Journal of the
Korean Mathematical Society JKMS

A pair of quasi-definite linear functionals $\{u_0,u_1\}$ is a generalized $\Delta$-coherent pair if monic orthogonal polynomials $$\{P_n(x)\}_{n=0}^\infty$$ and $$\{R_n(x)\}_{n=0}^\infty$$ relative to $u_0$ and $u_1$, respectively, satisfy a relation $$ R_n(x) = \frac{1}{n+1}\Delta P_{n+1}(x)-\frac{\sigma_n}{n}\Delta P_n(x)- \frac{\tau_{n-1}}{n-1}\Delta P_{n-1}(x), ~~ n\geq 2,$$ where $\sigma_n$ and $\tau_n$ are arbitrary constants and $\Delta p=p(x+1)-p(x)$ is the difference operator. We show that if $\{u_0,u_1\}$ is a generalized $\Delta$-coherent pair, then $u_0$ and $u_1$ must be discrete-semiclassical linear functionals. We also find conditions under which either $u_0$ or $u_1$ is discrete-classical.

Keywords: discrete orthogonal polynomials, $\Delta$-coherent pairs

MSC numbers: 33C45