Equality of the Casoratian and the Wronsikian for some Sets of Functions and an Application in Difference Equations

Abstract

The Casoratian determinants are very important in the study of linear difference equation, just as the Wronskian determinants are very important in the study of linear ordinary differential equations. The Casoratian and Wronskian determinants of a given set of functions are generally different. In this paper, we show that the Wroskian $ W[1, x, x^2,...,x^n ]$ and the Casoratian $ C[1,x, x^2,...,x^n ] $ of the set of functions $ \{1, x, x^2,...,x^n: \, n \in \mathbb{N} \}$ are equal to each other and independent of the variable $x$. Furthermore, we show that $ W[p_1, p_2,...,p_n, p_{n+1} ]= C[p_1, p_2,..., p_n, p_{n+1} ]$ for any basis $ \mathcal{B} = \{ p_1, p_2,..., p_n, p_{n+1} \}$ of the vector space spanned by $ \{ 1, x, x^2,...,x^n \}$. We also explore applications of such Casoratians in deriving the general solutions of certain classes of homogeneous linear difference equations.Additionally, the paper discusses scenarios where the Casoratian and Wronskian are proportional, as well as cases where they differ for finite sets of functions.