Suppose that y₀ = 1, y₁, ... 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

${\displaystyle y_{n+1}=(x-c_{n})y_{n}-d_{n}y_{n-1}}$ 

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_my_n) = 0 if m ≠ n.

The linear functional Λ is unique, and is given by Λ(1) = 1, Λ(y_n) = 0 if n > 0.

The functional Λ satisfies Λ(y²
_n) = d_n Λ(y²
_n–1), which implies that Λ is positive definite if (and only if) the numbers c_n are real and the numbers d_n are positive.

• Jacobi operator
• James Alexander Shohat

• Chihara, Theodore Seio (1978), An introduction to orthogonal polynomials, Mathematics and its Applications, vol. 13, New York: Gordon and Breach Science Publishers, ISBN 978-0-677-04150-6, MR 0481884 Reprinted by Dover 2011, ISBN 978-0-486-47929-3
• Favard, J. (1935), "Sur les polynomes de Tchebicheff.", C. R. Acad. Sci. Paris (in French), 200: 2052–2053, JFM 61.0288.01
• Rahman, Q. I.; Schmeisser, G. (2002), Analytic theory of polynomials, London Mathematical Society Monographs. New Series, vol. 26, Oxford: Oxford University Press, pp. 15–16, ISBN 0-19-853493-0, Zbl 1072.30006
• Subbotin, Yu. N. (2001) [1994], "Favard theorem", Encyclopedia of Mathematics, EMS Press
• Shohat, J. (1938), "Sur les polynômes orthogonaux généralises.", C. R. Acad. Sci. Paris (in French), 207: 556–558, Zbl 0019.40503