First, we provide characterizations of semi-classical orthogonal polynomials via structure relations of any order≥1. Then we also obtain a new characterization of classical orthogonal polynomials, which generalizes Al-Salam and Chihara's characterization of classical orthogonal polynomials. We discuss the structure of distributional or hyperfunctional representation of semi-classical moment functionals. Next, we find a necessary and sufficient condition for the two point masses perturbation of a quasi-definite moment functional to have orthogonal polynomials and investigate zero properties of the corresponding orthogonal polynomials. Semi-classical characters of such orthogonal polynomials are also discussed.
Secondly, we find the necessary and sufficient conditions for a second order difference equation of hypergeometric type:
α(x)△▽y(x)+β(x)△y(x)=$λ_ny(x)$
to have discrete classical orthogonal polynomials as solutions. By the conditions, we show that the restrictions on parameters for each discrete classical orthogonal polynomials can be relaxed when we consider orthogonal polynomials relative to quasi-definite moment functionals and also derive functional Rodrigues' formula for discrete classical orthogonal polynomials.
Finally, we discuss various characterizations of Hahn type orthogonal polynomials, which are first introduced by Hahn in 1949. He introduced a operator, now called Hahn operator:
δf(x):=$\frac{f(qx+ω)-f(x)}{(q-1)x+ω}$
which include the differential and finite difference operators as limiting cases, and operator equations which are satisfied by orthogonal polynomials. Here, we introduce new operator equations which are equivalent to Hahn operator equations and then obtain unified characterizations extending all results of classical and discrete classical orthogonal polynomials.
