Restricted Weighted Integer Compositions and Extended Binomial Coefficients

We prove a simple relationship between extended binomial coefficients — natural extensions of the well-known binomial coefficients — and weighted restricted integer compositions. Moreover, we give a very useful interpretation of extended binomial coefficients as representing distributions of sums of independent discrete random variables. We apply our results, e.g., to determine the distribution of the sum of k logarithmically distributed random variables, and to determining the distribution, specifying all moments, of the random variable whose values are part-products of random restricted integer compositions. Based on our findings and using the central limit theorem, we also give generalized Stirling formulae for central extended binomial coefficients. We enlarge the list of known properties of extended binomial coefficients.