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

TL;DR

This paper studies higher-order Kravchuk-Sobolev polynomials, deriving explicit formulas, ladder operators, and second-order difference equations, expanding understanding of their algebraic and analytical properties.

Contribution

It introduces a new class of Kravchuk-Sobolev polynomials of higher order, providing explicit representations and associated ladder operators and difference equations.

Findings

Explicit polynomial representations derived.

Ladder operators constructed for the polynomials.

Second-order linear difference equations established.

Abstract

In this contribution we consider sequences of monic polynomials orthogonal with respect to Sobolev-type inner product \[ \left\langle f,g\right\rangle _{\lambda,\mu}\!=\!\sum_{x=0}^Nf(x)g(x)\frac{\Gamma(N+1) p^x(1-p)^{N-x} }{\Gamma (N-x+1) \Gamma(x+1) }+\lambda\Delta^j f(0)\Delta^j g(0)+\mu\Delta^j f(N)\Delta^j g(N), \] where $0<p <1$, $\lambda,\mu\in \mathbb R_{+}$, $n\leq N\in \mathbb Z_{+}$, $j\in \mathbb Z_{+}$ 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.