On difference equations of Kravchuk-Sobolev type polynomials of higher order

Abstract

In this contribution we consider sequences of monic polynomials orthogonal with respect to Sobolev-type inner product \[ \left\langle f,g\right\rangle _{λ,μ}\!=\!\sum_{x=0}^Nf(x)g(x)\frac{Γ(N+1) p^x(1-p)^{N-x} }{Γ(N-x+1) Γ(x+1) }+λΔ^j f(0)Δ^j g(0)+μΔ^j f(N)Δ^j g(N), \] where $0<p<1$, $\lambda ,\mu \in {\mathbb{R}}_{+}$, $n\le N\in {\mathbb{Z}}_{+}$, $j\in {\mathbb{Z}}_{+}$ and $\Delta$ denotes the forward difference operators. We derive an explicit representation for these polynomials. In addition, the ladder operators associated with these polynomials are obtained. As a consequence, the linear difference equations of second order are also given.

11 pages