On the Total Score for a Negative-Binomially Roll of a Truncated Turn-Up Side of (v-u+1) - Sided Die

Let u,v∈N(u<v) and {T(x,y)=p^x q^y:x,y=u,u+1,u+2,u+3,…,v} be a string of sequence of success-failure events constituting the Bernoulli trials, with success p and failure q. Several probability distributions have derived their roots from the sequences of this form. However, it is our purpose to introduce new probability distribution functions that unify some of the existing ones generated by sets of this form mentioned in the literature and then give some of the statistics associated with it.