Applications of Fuss-Catalan numbers to success runs of Bernoulli trials (Q1657992)

Summary: In a recent paper, the authors derived the exact solution for the probability mass function of the geometric distribution of order \(k\), expressing the roots of the associated auxiliary equation in terms of generating functions for Fuss-Catalan numbers. This paper applies the above formalism for the Fuss-Catalan numbers to treat additional problems pertaining to occurrences of success runs. New exact analytical expressions for the probability mass function and probability generating function and so forth are derived. First, we treat sequences of Bernoulli trials with \(r \geq 1\) occurrences of success runs of length \(k\) with \(\ell\)-overlapping. The case \(\ell < 0\), where there must be a gap of at least \(\left|\ell\right|\) trials between success runs, is also studied. Next we treat the distribution of the waiting time for the \(r\)th nonoverlapping appearance of a pair of successes separated by at most \(k - 2\) failures (\(k \geq 2\)).