Complete characterizations of an oscillatory neutral difference equation (Q1269561)

Consider the neutral difference equation \[ \Delta (x_n+px_{n-\tau}) +qx_{n-\sigma} =0,\;n=0,1, \dots \] where \(p,q\) are real numbers, \(\tau\) is a positive integer and \(\sigma\) is a nonnegative integer. By means of the theory of envelopes, the authors determine the exact range of the parameters \(\tau, \sigma,p\) and \(q\) for which the corresponding equation is oscillatory. For related results see the author's paper in J. Difference Equ. Appl. 2, No. 3, 301-313 (1996; Zbl 0884.39006).