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

TL;DR

This paper extends the family of big $-1$ Jacobi polynomials to non-standard parameters, establishing their orthogonality through a novel factorization and bilinear form, even when classical conditions for orthogonality are not met.

Contribution

It introduces a broader parameter set for big $-1$ Jacobi polynomials and develops a new orthogonality framework via polynomial factorization.

Findings

Extended parameter range for big $-1$ Jacobi polynomials.

Established orthogonality through polynomial factorization.

Analyzed cases where classical orthogonality conditions fail.

Abstract

The big $-1$ Jacobi polynomials $(Q_n^{(0)}(x;\alpha,\beta,c))_n$ 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 \[ xQ^{(0)}_n(x)=Q^{(0)}_{n+1}(x)+b_{n} Q^{(0)}_n(x)+ u_{n} Q^{(0)}_{n-1}(x), \quad n=0, 1, 2,\ldots. \] 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.