Hankel continued fractions and Hankel determinants of the Euler numbers

Han, Guo-Niu

doi:10.1090/tran/8031

Cited by 15 publications

(12 citation statements)

References 44 publications

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“…The identity (7.14) was slightly changed from its original form. The following three identities were adapted from (H21), (H23), and (H24), respectively, in [9].…”

Section: A Collection Of Hankel Determinant Formulasmentioning

confidence: 99%

Hankel Determinants of sequences related to Bernoulli and Euler Polynomials

Dilcher¹,

Jiu²

2020

Preprint

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We evaluate the Hankel determinants of various sequences related to Bernoulli and Euler numbers and special values of the corresponding polynomials. Some of these results arise as special cases of Hankel determinants of certain sums and differences of Bernoulli and Euler polynomials, while others are consequences of a method that uses the derivatives of Bernoulli and Euler polynomials. We also obtain Hankel determinants for sequences of sums and differences of powers and for generalized Bernoulli polynomials belonging to certain Dirichlet characters with small conductors. Finally, we collect and organize Hankel determinant identities for numerous sequences, both new and known, containing Bernoulli and Euler numbers and polynomials.

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“…The identity (7.14) was slightly changed from its original form. The following three identities were adapted from (H21), (H23), and (H24), respectively, in [9].…”

Section: A Collection Of Hankel Determinant Formulasmentioning

confidence: 99%

Hankel Determinants of sequences related to Bernoulli and Euler Polynomials

Dilcher¹,

Jiu²

2020

Preprint

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“…For the initial elements, we have a 1,k = −k + 1 for 1 ≤ k ≤ t 1 by the same argument as in Eq. (10). Moreover, for 1 ≤ k ≤ t 2 , we have…”

Section: Proof Of Theorem 11mentioning

confidence: 92%

“…There are a number of methods developed for evaluating Hankel determinants, such as continued fractions, orthogonal polynomials, S-and J-fractions (see [12,15]), and H-fractions [9]. Recently, Han derived an explicit formula for the Hankel determinants of the Euler numbers by using H-fractions [10].…”

mentioning

confidence: 99%

On Hankel Determinants for Dyck Paths with Peaks Avoiding Multiple Classes of Heights

Chien¹,

Eu²,

Fu³

2021

Preprint

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For any integer m ≥ 2 and a set V ⊂ {1, . . . , m}, let (m, V ) denote the union of congruence classes of the elements in V modulo m. We study the Hankel determinants for the number of Dyck paths with peaks avoiding the heights in the set (m, V ). For any set V of even elements of an even modulo m, we give an explicit description of the sequence of Hankel determinants in terms of subsequences of arithmetic progression of integers. There are numerous instances for varied (m, V ) with periodic sequences of Hankel determinants. We present a sufficient condition for the set (m, V ) such that the sequence of Hankel determinants is periodic, including even and odd modulus m.

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“…Their ordinary generating function is a continued fraction [Bar09]. More recently, Han showed [Han20] this generating function is the q " ´1 evaluation of a q-analogue of the Euler numbers. Extending Han's work, Pan and Zeng gave the first organic combinatorial interpretation of these integers as certain labeled Motzkin paths called André paths [PZ19].…”

Section: Introductionmentioning

confidence: 99%

On pattern avoidance in matchings and involutions

Fang,

Hamaker,

Troyka

2020

Preprint

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In this paper, we study the relationship between two notions of pattern avoidance for involutions in the symmetric group and their restriction to fixed-pointfree involutions. The first is classical, while the second appears in the geometry of certain spherical varieties and generalizes the notion of pattern avoidance for perfect matchings studied by Jelínek. The first notion can always be expressed in terms of the second, and we give an effective algorithm to do so. We also give partial results characterizing the families of involutions where the converse holds. As a consequence, we prove two conjectures of McGovern characterizing (rational) smoothness of certain varieties. We also give new enumerative results, and conclude by proposing several lines of inquiry that extend our current work.

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Hankel continued fractions and Hankel determinants of the Euler numbers

Cited by 15 publications

References 44 publications

Hankel Determinants of sequences related to Bernoulli and Euler Polynomials

Hankel Determinants of sequences related to Bernoulli and Euler Polynomials

On Hankel Determinants for Dyck Paths with Peaks Avoiding Multiple Classes of Heights

On pattern avoidance in matchings and involutions

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