## Orthogonality of the big -1 Jacobi polynomials for non-standard parameters

Cohl, H. S. and Costas-Santos, R. S.**Contemporary Mathematics**. Accepted, 2023

## 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.

## Download

Link | Size | Description |
---|---|---|

paper_34.pdf | 352 KB | PDF, 7 Pages |