Continued Fractions for Rogers–Szegö Polynomials

We evaluate different Hankel determinants of Rogers–Szegö polynomials, and deduce from it continued fraction expansions for the generating function of RS polynomials. We also give an explicit expression of the orthogonal polynomials associated to moments equal to RS polynomials, and a decomposition of the Hankel form with RS polynomials as coefficients.     

Generalized harmonic numbers,Jacobi numbers and a Hankel determinant evaluation

In this paper we will introduce a sequence of complex numbers that are called the Jacobi numbers. This sequence generalizes in a natural way several sequences that are known in the literature, such as Catalan numbers, central binomial numbers, generalized catalan numbers, the coefficient of the Hilbert matrix and others. Subsequently, using a study of the polynomial of Jacobi, we give an evaluation of the Hankel determinants that associated with the sequence of Jacobi numbers. Finally, by finding a relationship between the Jacobi numbers and generalized harmonic numbers, we determine the evaluation of the Hankel determinants that are associated with generalized harmonic numbers.     

Two Kinds of Numbers and Their Applications

C. Radoux （J. Comput. Appl. Math., 115 （2000） 471-477） obtained a computational formula of Hankel determinants on some classical combinatorial sequences such as Catalan numbers and polynomials, Bell polynomials, Hermite polynomials, Derangement polynomials etc. From a pair of matrices this paper introduces two kinds of numbers. Using the first kind of numbers we give a unified treatment of Hankel determinants on those sequences, i.e., to consider a general representation of Hankel matrices on the first kind of numbers. It is interesting that the Hankel determinant of the first kind of numbers has a close relation that of the second kind of numbers.     

On the Determinant of a Certain Wiener-Hopf + Hankel Operator

We establish an asymptotic formula for determinants of truncated Wiener-Hopf+Hankel operators with symbol equal to the exponential of a constant times the characteristic function of an interval. This is done by reducing it to the corresponding (known) asymptotics for truncated Toeplitz+Hankel operators. The determinants in question arise in random matrix theory in determining the limiting distribution for the number of eigenvalues in an interval for a scaled Laguerre ensemble of positive Hermitian matrices.     

Sobolev orthogonal polynomials in computing of Hankel determinants

In this paper, we study closed form evaluation for some special Hankel determinants arising in combinatorial analysis, especially for the bidirectional number sequences. We show that such problems are directly connected with the theory of quasi-definite discrete Sobolev orthogonal polynomials. It opens a lot of procedural dilemmas which we will try to exceed. A few examples deal with Fibonacci numbers and power sequences will illustrate our considerations. We believe that our usage of Sobolev orthogonal polynomials in Hankel determinant computation is quite new.     

A complete solution of the normal Hankel problem

The normal Hankel problem is the one of characterizing the matrices that are normal and Hankel at the same time. We give a complete solution of this problem.     

Hankel operators, the Segal-Bargmann space, and symmetrically-normed ideals

Starting with the Segal-Bargmann space, we investigate the Hankel operators with symbol functions in a certain linear space. Given an appropriate symbol function, we consider the associated Hankel operator together with the Hankel operator associated with that symbol function's complex conjugate. We give a necessary and sufficient condition for the simultaneous membership of these two operators in the symmetrically-normed ideal associated with any given symmetric norming function.     

Hankel determinants of order four for a set of functions with bounded turning of order α

Lithuanian Mathematical Journal - We investigate the bounds of Hankel determinants H4,1(f), H4,2(f), H4,3(f) for the set of functions with bounded turning of order alpha. We also study these bounds...     

On a Toeplitz determinant identity of Borodin and Okounkov

In this note we give two other proofs of an identity of A. Borodin and A. Okounkov which expresses a Toeplitz determinant in terms of the Fredholm determinant of a product of two Hankel operators. The second of these proofs yields a generalization of the identity to the case of block Toeplitz determinants.Supported by National Science Foundation grant DMS-9970879.Supported by National Science Foundation grant DMS-9732687.     

