ with Λ(1)=1; in other words Λ("y""m""y""n") = 0 if "m" ≠ "n".