We give a combinatorial proof that k l-k-1 l + t q q q q a polynomial in q with nonnegative coefficients for nonnegative integers a, b, k, lwith a>~b and l~>k. In particular, for a=b=n and l=k, this implies the q-log-concavity of the Gaussian binomial coefficients k , which was conjectured q by BUTLER (Proc.

This paper presents lemmas and its corollaries on the combinatorial geometric series summation of binomial coefficients. Also, coefficient for each term in refers to a coefficient. These ideas can enable scientific researchers solve real life problems.

We derive asymptotic formulae for central polynomial coefficients, a generalization of binomial coefficients, using the distribution of the sum of independent uniform random variables and the CLT. 1 Stirling’s formula and binomial coefficients For a nonnegative integer m, Stirling’s formula m! ∼ √ 2πm (m e )m , where e is Euler’s number, yields an approximation of the central binomial coefficie...

As in cross sectional studies, longitudinal studies involve non-Gaussian data such as binomial, Poisson, gamma, and inverse-Gaussian distributions, and multivariate exponential families. A number of statistical tools have thus been developed to deal with non-Gaussian longitudinal data, including analytic techniques to estimate parameters in both fixed and random effects models. However, as yet ...

Inhomogeneous random graphs are commonly used models for complex networks where nodes have varying degrees of connectivity. Computing the degree distribution such is a fundamental problem and has important applications in various fields. We define inhomogeneous graph as model edges drawn independently probability link between any two vertices can be different each node pair. In this paper, we p...

A generalized central trinomial coefficient Tn(b, c) is the coefficient of x in the expansion of (x+bx+c) with b, c ∈ Z. In this paper we investigate congruences and series for sums of terms related to central binomial coefficients and generalized central trinomial coefficients. The paper contains many conjectures on congruences related to representations of primes by certain binary quadratic f...

3 Baum-Welch Algorithm (EM) 5 3.1 Outline of Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.2 First Term of Lc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.3 Second Term of Lc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.4 Third Term of Lc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.4.1 Pois...

We consider sums of the Gaussian q-binomial coefficients with a parametric rational weight function. We use the partial fraction decomposition technique to prove the claimed results. We also give some interesting applications of our results to certain generalized Fibonomial sums weighted with finite products of reciprocal Fibonacci or Lucas numbers.

During his lifetime, Ramanujan provided many formulae relating binomial sums to special values of the Gamma function. Based on numerical computations, Van Hamme recently conjectured p-adic analogues to such formulae. Using a combination of ordinary and Gaussian hypergeometric series, we prove one of these conjectures. Mathematics Subject Classification (2000). Primary: 33C20; Secondary: 11S80.