Criterion of \(p\)-criticality for one term \(2n\)-order difference operators.
The self-adjoint even order difference equation \((-1)^n\Delta ^n(r_k\Delta ^n y_k)+q_ky_{k+n}=0\), \(k\in \mathbb {Z}\) with a positive sequence \(r_k\) is investigated. The concept of \(p\)-criticality, \(p\in \{0,\dots ,n\}\), of the one term difference operator \(l(x)_k:=\Delta ^n(r_k\Delta ^n x_k)\) is introduced, which ``measures'' linear dependence of the recessive system of solutions \(l(y)=0\) at \(-\infty \) and \(\infty \). It is shown, among others, that the number \(p\) in the definition of \(p\)-criticality depends on the convergence/divergence of the infinite series \(\sum _{k=0}^\infty k^{2p}r_k^{-1}\), \(\sum _{k=-\infty }^{0}k^{2p}r_k^{-1}\).
scientific article; zbMATH DE number 6054250