AN INEQUALITY FOR FINITE-DIFFERENCES VIA ASYMPTOTICS OF RICCATI MATRIX DIFFERENCE-EQUATIONS

The paper deals with the Riccati matrix difference equation, which cor responds to the Sturm-liouville equation of even order, namely: (-1)(n )Delta(2n)y(k+1-n) = lambda y(k+1) with real parameter lambda. The mai n theorem states, in essence, the asymptotic expansion of the principa l solution of Riccati's equation as lambda --> -infinity in terms of n egative powers of lambda. Via a discrete Picone type identity the asym ptotic result is used to derive an inequality for a discrete quadratic functional containing n(th) order differences.