On the characterization theorem of semi-classical orthogonal polynomial sequences

Abstract

The purpose of this paper is to remove vagueness and inconsistence in the definition of semi-classical orthogonal polynomial sequences and semi-classical moment functionals. In chapter 3, the exact statement of the characterization theorem of semi-classical OPS is presented and proved. In chapter 4, we show that the number min($\alpha$ , $\beta$) max (deg $\alpha$-2, deg $\beta$-1) equals the smallest order of quasi-orthogonality of the polynomial sequence resulting from differenting semi-classical OPS relative to $\sigma$, where ($\alpha$ , $\beta$) runs over all polynomials such that $(\alpha\sigma)`` = \beta \sigma,\; \deg\; \alpha \ge 0,\; \deg\; $\beta \ge 1$.