Authors
Cohl, H.S. , COSTAS SANTOS, ROBERTO SANTIAGO
Abstract
The big -1 Jacobi polynomials (Qn(0) (x; a, ß, c))n have been classically defined for a, ß ? (-1,8), c ? (-1, 1). We extend this family so that wider parameter values are allowed, i.e., the parameters may be non-standard. Assuming initial conditions Q0(0) (x) = 1, Q-1(0) (x) = 0, we consider the big -1 Jacobi polynomials as monic orthogonal polynomials which satisfy the threeterm recurrence relation xQn(0) (x) = Qn+1(0)(x) + bnQn(0) (x) + unQn-1(0)(x), n = 0, 1, 2,…. For standard parameters, the coefficients un are positive for all n. We discuss the situation when Favard’s theorem cannot be directly applied, as there is some positive integer n such that un = 0. We express the big -1 Jacobi polynomials for non-standard parameters as a product of two polynomials. Using this factorization, we obtain a bilinear form with respect to which these polynomials are orthogonal. © 2025 American Mathematical Society.
