An identity involving partitional generalized binomial coefficients

Define coefficients (_κ^λ) by C_λ(I_p + Z)/C_λ(I_p) = Σ_k=0^l Σ_{?∈$\text{P}$k} (_?^λ) C_κ(Z)/C_κ(I_p), where the C_λ's are zonal polynomials in p by p matrices. It is shown that C_?(Z) etr(Z)/k! = Σ_l=k^∞ Σ_{λ∈$\text{P}$l} (_?^λ) C_λ(Z)/l!. This identity is extended to analogous identities involving generalized Laguerre, Hermite, and other polynomials. Explicit expressions are given for all (_?^λ), ? ∈ $\text{P}$_k, k ≤ 3. Several identities involving the (_?^λ)'s are derived. These are used to derive explicit expressions for coefficients of $\text{C}{}_{\mathrm{\lambda }}\text{(Z)}\text{l}\text{!}$ in expansions of P(Z), $\text{etr}\text{(Z)}\text{k!}$ for all monomials P(Z) in s_j = tr Z^j of degree k ≤ 5.     

On the expansion of Cϱ1(V + I) as a sum of zonal polynomials

The coefficients a_τ?, sometimes called “generalized binomial coefficients” in the expansion $\text{C}{}_{\mathrm{?}}{}^1\text{(V +I) = Σ}{}_{\mathrm{\tau }}\text{a}{}_{\mathrm{?\tau }}\text{C}{}_{\mathrm{\tau }}{}^1\text{(V)}$, are computed explicitly when t = r + 1, where ? is a partition of r and τ a partition of t. A recursion formula permits the calculation of the general a_τ?. Several properties of a_τ? are proved. A connection between the a_τ? and other coefficients is established. The main tools used are Bingham's identity, results from the theory of invariant differential operators, and a lemma concerning zonal polynomials.     

Generalized Levinson-Durbin sequences, binomial coefficients and autoregressive estimation

For a discrete time second-order stationary process, the Levinson-Durbin recursion is used to determine the coefficients of the best linear predictor of the observation at time k+1, given k previous observations, best in the sense of minimizing the mean square error. The coefficients determined by the recursion define a Levinson-Durbin sequence. We also define a generalized Levinson-Durbin sequence and note that binomial coefficients form a special case of a generalized Levinson-Durbin sequence. All generalized Levinson-Durbin sequences are shown to obey summation formulas which generalize formulas satisfied by binomial coefficients. Levinson-Durbin sequences arise in the construction of several autoregressive model coefficient estimators. The least squares autoregressive estimator does not give rise to a Levinson-Durbin sequence, but least squares fixed point processes, which yield least squares estimates of the coefficients unbiased to order 1/T, where T is the sample length, can be combined to construct a Levinson-Durbin sequence. By contrast, analogous fixed point processes arising from the Yule-Walker estimator do not combine to construct a Levinson-Durbin sequence, although the Yule-Walker estimator itself does determine a Levinson-Durbin sequence. The least squares and Yule-Walker fixed point processes are further studied when the mean of the process is a polynomial time trend that is estimated by least squares.     

Asymptotic prime-power divisibility of binomial, generalized binomial, and multinomial coefficients

This paper presents asymptotic formulas for the abundance of binomial, generalized binomial, multinomial, and generalized multinomial coefficients having any given degree of prime-power divisibility. In the case of binomial coefficients, for a fixed prime , we consider the number of with for which is divisible by (but not ) when is an integer and , say. By means of a classical theorem of Kummer and the probabilistic theory of large deviations, we show that this number is approximately , where and is given by an explicit formula. We also develop a ``-adic multifractal' theory and show how may be interpreted as a multifractal spectrum of divisibility dimensions. We then prove that essentially the same results hold for a large class of the generalized binomial coefficients of Knuth and Wilf, including the -binomial coefficients of Gauss and the Fibonomial coefficients of Lucas, and finally we extend our results to multinomial coefficients and generalized multinomial coefficients.

     

Prime power divisors of binomial coefficients

It is known that for sufficiently large n and m and any r the binomial coefficient (ⁿ_m) which is close to the middle coefficient is divisible by p^r where p is a ‘large’ prime. We prove the exact divisibility of (ⁿ_m) by p^r for p> c(n). The lower bound is essentially the best possible. We also prove some other results on divisibility of binomial coefficients.     

Negation of binomial coefficients

We extend the concept of a binomial coefficient to all integer values of its parameters. Our approach is purely algebraic, but we show that it is equivalent to the evaluation of binomial coefficients by means of the Γ-function. In particular, we prove that the traditional rule of “negation” is wrong and should be substituted by a slightly more complex rule. We also show that the “cross product” rule remains valid for the extended definition.     

Modular periodicity of binomial coefficients

We prove that if the signed binomial coefficient viewed modulo p is a periodic function of i with period h in the range 0?i?k, then k+1 is a power of p, provided h is not too large compared to k. (In particular, 2h?k suffices). As an application, we prove that if G and H are multiplicative subgroups of a finite field, with H<G, and such that 1-α∈G for all α∈G?H, then G∪{0} is a subfield.     

Congruences for sums of binomial coefficients

Let q>1 and m>0 be relatively prime integers. We find an explicit period νm(q) such that for any integers n>0 and r we have

     

On a kind of curious binomial identity

As a generalization of Calkin's identity and its alternating form, we compute a kind of binomial identity involving some real number sequences and a partial sum of the binomial coefficients, from which many interesting identities follow.     

