A distinction of k-hyponormal and weakly k-hyponormal weighted shifts

Let ?(x) : ?x/2, ?2/3, ?3/4, ?4/5, ... be a sequence with real variable x > 0 and let W?(x) the associated weighted shift weight ?(x). In [17], Exner-Jung-Park provided an algorithm to distinguish weak k-hyponormality of W?(x), obtained sn for some low numbers n = 4,..., 10, such that W?(sn) is weakly n-hyponormal but not n-hyponormal. this paper, we obtain formula (for all positive integer n) n-hyponormal, which improves Exner-Jung-Park?s result above.