Abstract
This paper is concerned with the deficiency indices of a polynomial say of a formally self-adjoint difference expression . Since may be not formally self-adjoint, we extend the definitions of the limit point case and limit circle case in a natural way to formally non-self-adjoint difference expressions. It is shown that is in limit-circle case if and only if is in limit-circle case, and if is in limit point case, then so is , where is any polynomial with its degree no more than that of . In addition, provided that is in limit point case, some conditions for to be in limit-point case are obtained.
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