Orthogonal polynomials and connection to generalized Motzkin numbers for higher-order Euler polynomials

Jiu, Lin; Shi, Diane Y. H.

doi:10.1016/j.jnt.2018.11.021

Cited by 5 publications

(2 citation statements)

References 15 publications

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“…Already Al-Salam and Carlitz [1] found the Hankel determinants of the sequence of these higher-order Euler numbers. More recently, Han [9] dealt with other related sequences of higherorder Euler numbers, and the second author and Shi [11] determined the orthogonal polynomials of higher-order Euler polynomials, which also led to relevant Hankel determinants. Higher-order Bernoulli numbers, however, are more challenging; see the remarks in [11, p. 401].…”

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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Section: A Collection Of Hankel Determinant Formulasmentioning

confidence: 99%

Hankel Determinants of sequences related to Bernoulli and Euler Polynomials

Dilcher¹,

Jiu²

2020

Preprint

Self Cite

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“…By using the special case x = 1 in Theorem 1 of [12], we see that (2.4) holds for c k = E k (1) with…”

Section: Hankel Determinant Identities Imentioning

confidence: 99%

Hankel Determinants of shifted sequences of Bernoulli and Euler numbers

Dilcher,

Jiu

2023

CDM

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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 14 sequences related to Bernoulli and Euler numbers, one of which concerns Euler polynomials.

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Orthogonal polynomials and Hankel determinants for certain Bernoulli and Euler polynomials

Dilcher

Jiu

2021

Journal of Mathematical Analysis and Applications

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Orthogonal polynomials and connection to generalized Motzkin numbers for higher-order Euler polynomials

Cited by 5 publications

References 15 publications

Hankel Determinants of sequences related to Bernoulli and Euler Polynomials

Hankel Determinants of sequences related to Bernoulli and Euler Polynomials

Hankel Determinants of shifted sequences of Bernoulli and Euler numbers

Orthogonal polynomials and Hankel determinants for certain Bernoulli and Euler polynomials

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