Restricted Weighted Integer Compositions and Extended Binomial Coefficients
School of Computer Science
Carnegie Mellon University
Pittsburgh, PA 15213
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
determine 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.
Full version: pdf,
(Concerned with sequence
Received April 16 2012;
revised versions received April 23 2012; September 7 2012; October
13 2012; January 3 2013.
Published in Journal of Integer Sequences, January 4 2013.
Journal of Integer Sequences home page