The largest missing value in a composition of an integer

Archibald and Knopfmacher recently considered the largest missing value in a composition of an integer and established the mean and variance. Our alternative, probabilistic approach produces (in principle) all moments in an almost automatic way. In order to show that our forms match the ones given by Archibald and Knopfmacher, we have to derive some identities which are interesting on their own. We construct a one-parameter family of identities, and the first one is (equivalent to) the celebrated identity due to Allouche and Shallit. We finally provide a simple direct analysis of the LMV(−1) case: if the largest missing value is exactly one smaller than the largest value, we say that the sequence has the LMV(−1) property.