We find all polynomials solutions Pn(x) of the abstract "hypergeometric"
equation LPn(x)=λnPn(x), where L is a linear operator sending
any polynomial of degree n to a polynomial of the same degree with the
property that L is two-diagonal in the monomial basis, i.e. Lxn=λnxn+μnxn−1 with arbitrary nonzero coefficients λn,μn . Under obvious nondegenerate conditions, the polynomial eigensolutions
LPn(x)=λnPn(x) are unique. The main result of the paper is a
classification of all {\it orthogonal} polynomials Pn(x) of such type, i.e.
Pn(x) are assumed to be orthogonal with respect to a nondegenerate linear
functional σ. We show that the only solutions are: Jacobi, Laguerre
(correspondingly little q-Jacobi and little q-Laguerre and other special
and degenerate cases), Bessel and little -1 Jacobi polynomials.Comment: 20 page