On the unlinking conjecture of independent polynomial functions

The celebrated U-conjecture states that under the N_n(0,I_n) distribution of the random vector X=(X₁,…,X_n) in Rⁿ, two polynomials P(X) and Q(X) are unlinkable if they are independent [see Kagan et al., Characterization Problems in Mathematical Statistics, Wiley, New York, 1973]. Some results have been established in this direction, although the original conjecture is yet to be proved in generality. Here, we demonstrate that the conjecture is true in an important special case of the above, where P and Q are convex nonnegative polynomials with P(0)=0.     

On fundamental skew distributions

A new class of multivariate skew-normal distributions, fundamental skew-normal distributions and their canonical version, is developed. It contains the product of independent univariate skew-normal distributions as a special case. Stochastic representations and other main properties of the associated distribution theory of linear and quadratic forms are considered. A unified procedure for extending this class to other families of skew distributions such as the fundamental skew-symmetric, fundamental skew-elliptical, and fundamental skew-spherical class of distributions is also discussed.     

On the minimization of multinomial tails and the Gupta-Nagel conjecture

This paper is primarily concerned with the open problem of minimizing the lower tail of the multinomial distribution. During the study of that specific problem, we have developed an approach which reveals itself useful for solving a general class of problems involving multinomial probabilities. Concerning the main problem, we provide a self-contained proof that the minimum of the multinomial lower tail is reached, as conjectured by Gupta and Nagel (Sankhya Ser. B 29 (1967) 1) (within the framework of subset-selection problems) at the equal probability configuration, i.e., when the cell probabilities are equal to one another. We also point out some novel inequalities and general properties involving multinomial probabilities and multinomial coefficients.     

二项式系数和序列的同余性质

利用同余理论研究了二项式系数和序列an（r,s）和bn（ε,a,b,c）分别在模p^2和模p^3下的同余性质,这将为研究它们的多项式递推公式提供有利的工具.     

On congruences related to central binomial coefficients

It is known that

     

On the recurrence coefficients of the generalized little q-Laguerre polynomials

In this paper we consider a semi-classical variation of the weight related to the little q-Laguerre polynomials and obtain a second order second degree discrete equation for the recurrence coefficients in the three-term recurrence relation.     

On the singularity of multivariate skew-symmetric models

In recent years, the skew-normal models introduced by Azzalini (1985) [1]-and their multivariate generalizations from Azzalini and Dalla Valle (1996) [4]-have enjoyed an amazing success, although an important literature has reported that they exhibit, in the vicinity of symmetry, singular Fisher information matrices and stationary points in the profile log-likelihood function for skewness, with the usual unpleasant consequences for inference. It has been shown (DiCiccio and Monti (2004) [23], DiCiccio and Monti (2009) [24] and Gómez et al. (2007) [25]) that these singularities, in some specific parametric extensions of skew-normal models (such as the classes of skew-t or skew-exponential power distributions), appear at skew-normal distributions only. Yet, an important question remains open: in broader semiparametric models of skewed distributions (such as the general skew-symmetric and skew-elliptical ones), which symmetric kernels lead to such singularities? The present paper provides an answer to this question. In very general (possibly multivariate) skew-symmetric models, we characterize, for each possible value of the rank of Fisher information matrices, the class of symmetric kernels achieving the corresponding rank. Our results show that, for strictly multivariate skew-symmetric models, not only Gaussian kernels yield singular Fisher information matrices. In contrast, we prove that systematic stationary points in the profile log-likelihood functions are obtained for (multi)normal kernels only. Finally, we also discuss the implications of such singularities on inference.     

Multivariate maximum entropy identification, transformation, and dependence

This paper shows that multivariate distributions can be characterized as maximum entropy (ME) models based on the well-known general representation of density function of the ME distribution subject to moment constraints. In this approach, the problem of ME characterization simplifies to the problem of representing the multivariate density in the ME form, hence there is no need for case-by-case proofs by calculus of variations or other methods. The main vehicle for this ME characterization approach is the information distinguishability relationship, which extends to the multivariate case. Results are also formulated that encapsulate implications of the multiplication rule of probability and the entropy transformation formula for ME characterization. The dependence structure of multivariate ME distribution in terms of the moments and the support of distribution is studied. The relationships of ME distributions with the exponential family and with bivariate distributions having exponential family conditionals are explored. Applications include new ME characterizations of many bivariate distributions, including some singular distributions.     

A note on cuts for contingency tables

In this note, we propose a general method to find cuts for a contingency table. Useful cuts are, in many cases, statistics S-sufficient for the nuisance parameter and S-ancillary for the parameter of interest. In general, cuts facilitate a strong form of parameter separation known to be useful for conditional inference [E.L. Lehmann, Testing Statistical Hypotheses, 2nd ed., Springer, New York, 1997, pp. 546–548]. Cuts also achieve significant dimension reduction, hence, increase computational efficiency. This is particularly true for the inference about cross-tabulated data, usually with a large number of parameters. Depending on the parameter of interest, we propose a flexible transformation to reparameterize the discrete multivariate response distribution. Inference on cell probabilities or odds ratios will require different parameterizations. The reparameterized distribution is not sum-symmetric. Thus, the finding in this paper expands the results in Barndorff-Nielsen [O.E. Barndorff-Nielsen, Information and Exponential Families in Statistical Theory, John Wiley, New York, 1978, pp. 202–206].