Criteria for right disfocality of linear difference equations (Q1082550)

Let be \(I=\{a,a+1,...,b\}\), \(n\geq 1\), and \(I^ j=\{a,a+1,...,b+j\}\), \(j=0,1,...,n\). Define the n-th order linear difference equation by (1) \(Pu(m)=\sum^{m}_{j=0}\alpha_ j(m)u(m+j)\), where the independent variable m ranges over I, \(\alpha_ n(m)\equiv 1\), \(\alpha_ 0(m)\neq 0\) on I, and the coefficients \(\alpha_ j(m)\), \(0\leq j\leq n\), are defined on I. The author defines the notion of right disfocality and gives criteria for right disfocality of linear difference equation (1). The main result of this paper refers to the equivalence of the following properties of equation (1): (a) (1) is right disfocal on \(I^ n\); (b) (1) has a D-Markov system of solutions \(u_ 1,u_ 2,...,u_ n\), on \(I^ n\) satisfying the partial set of initial conditions \(\Delta^{i-1} u_ k(a)=0\), \(1\leq i\leq n-k\), \((-1)^{k-1} \Delta^{n-k} u_ k(a)>0\), \(1\leq k\leq n\); (c) (1) has a D-Fekete system of solutions on \(I^ n\); (d) (1) has a D-Descartes system of solutions on \(I^ n\); (e) u(m)\(\equiv 0\) is the only solution of (1) such that for each \(0\leq k\leq n-1\), \(u(a)=...=\Delta^{n-k-1} u(m)=0\), \(\Delta^{n-k+1} u(m)=...=\Delta^{n-1} u(m)=0\), \(a+1\leq m\in I^ 1\), and \(\Delta^{n- k}u\) has a node at \(\mu\) for some \(\mu\in \{a,...,m\}\).