Chebyshev polynomials of the second kind via raising operator preserving the orthogonality

It is well known that monic orthogonal polynomial sequences and , the Chebyshev polynomials of the first and second kind, satisfy the relation (). One can also easily check that the following "inverse" of the mentioned formula holds: (), where with being an arbitrary nonzero parameter and representing the identity operator. Note that whereas the first expression involves the operator D which lowers the degree by one, the second one involves which raises the degree by one (i.e. it is a "raising operator"). In this paper it is shown that the scaled Chebyshev polynomial sequence where , is actually the only monic orthogonal polynomial sequence which is -classical (i.e. for which the application of the raising operator turns the original sequence into another orthogonal one).