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

The big $-1$ Jacobi polynomials $(Q_n^{(0)}(x;\alpha ,\beta ,c){)}_n$ have been classically defined for $\alpha ,\beta \in (-1,\mathrm{\infty })$, $c\in (-1,1)$. We extend this family so that wider sets of parameters are allowed, i.e., they are non-standard. Assuming initial conditions $Q_0^{(0)}(x)=1$, $Q_{-1}^{(0)}(x)=0$, we consider the big $-1$ Jacobi polynomials as monic orthogonal polynomials which therefore satisfy the following three-term recurrence relation $$x{Q}_n^{(0)}(x)=Q_{n+1}^{(0)}(x)+b_n{Q}_n^{(0)}(x)+u_n{Q}_{n-1}^{(0)}(x),n=0,1,2,\dots \mathrm{.}$$ 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.