Evaluation of some Hankel determinants

We evaluate some Hankel determinants of Meixner polynomials, associated to the series exp(∑α[i]zⁱ/i), where [1],[2],… are the q-integers.     

A Widom like formula for some Toeplitz plus Hankel determinants

A well known Widom formula expresses the determinant of a Toeplitz matrix _Tn$T_n$ with Laurant polynomial symbol f in terms of the zeros of f. We give similar formulae for some even Toeplitz plus Hankel matrices. The formulae are based on an analytic representation of the determinant of such matrices in terms of Chebyshev polynomials.     

加权Motzkin序列的Hankel行列式

基于经典的Motzkin路引入了一类新的加权Motzkin路的定义,用这种路给出了一类指数型Riordan矩阵的组合解释,得到了相应的Riordan矩阵第0列元素(加权Motzkin序列)的加法公式.作为应用,得到了一类加权Motzkin序列的Hankel行列式的计算方法.     

Hankel determinants of some polynomials arising in combinatorial analysis

In this paper we investigate Hankel determinants of the form , where c _n (t) is one of a number of polynomials of combinatorial interest. We show how some results due to Radoux may be generalized, and also show how “stepped up” Hankel determinants of the form may be evaluated. This revised version was published online in June 2006 with corrections to the Cover Date.     

Counter-examples for refinement of conjectures and proofs in primary school mathematics

The purpose of this study is to explore how primary school students reexamine their conjectures and proofs when they confront counter-examples to the conjectures they have proved. In the case study, a pair of Japanese fifth graders thought that they had proved their primitive conjecture with manipulative objects (that is, they constructed an action proof), and then the author presented a counter-example to them. Confronting the counter-example functioned as a driving force for them to refine their conjectures and proofs. They understood the reason why their conjecture was false through their analysis of its proof and therefore could modify their primitive conjecture. They also identified the part of the proof which was applicable to the counter-example. This identification and their action proof were essential for their invention of a more comprehensive conjecture.     

A multilinear operator for almost product evaluation of Hankel determinants

In a recent paper we have presented a method to evaluate certain Hankel determinants as almost products; i.e. as a sum of a small number of products. The technique to find the explicit form of the almost product relies on differential-convolution equations and trace calculations. In the trace calculations a number of intermediate nonlinear terms involving determinants occur, but only to cancel out in the end.In this paper, we introduce a class of multilinear operators γ acting on tuples of matrices as an alternative to the trace method. These operators do not produce extraneous nonlinear terms, and can be combined easily with differentiation.The paper is self contained. An example of an almost product evaluation using γ-operators is worked out in detail and tables of the γ-operator values on various forms of matrices are provided. We also present an explicit evaluation of a new class of Hankel determinants and conjectures.     

Automatic proofs of some conjectures of Sun on the relations between sums of squares and sums of triangular numbers

The Ramanujan Journal - Recently, Sun posed a number of conjectures on the relations between sums of squares and sums of triangular numbers. Some of these conjectures were confirmed by Baruah,...     

Hankel determinants for convolution powers of Catalan numbers

The Hankel determinants ${\left(\frac{r}{2(i+j)+r}\left(\genfrac{}{}{0ex}{}{2(i+j)+r}{i+j}\right)\right)}_{0\le i,j\le n?1}$ of the convolution powers of Catalan numbers were considered by Cigler and Krattenthaler. We evaluate these determinants for $r\le 31$ by finding shifted periodic continued fractions, which arose in application of Sulanke and Xin’s continued fraction method. These include some of the conjectures of Cigler as special cases. We also conjecture a polynomial characterization of these determinants. The same technique is used to evaluate the Hankel determinants ${\left(\left(\genfrac{}{}{0ex}{}{2(i+j)+r}{i+j}\right)\right)}_{0\le i,j\le n?1}$. Similar results are obtained.     

Closed-form expression for Hankel determinants of the Narayana polynomials

We considered a Hankel transform evaluation of Narayana and shifted Narayana polynomials. Those polynomials arises from Narayana numbers and have many combinatorial properties. A mainly used tool for the evaluation is the method based on orthogonal polynomials. Furthermore, we provided a Hankel transform evaluation of the linear combination of two consecutive shifted Narayana polynomials, using the same method (based on orthogonal polynomials) and previously obtained moment representation of Narayana and shifted Narayana polynomials.     

