Paper: Orthogonality of the big \(-1\) Jacobi polynomials for non-standard parameters

Orthogonality of the big \(-1\) Jacobi polynomials for non-standard parameters

Abstract

The big \(−1\) Jacobi polynomials \((Q^{(0)}(x; \alpha, \beta, c))_n\) have been classically defined for \(\alpha, \beta \in (-1, +\infty)\), \(c \in(-1, 1)\). We extend this family so that wider sets of parameters are allowed, i.e., they are non-standard. Assuming initial conditions \(Q^{(0)}_0(x) = 1\), \(Q^{(0)}_{-1}(x) = 0\), we consider the big \(-1\) Jacobi polynomials as monic orthogonal polynomials which therefore satisfy the following three-term recurrence relation \[ xQ^{(0)}_n(x) = Q^{(0)}_{n+1}(x) + b_nQ^{(0)}_n(x) + u_nQ^{(0)}_{n-1} (x), n = 0,1,2,.... n. \] For standard parameters, the coefficients \(u_n > 0\) for all \(n\). We discuss the situation where Favard’s theorem cannot be directly applied for some positive integer \(n\) such that \(u_n = 0\). 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.

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