ORTHOGONAL POLYNOMIALS AND A DISCRETE BOUNDARY-VALUE PROBLEM-II

Let {P(n)}n=0 infinity be a system of polynomials orthogonal with respect to a measure-mu on the real line. Then P(n) satisfy the three-term recurrence formula xP(n) = gamma(n)P(n+1) + beta(n)P(n) + alpha(n)P(n-1). Conditions are given on the sequence alpha(n), beta(n), and gamma(n) under which any product P(n)P(m) is a linear combination of P(k) with positive coefficients. The result is applied to the measures d-mu(x) = (1-x2)alpha Absolute value of x 2-beta + 1 dx and d-mu(x) = Absolute value of x 2-alpha + 1 e-x2 dx, alpha, beta > -1. As a corollary, a Gasper result is derived on the Jacobi polynomials P(n)(alpha,beta) with alpha greater-than-or-equal-to beta and alpha + beta + 1 greater-than-or-equal-to 0.