Perturbations of Jacobi polynomials and piece-wise hypergeometric   orthogonal systems
  We construct noncomplete orthogonal systems on the ray $[0,\infty)$ 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. 
Yurii A. Neretin