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

     

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.     

Linear recurrence relations for binomial coefficients modulo a prime

We investigate when the sequence of binomial coefficients modulo a prime p, for a fixed positive integer k, satisfies a linear recurrence relation of (positive) degree h in the finite range 0?i?k. In particular, we prove that this cannot occur if 2h?k<p−h. This hypothesis can be weakened to 2h?k<p if we assume, in addition, that the characteristic polynomial of the relation does not have −1 as a root. We apply our results to recover a known bound for the number of points of a Fermat curve over a finite field.     

On congruences related to central binomial coefficients

It is known that

     

Factors of binomial sums from the Catalan triangle

By using the Newton interpolation formula, we generalize the recent identities on the Catalan triangle obtained by Miana and Romero as well as those of Chen and Chu. We further study divisibility properties of sums of products of binomial coefficients and an odd power of a natural number. For example, we prove that for all positive integers n₁,…,nm, nm+1=n₁, and any nonnegative integer r, the expression

     

On sums of binomial coefficients and their applications

In this paper we study recurrences concerning the combinatorial sum and the alternate sum , where m>0, n?0 and r are integers. For example, we show that if n?m-1 then

     

On the calculation of generalized binomial coefficients

The generalized binomial coefficients (_κ^λ) are defined by

$\text{Cλ(I}{}_{\mathrm{m}}\text{+ R)}\text{Cλ(I}{}_{\mathrm{m}}\text{)}\text{=}\text{Σ}\text{k=0}\text{l}\text{Σ}\text{κ}\text{(}\text{λ}\text{κ}\text{)}\text{C}{}_{\mathrm{\kappa }}\text{(R)}\text{C}{}_{\mathrm{\kappa }}\text{(I}{}_{\mathrm{m}}\text{)}$

, where the C_k(R) are the zonal polynomials of the m × m matrix R. In this paper some simple expressions are derived which allow straightforward calculation of a large number of these coefficients.     

On some identities for the Fibonomial coefficients via generating function

Some new identities for the Fibonomial coefficients are derived. These identities are related to the generating function of the kth powers of the Fibonacci numbers. Proofs are based on manipulation with the generating function of the sequence of “signed Fibonomial triangle”.     

A congruence involving products of q-binomial coefficients

In this paper we establish a q-analogue of a congruence of Sun concerning the products of binomial coefficients modulo the square of a prime.     

Wavelet methods for continuous-time prediction using Hilbert-valued autoregressive processes

We consider the prediction problem of a continuous-time stochastic process on an entire time-interval in terms of its recent past. The approach we adopt is based on the notion of autoregressive Hilbert processes that represent a generalization of the classical autoregressive processes to random variables with values in a Hilbert space. A careful analysis reveals, in particular, that this approach is related to the theory of function estimation in linear ill-posed inverse problems. In the deterministic literature, such problems are usually solved by suitable regularization techniques. We describe some recent approaches from the deterministic literature that can be adapted to obtain fast and feasible predictions. For large sample sizes, however, these approaches are not computationally efficient.With this in mind, we propose three linear wavelet methods to efficiently address the aforementioned prediction problem. We present regularization techniques for the sample paths of the stochastic process and obtain consistency results of the resulting prediction estimators. We illustrate the performance of the proposed methods in finite sample situations by means of a real-life data example which concerns with the prediction of the entire annual cycle of climatological El Niño-Southern Oscillation time series 1 year ahead. We also compare the resulting predictions with those obtained by other methods available in the literature, in particular with a smoothing spline interpolation method and with a SARIMA model.     

Generalized Fermat, double Fermat and Newton sequences

In this paper, we discuss the relationship among the generalized Fermat, double Fermat, and Newton sequences. In particular, we show that every double Fermat sequence is a generalized Fermat sequence, and the set of generalized Fermat sequences, as well as the set of double Fermat sequences, is closed under term-by-term multiplication. We also prove that every Newton sequence is a generalized Fermat sequence and vice versa. Finally, we show that double Fermat sequences are Newton sequences generated by certain sequences of integers. An approach of symbolic dynamical systems is used to obtain congruence identities.     

