A formula for the coefficients of orthogonal polynomials from the three-term recurrence relations

In this work, the coefficients of orthogonal polynomials are obtained in closed form. Our formula works for all classes of orthogonal polynomials whose recurrence relation can be put in the form Rn(x) = x Rn−1(x) − αn−2 Rn−2(x). We show that Chebyshev, Hermite and Laguerre polynomials are all members of the class of orthogonal polynomials with recurrence relations of this form. Our formula unifies the previously known formulas for the coefficients of these familiar polynomial families. c © 2005 Elsevier Ltd. All rights reserved.