Wavelets associated with Hankel transform and their Weyl transforms

The Hankel transform is an important transform. In this paper, we study thewavelets associated with the Hankel transform, then define the Weyl transform of thewavelets. We give criteria of its boundedness and compactness on the L~p-spaces.     

On m-step Fibonacci sequence in discrete Lotka-Volterra system

The integrable Lotka-Volterra (LV) system stands for a prey-predator model in mathematical biology. The discrete LV (dLV) system is derived from a time-variable discretization of the LV system. The solution to the dLV system is known to be represented by using the Hankel determinants. In this paper, we show that, if the entries of the Hankel determinants become m-step Fibonacci sequences at the initial discrete-time, then those are also so at any discrete time. In other words, the m-step Fibonacci sequences always arrange in the entries of the Hankel determinants under the time-variable evolution of the dLV system with suitable initial setting. Here the 2-step and the 3-step Fibonacci sequences are the famous Fibonacci and Tribonacci sequences, respectively. We also prove that one of the dLV variables converges to the ratio of two successive and sufficiently large m-step Fibonacci numbers, for example, the golden ratio in the case where m=2, as the discrete-time goes to infinity. Some examples are numerically given.     

Hankel Determinants of a Linear Combination of Three Successive Catalan Numbers

We evaluate the determinants of Hankel matrices, whose elements are a linear combination of three successive shifted Catalan numbers. This is done by finding a Jacobi linear functional, such that their moments are, up to a multiplicative constant, the Catalan numbers. The values of such determinants are then expressed in terms of Jacobi polynomials.     

Essential Norm Estimates for Little Hankel Operators on Weighted Bergman Spaces of the Unit Ball

We give estimates for the essential norm of a bounded little Hankel operator with $L^2$ symbol on weighted Bergman spaces of the unit ball in terms of a certain integral transform of the symbol. As an application of these estimates, we also give a necessary and sufficient condition for the little Hankel operators to be compact.     

Bigradients,Hankel determinants and the Newton-Padé table

In [9], Warner introduced generalized bigradients in the study of the Newton-Padé table. In this paper we introduce generalized Hankel determinants and derive, using the framework of Newton-Padé approximation, a relationship between these generalized Hankel determinants and generalized bigradients. This generalizes a determinantal identity obtained by Householder and Stewart [7, p. 136].     

On a Two-Point Version of Transfinite Diameter

We study properties of a two-point version of the transfinite diameter of a set. By using relations obtained for its calculation, we prove a two-point version of the well-known Pólya theorem on an estimate from above for the Hankel determinants of a holomorphic function.     

Model order determination using the Hankel matrix of impulse responses

This letter studies identification problems of model orders using the Hankel matrix of impulse responses of a system and presents two order identification methods: one is based on the singularities or ratios of the Hankel matrix determinants and the other is based on the singular value decomposition of the Hankel matrix. A numerical example verifies the proposed methods.     

Sequence transformations as statistical tools

A number of sequence transformations are actually used in statistics to solve different kinds of problems. In the two first parts of this paper we set the statistical problem of estimating the unknown orders of an ARMA process and we give its equivalent formulation in terms of invariance properties of sequence transformations: the most used among them are reviewed in the third part. The study of this estimation problem occuring in the Box-Jenkins method has led to new properties of Hankel determinants which are given in the part four.     

A note on p-supersolvable groups

The notion of Hankel operators associated with analytic crossed products were introduced and researched in [2]. In this paper, we study the adjoint of Hankel operators and give necessary and sufficient condition that the adjoint of a Hankel operator is again a Hankel operator. This work was supported in part by a Grant-in-Aid for Scientific Research, Japan Society for the Promotion of Science.     

Оценка емкости множества особенностей функций, заданных своим раэложением в непрерывную дробь

We obtain representations of Hankel’s determinants of functions defined by continued fraction expansions, via the parameters of the fraction. As a corollary of these representations, we prove that functions defined by continued fraction expansions of a certain type cannot be (uniquely) meromorphic continued beyond the convergence domain.     

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.