A commutator method for the diagonalization of Hankel operators

A method for the explicit diagonalization of some Hankel operators is presented. This method makes it possible to give new proofs of classical results on the diagonalization of Hankel operators with absolutely continuous spectrum and obtain new results. The approach relies on the commutation of a Hankel operator with a certain second-order differential operator.     

Alternative proofs for inequalities of some trigonometric functions

By using an identity relating to Bernoulli's numbers and power series expansions of cotangent function and logarithms of functions involving sine function, cosine function and tangent function, four inequalities involving cotangent function, sine function, secant function and tangent function are established.     

A fast algorithm for evaluating nth order tri-diagonal determinants

The cost of all existing algorithms for evaluating the nth order determinants (Numerical Analysis, 7th Edition, Brooks & Cole Publishing, Pacific Grove, CA, 2001) is at most O(n³). In the current article we present a new efficient computational algorithm for evaluating the nth order tri-diagonal determinants with cost O(n) only. The algorithm is suited for implementation using Computer Algebra Systems such as MAPLE and MACSYMA. Some examples are given to illustrate the algorithm.     

A direct method for the stabilization of some locally damped semilinear wave equations

First, we consider a semilinear wave equation with a locally distributed damping in a bounded domain. Using the Carleman estimate, we devise an elementary proof of the exponential decay of the energy of this system. Afterwards we apply the same technique to the stabilization of the same type of equation in the whole space. Our proofs are constructive, and much simpler than those found in the literature. To cite this article: L. Tcheugoué Tébou, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Algebraic/combinatorial proofs of Cayley-type identities for derivatives of determinants and pfaffians

The classic Cayley identity states that

det(∂)(detX)^s=s(s+1)?(s+n−1)(detX)^s−1$\det (\partial ){(\det X)}^s=s(s+1)?(s+n-1){(\det X)}^{s-1}$

where X=(_xij)$X=(x_{ij})$ is an n×n$n\times n$ matrix of indeterminates and ∂=(∂/∂_xij)$\partial =(\partial /\partial x_{ij})$ is the corresponding matrix of partial derivatives. In this paper we present straightforward algebraic/combinatorial proofs of a variety of Cayley-type identities, both old and new. The most powerful of these proofs employ Grassmann algebra (= exterior algebra) and Grassmann–Berezin integration. Among the new identities proven here are a pair of “diagonal-parametrized” Cayley identities, a pair of “Laplacian-parametrized” Cayley identities, and the “product-parametrized” and “border-parametrized” rectangular Cayley identities.     

A note on the maximization of matrix valued Hankel determinants with applications

In this note, we consider the problem of maximizing the determinant of moment matrices of matrix measures. The maximizing matrix measure can be characterized explicitly by having equal (matrix valued) weights at the zeros of classical (one-dimensional) orthogonal polynomials. The results generalize classical work of Schoenberg (Indag. Math. 62 (1959) 282) to the case of matrix measures. As a statistical application we consider several optimal design problems in linear models, which generalize the classical weighing design problems.     

A proof of some conjectures of Mao on partition rank inequalities

Based on work of Atkin and Swinnerton-Dyer on partition rank difference functions, and more recent work of Lovejoy and Osburn, Mao has proved several inequalities between partition ranks modulo 10, and additional results modulo 6 and 10 for the \(M_2\) rank of partitions without repeated odd parts. Mao conjectured some additional inequalities. We prove some of Mao’s rank inequality conjectures for both the rank and the \(M_2\) rank modulo 10 using elementary methods.     

Jacobi continued fraction and Hankel determinants of the Thue-Morse sequence

We study the Jacobi continued fraction and the Hankel determinants of the Thue-Morse sequence and obtain several interesting properties. In particular, a formal power series φ(x) is being discovered, having the property that the Hankel transforms of φ(x) and of φ(x²) are identical.     

Proof of some congruence conjectures of Guo and Liu

The Ramanujan Journal - Let n and r be positive integers. Define the numbers $$S_{n}^{(r)}$$ by $$S_{n}^{(r)}=\sum _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) ^2\left(...