Almost Product Evaluation of Hankel Determinants

Eğecioǧlu, Ömer; Redmond, Timothy; Ryavec, Charles

doi:10.37236/730

Cited by 14 publications

(17 citation statements)

References 14 publications

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“…Furthermore, Zudilin [9] used Hankel determinants to obtain a new irrationality criterion. Finally, we note that although evaluating Hankel determinants can be difficult, there are many methods for doing so (see [2][3][4][5]). Now we describe the setup for our method.…”

Section: Introduction and Main Theoremmentioning

confidence: 99%

Rational Approximations via Hankel Determinants

Ferguson¹

2020

Preprint

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Define the monomials e n (x) := x n and let L be a linear functional. In this paper we describe a method which, under specified conditions, produces approximations for the value L(e 0 ) in terms of Hankel determinants constructed from the values L(e 1 ), L(e 2 ), . . . . Many constants of mathematical interest can be expressed as the values of integrals. Examples include the Euler-Mascheroni constant γ, the Euler-Gompertz constant δ, and the Riemann-zeta constants ζ(k) for k ≥ 2. In many cases we can use the integral representation for the constant to construct a linear functional for which L(e 0 ) equals the given constant and L(e 1 ), L(e 2 ), . . . are rational numbers. In this case, under the specified conditions, we obtain rational approximations for our constant. In particular, we execute this procedure for the previously mentioned constants γ, δ, and ζ(k). We note that our approximations are not strong enough to study the arithmetic properties of these constants.

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Section: Introduction and Main Theoremmentioning

confidence: 99%

Rational Approximations via Hankel Determinants

Ferguson¹

2020

Preprint

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“…is not amenable to standard methods since it does not have a product evaluation. In a recent paper [2] we proved that for the entries (1), the evaluation is an almost product; in this case a sum of n + 1 products of simple factors: det 3(i + j) + 1 i + j n! (4n + 3)!!…”

Section: Introductionmentioning

confidence: 99%

“…The technique presented in [2] to find the explicit form of the almost product for this particular Hankel determinant relies on the following steps:…”

Section: Introductionmentioning

confidence: 99%

“…If it had just been a matter of calculating the differential equation in the (3, 1)-case as we did in [2], then which technique we used might not have mattered much. However, we wanted to try to extend the differential equation method to a larger class of (β, α)-cases, and we found that already in the (2, 2)-case, the γ-operators simplified the calculations significantly.…”

Section: Introductionmentioning

confidence: 99%

“…To be specific, in the trace approach some nonlinear terms occur in the calculations, which get canceled in the end. For example the following ratio of determinants (using the notation in [2])…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A multilinear operator for almost product evaluation of Hankel determinants

Eğecioǧlu

Redmond

Ryavec

2010

Journal of Combinatorial Theory, Series A

Self Cite

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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.

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Non-Self-Adjoint Toeplitz Matrices Whose Principal Submatrices Have Real Spectrum

Shapiro

Štampach

2017

Constr Approx

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We introduce and investigate a class of complex semi-infinite banded Toeplitz matrices satisfying the condition that the spectra of their principal submatrices accumulate onto a real interval when the size of the submatrix grows to ∞. We prove that a banded Toeplitz matrix belongs to this class if and only if its symbol has real values on a Jordan curve located in C\{0}. Surprisingly, it turns out that, if such a Jordan curve is present, the spectra of all the principal submatrices have to be real. The latter claim is also proved for matrices given by a more general symbol. The special role of the Jordan curve is further demonstrated by a new formula for the limiting density of the asymptotic eigenvalue distribution for banded Toeplitz matrices from the studied class. Certain connections between the problem under investigation, Jacobi operators, and the Hamburger moment problem are also discussed. The main results are illustrated by several concrete examples; some of them allow an explicit analytic treatment, while some are only treated numerically.

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Almost Product Evaluation of Hankel Determinants

Cited by 14 publications

References 14 publications

Rational Approximations via Hankel Determinants

Rational Approximations via Hankel Determinants

A multilinear operator for almost product evaluation of Hankel determinants

Non-Self-Adjoint Toeplitz Matrices Whose Principal Submatrices Have Real Spectrum

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