Journal of the
Korean Mathematical Society JKMS

Let $\rn$ be a discrete Sobolev orthogonal polynomials (DSOPS) relative to a symmetric bilinear form \begin{equation*}\label{eq0.1} \phi (p,q) = \int_{\mathbb R} pq d\mu_0 + \int_{\mathbb R} \Delta p \Delta q \, d\mu_1 , \end{equation*} where $d\mu_0$ and $d\mu_1$ are signed Borel measures on $\mathbb R$. We find necessary and sufficient conditions for $\rn$ to satisfy a second order difference equation \begin{equation*}\label{eq0.2} \ell_2(x) \Delta \nabla y(x) + \ell_1(x) \Delta y(x) = \lambda_n y(x) \end{equation*} and classify all such $\rn$. Here, $\Delta$ and $\nabla$ are forward and backward difference operators defined by $\Delta f(x) = f(x+1) - f(x)$ and $\nabla f(x) = f(x) - f(x-1)$.

Keywords: second order difference equations, discrete Sobolev orthogonal polynomials

MSC numbers: 33C45