On binomial and circular binomial distributions of order k for l-overlapping success runs of length k

The number of l-overlapping success runs of length k in n trials, which was introduced and studied recently, is presently reconsidered in the Bernoulli case and two exact formulae 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 of l-overlapping success runs of length k in n Bernoulli trials arranged on a circle is presently considered for the first time and it probability distribution function and mean are derived. Finally, the latter distribution is related to the first, tow 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 binomially and circular binomially distributions of order k. [PUBLICATION ABSTRACT]