On binomial and circular binomial distributions of orderk forl-overlapping success runs of lengthk

The number ofl-overlapping success runs of lengthk inn trials, which was introduced and studied recently, is presently reconsidered in the Bernoulli case and two exact formulas are derived for its probability distribution function in terms of multinomial and binomial coefficients respectively. A recurrence relation concerning this distribution, as well as its mean, is also obtained. Furthermore, the number ofl-overlapping success runs of lengthk inn Bernoulli trials arranged on a circle is presently considered for the first time and its probability distribution function and mean are derived. Finally, the latter distribution is related to the first, two open problems regarding limiting distributions are stated, and numerical illustrations are given in two tables. All results are new and they unify and extend several results of various authors on binomial and circular binomial distributions of orderk.