The roots of polynomials and the operator $\Delta _i^3$ on the Hahn sequence space h

Abstract

In this paper, we define the third order generalized difference operator $\Delta _i^3$, where

$$\begin{aligned} (\Delta _i^3x)_k=\sum _{i=0}^3\frac{(-1)^i}{i+1}\left( {\begin{array}{c}3\\ i\end{array}}\right) x_{k-i}= x_k-\frac{3}{2}x_{k-1}+x_{k-2}-\frac{1}{4}x_{k-3}, \end{aligned}$$

and show that it is a linear bounded operator on the Hahn sequence space h. Then we study the spectrum and point spectrum of the operator $\Delta _i^3$ on h. Furthermore, we determine the point spectrum of the adjoint of this operator. This is achieved by studying some properties of the roots of certain third order polynomials.