In this article, we give a formula for the generalization of binomial coefficient to complex numbers as linear combination sinc functions. We then general compute integral on real line product and given function, which, in some cases, turns out be equal series their values integers. Finally, establish list identities obtained by applying these formulas.

This work is concerned with nonlinear time series models and, in particular, with nonparametric models for the dynamics of the mean of the time series. We build on the functional-coefficient autoregressive (FAR) model of Chen and Tsay (1993) which is a generalization of the autoregressive (AR) model where the coefficients are varying and are given by functions of the lagged values of the series...

A new bijection between the diagonally convex directed (dai-) polyominoes and ternary trees makes it possible to enumerate the dcd-polyominoes according to several parameters (sources, diagonals, horizontal and vertical edges, target cells). For a part of these results we also give another proof, which is based on Raney’s generalized lemma. Thanks to the fact that the diagonals of a dcd-polyomi...

The authors study the asymptotic behaviour of the likelihood ratio statistic for testing homogeneity in the finite mixture models of a general parametric distribution family. They prove that the limiting distribution of this statistic is the squared supremum of a truncated standard Gaussian process. The autocorrelation function of the Gaussian process is explicitly presented. A re-sampling proc...

Letting Bn,r be the n-th r-Bell polynomial, it is well known that Bn(x) admits specific integer coordinates in the two bases {x}i and {xBi(x)}i according to, respectively, the Stirling numbers and the binomial coefficients. Our aim is to prove that the sequences Bn+m,r(x) and Bn,r+s(x) admit a binomial recurrence coefficient in different bases of the Q-vector space formed by polynomials of Q[X].

The latter is an especially attractive characterization of the nth central binomial coefficient. Contrast this with the nth central trinomial coefficient, B(n), defined to be the largest coefficient of the polynomial (1 + x + x). There is no simple closed-form expression for B(n) [2]. It possesses recursion (n+ 1)B(n+ 1) = (2n+ 1)B(n) + 3nB(n− 1), B(0) = B(1) = 1 and asymptotics B(n) ∼ r 3 4π n...

We study bias arising as a result of nonlinear transformations of random variables in random or mixed effects models and its effect on inference in group-level studies or in meta-analysis. The findings are illustrated on the example of overdispersed binomial distributions, where we demonstrate considerable biases arising from standard log-odds and arcsine transformations of the estimated probab...