The boundary value problem (BVP) for the second order difference equation
with and was recently discussed by the authors [see J. Ji and B. Yang, Linear Algebra Appl. 420, 218–227 (2007; Zbl 1113.39021)]. In this paper, they extend their earlier results to the special case . The difficulty lies in the singularity of some matrix. A new approach is developed to handle the case. As in their earlier paper, they obtain the complete structure of the eigenvalues of the BVP
and describe the monotonic behavior of all eigenvalues as the coefficients and change.