Perturbations of Jacobi polynomials and piece-wise hypergeometric orthogonal systems

We construct noncomplete orthogonal systems on the ray $[0,\mathrm{\infty })$ that look like Jacobi polynomials $P_n(x)$ after a shift of degree $n\mapsto n+a$, where $a$ is a real constant. These systems are solutions of some exotic Sturm-Liouville problem for hypergeometric differential operators. We obtain the explicit spectral decomposition for these problems.