Orthogonal polynomials and Hankel determinants for certain Bernoulli and Euler polynomials

Dilcher, Karl; Jiu, Lin

doi:10.1016/j.jmaa.2020.124855

Cited by 5 publications

(22 citation statements)

References 14 publications

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“…It is the purpose of this paper, later in Section 5, to give a direct computational proof, based on the corresponding Hankel determinant of certain Bernoulli polynomials B n (x). These sequences are different from the recent work [7,8,9], by Dilcher and the first author, on the Hankel determinants of sequences related to Bernoulli and Euler polynomial.…”

Section: Introductioncontrasting

confidence: 85%

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Hankel Determinants of Certain Sequences Of Bernoulli Polynomials: A Direct Proof of an Inverse Matrix Entry from Statistics

Jiu¹,

Li²

2021

Preprint

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We calculate the Hankel determinants of sequences of Bernoulli polynomials. This corresponding Hankel matrix comes from statistically estimating the variance in nonparametric regression. Besides its entries' natural and deep connection with Bernoulli polynomials, a special case of the matrix can be constructed from a corresponding Vandermonde matrix. As a result, instead of asymptotic analysis, we give a direct proof of calculating an entry of its inverse.

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Section: Introductioncontrasting

confidence: 85%

“…All the necessary background stated here in this section can be found in [7,8,9], in concise form. We repeat this material here for easy reference, and to make this paper self-contained.…”

Section: Preliminariesmentioning

confidence: 99%

Hankel Determinants of Certain Sequences Of Bernoulli Polynomials: A Direct Proof of an Inverse Matrix Entry from Statistics

Jiu¹,

Li²

2021

Preprint

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“…We begin this section with some necessary background on the connection between orthogonal polynomials and Hankel determinants. All this is well-known and can also be found in concise form in [4] and [5]. We repeat this material here for easy reference, and to make this paper self-contained.…”

Section: Orthogonal Polynomials and A Fundamental Lemmamentioning

confidence: 88%

“…Finding the second main ingredient, namely the pair of coefficient sequences (s n ) and (t n ), is a bit less straightforward than in the previous proofs. In Theorem 5.1 of [4] with ν = 2, the orthogonal polynomials belonging to the polynomial sequence E 2k+2 ( 1+x 2 ) k≥0 is given as P n+1 (y; x) = y + s n (x) P n (y; x) − t n (x)P n−1 (y; x), with appropriate initial conditions, and with…”

Section: Hankel Determinant Identities Imentioning

confidence: 99%

“…In the recent paper [4] we used the connection with orthogonal polynomials and continued fractions to find new evaluations of Hankel determinants of certain subsequences of Bernoulli and Euler polynomials. This was followed in [5] by evaluations of Hankel determinants of various other sequences related to Bernoulli and Euler numbers and polynomials.…”

Section: Introductionmentioning

confidence: 99%

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Hankel Determinants of shifted sequences of Bernoulli and Euler numbers

Dilcher,

Jiu

2021

Preprint

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Hankel determinants of sequences related to Bernoulli and Euler numbers have been studied before, and numerous identities are known. However, when a sequence is shifted by one unit, the situation often changes significantly. In this paper we use classical orthogonal polynomials and related methods to prove a general result concerning Hankel determinants for shifted sequences. We then apply this result to obtain new Hankel determinant evaluations for a total of 13 sequences related to Bernoulli and Euler numbers, one of which concerns Euler polynomials.

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Hankel determinants of sequences related to Bernoulli and Euler polynomials

Dilcher

Jiu

2021

Int. J. Number Theory

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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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Orthogonal polynomials and Hankel determinants for certain Bernoulli and Euler polynomials

Cited by 5 publications

References 14 publications

Hankel Determinants of Certain Sequences Of Bernoulli Polynomials: A Direct Proof of an Inverse Matrix Entry from Statistics

Hankel Determinants of Certain Sequences Of Bernoulli Polynomials: A Direct Proof of an Inverse Matrix Entry from Statistics

Hankel Determinants of shifted sequences of Bernoulli and Euler numbers

Hankel determinants of sequences related to Bernoulli and Euler polynomials

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