Development of the Levin-type algorithms for accelerating convergence of sequences

There are two aims of this paper. Firstly we shall introduce the determinantal representations of the new Levin-type algorithms and secondly we shall demonstrate further development of the Levin-type algorithms. We consider the use of the Levin-type algorithms to accelerate the convergence of scalar sequence and their effectiveness for approximating the solution of a given power series is illustrated. In process we shall demonstrate the convergence of each of the methods considered. The approximate solution of the super enhanced Levin algorithm and the efficient Levin algorithm are found to be substantially more accurate than the Cizek, Zamastil and Skala transformation and the iterated Aitken Δ²${\mathrm{\Delta }}^2$ algorithm.     

Fundamental solution of a multidimensional difference equation with periodical and matrix coefficients

Summary The multidimensional (partial) difference equation with periodical coefficients is transformed into an equation for a vector sequence. Integral formulae for the vector fundamental solution are developed and some results about its asymptotic properties are explained. As an example, the results are used for a simple difference equation on a hexagonal grid.     

A Note on Log-Convexity of q-Catalan Numbers

The q-Catalan numbers studied by Carlitz and Riordan are polynomials in q with nonnegative coefficients. They evaluate, at q = 1, to the Catalan numbers: 1, 1, 2, 5, 14,…, a log-convex sequence. We use a combinatorial interpretation of these polynomials to prove a q-log-convexity result. The sequence of q-Catalan numbers is not q-log-convex in the narrow sense used by other authors, so our work suggests a more flexible definition of q-log convex be adopted. Received January 2, 2007     

Generalized Eulerian numbers and the topology of the Hessenberg variety of a matrix

Let Agl(n, C) and let p be a positive integer. The Hessenberg variety of degree p for A is the subvariety Hess(p, A) of the complete flag manifold consisting of those flags S ₁ S _n–1 in ⁿ which satisfy the condition AS _i S _i+p ,for all i. We show that if A has distinct eigenvalues, then Hess(p, A) is smooth and connected. The odd Betti numbers of Hess(p, A) vanish, while the even Betti numbers are given by a natural generalization of the Eulerian numbers. In the case where the eigenvalues of A have distinct moduli, |₁|<<|₁|, these results are applied to determine the dimension and topology of the submanifold of U(n) consisting of those unitary matrices P for which A ₀=P ^-1 AP is in Hessenberg form and for which the diagonal entries of the QR-iteration initialized at A ₀ converge to a given permutation of ₁,_n.Research partially supported by the National Science Foundation under Grant ECS-8696108.     

On unit roots for spatial autoregressive models

In this paper we consider the unit root problem for one rather simple autoregressive model Y_t,s=aY_t-1,s+bY_t,s-1+?_t,s on a two-dimensional lattice. We show that the growth of variance of Y_t,s is essentially different from corresponding growth in the unit root case for AR(1) or AR(2) time series models. We also show that the dimension of the lattice plays an important role: the growth of variance of autoregressive field on a d-dimensional lattice is different for d=2,3 and d≥4.     

Efficiencies of tests and estimators forp-order autoregressive processes when the error distribution is nonnormal

Summary We considerpth order autoregressive time series where the shocks need not be normal. By employing the concept of contiguity, we obtain the sysmptotic power for tests of hypothesis concerning the autoregressive parameters. Our approach allows consideration of the double exponential and other thicker-tailed distributions for the shocks. We derive a new result in the contiguity framework that leads directly to an expression for the Pitman efficiencies of tests as well as estimators. The numerical values of the efficiencies suggest a lack of robustness for the normal theory least squares estimators when the shock distribution is thick tailed or an outlier prone mixed normal. An important alternative test statistic is proposed that competes with the normal theory tests. This research was supported by the Office of Naval Research under Grant No. N00014-78-C-0722 and by the Army Research Office.     

Nonexistence of complete sufficient statistics for stationaryk-state Markov chains,k≧3

It is shown that there does not exist a complete sufficient statistic for the class of allk-state Markov chains,k≧3. Whenk=2 there is a complete sufficient statistic.     

Quasi-maximum likelihood estimation in GARCH processes when some coefficients are equal to zero

The asymptotic distribution of the quasi-maximum likelihood (QML) estimator is established for generalized autoregressive conditional heteroskedastic (GARCH) processes, when the true parameter may have zero coefficients. This asymptotic distribution is the projection of a normal vector distribution onto a convex cone. The results are derived under mild conditions. For an important subclass of models, no moment condition is imposed on the GARCH process. The main practical implication of these results concerns the estimation of overidentified GARCH models.