Hankel determinants of linear combinations of consecutive Catalan-like numbers

Mu, Lili; Wang, Yi; Yeh, Yeong‐Nan

doi:10.1016/j.disc.2017.07.004

Cited by 19 publications

(14 citation statements)

References 14 publications

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“…., be the polynomials orthogonal with respect to c, satisfying the recurrence relation (2.4). Following [13], we consider the infinite band matrix (4.1)…”

Section: Further Auxiliary Resultsmentioning

confidence: 99%

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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“…., be the polynomials orthogonal with respect to c, satisfying the recurrence relation (2.4). Following [13], we consider the infinite band matrix (4.1)…”

Section: Further Auxiliary Resultsmentioning

confidence: 99%

Hankel Determinants of shifted sequences of Bernoulli and Euler numbers

Dilcher,

Jiu

2021

Preprint

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“…. , 6: If we factor a somewhat larger term, for instance 15 • 5 4 11 11 • 13 9 • 17 5 • 19 3 , we see, especially in the denominator, that a definite pattern emerges. In fact, these are special cases of the following known result: If b = (B k ) k≥0 then for all n ≥ 0 we have (1.4) H n (B k ) = (−1) ( n+1 2 ) n ℓ=1 ℓ 4 4(2ℓ + 1)(2ℓ − 1)…”

Section: Introductionmentioning

confidence: 91%

“…We now take a more general approach, based on results in [8] and [15]. Given a sequence a = (a 0 , a 1 , .…”

Section: Shifted Sequencesmentioning

confidence: 99%

Orthogonal polynomials and Hankel Determinants for certain Bernoulli and Euler Polynomials

Dilcher

Jiu

2020

Preprint

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Using continued fraction expansions of certain polygamma functions as a main tool, we find orthogonal polynomials with respect to the oddindex Bernoulli polynomials B 2k+1 (x) and the Euler polynomials E 2k+ν (x), for ν = 0, 1, 2. In the process we also determine the corresponding Jacobi continued fractions (or J-fractions) and Hankel determinants. In all these cases the Hankel determinants are polynomials in x which factor completely over the rationals.

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“…preserves the SM property. Finally, for (iv), it follows the next general criterion (see [28] for instance). If the generating function of (u i ) i≥0 can be expressed by…”

Section: Stieltjes Moment Property and Continued Fractionsmentioning

confidence: 99%

On a Stirling-Whitney-Riordan triangle

Zhu

2020

Preprint

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Motivated by the Stirling triangle of the second kind, the Whitney triangle of the second kind and a triangle of Riordan, we study a Stirling-Whitney-Riordan triangle [T n,k ] n,k satisfying the recurrence relation:where initial conditions T n,k = 0 unless 0 ≤ k ≤ n and T 0,0 = 1. Let its rowgenerating function T n (q) = k≥0 T n,k q k for n ≥ 0.We prove that the Stirling-Whitney-Riordan triangle [T n,k ] n,k is x-totally positive with x = (a 1 , a 2 , b 1 , b 2 , λ). We show real rootedness and log-concavity of T n (q) and stability of the Turán-type polynomial T n+1 (q)T n−1 (q)−T 2 n (q). We also present explicit formulae of T n,k and the exponential generating function of T n (q), and the ordinary generating function of T n (q) in terms of a Jacobi continued fraction expansion. Furthermore, we get the x-Stieltjes moment property and 3-x-log-convexity of T n (q) and that the triangular convolution z n = n i=0 T n,k x i y n−i preserves Stieltjes moment property of sequences. Finally, for the first column (T n,0 ) n≥0 , we derive some similar properties to those of (T n (q)) n≥0 .

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Hankel determinants of linear combinations of consecutive Catalan-like numbers

Cited by 19 publications

References 14 publications

Hankel Determinants of shifted sequences of Bernoulli and Euler numbers

Hankel Determinants of shifted sequences of Bernoulli and Euler numbers

Orthogonal polynomials and Hankel Determinants for certain Bernoulli and Euler Polynomials

On a Stirling-Whitney-Riordan triangle

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