Suppose that y0 = 1, y1, ... is a sequence of polynomials where yn 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 cn and dn, then the polynomials yn form an orthogonal sequence for some linear functional Λ with Λ(1)=1; in other words Λ(ymyn) = 0 if m ≠ n.
The linear functional Λ is unique, and is given by Λ(1) = 1, Λ(yn) = 0 if n > 0.
The functional Λ satisfies Λ(y2n) = dn Λ(y2n–1), which implies that Λ is positive definite if (and only if) the numbers cn are real and the numbers dn are positive.