Generalized harmonic numbers via poly-Bernoulli polynomials

Kargın, Levent; Cenkci, Mehmet; Dil, Ayhan; Can, Mümün

doi:10.5486/pmd.2022.9074

Cited by 9 publications

(8 citation statements)

References 23 publications

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“…Moreover, by virtue of Theorem 5, it turns out that, whenever r ≥ 1 and m ≥ 1, the determinant of the lower Hessenberg matrix ( 9) is invariably an even or odd polynomial in N r of degree m − 1, according as m is odd or even. Consequently, whenever r ≥ 1 and m ≥ 1, the said determinantal formula (8) gives us S…”

Section: Discussionmentioning

confidence: 99%

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A determinantal formula for the hyper-sums of powers of integers

Cereceda¹

2022

Preprint

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Section: Discussionmentioning

confidence: 99%

“…i+n+1 n+1 S m+i (n) has been derived recently and independently by Kargın et al [8,Equation (4.7)]. )…”

Section: Introductionmentioning

confidence: 99%

A determinantal formula for the hyper-sums of powers of integers

Cereceda¹

2022

Preprint

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“…where, for any integer j ≥ 1, the j-th order difference operator ∆ j is defined by ∆ j f (x) = ∆(∆ j−1 f (x)) = ∆ j−1 (∆f (x)) and ∆ 0 f (x) = f (x), and where ∆ j f (a) = ∆ j f (x)| x=a . Hence, applying the Newton-Gregory expansion to the power sum polynomial S k (x) and using (11) yields…”

Section: Proof Of Theoremmentioning

confidence: 99%

Sums of Powers of Integers and Stirling Numbers

Cereceda¹

2022

Reson

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By applying the Newton-Gregory expansion to the polynomial associated with the sum of powers of integerswe derive a couple of infinite families of explicit formulas for S k (n). One of the families involves the r-Stirling numbers of the second kind k j r , j = 0, 1, . . . , k, while the other involves their duals k j −r , with both families of formulas being indexed by the non-negative integer r. As a by-product, we obtain three additional formulas for S k (n) in terms of the numbers k j n+m , k j n−m (where m is any given non-negative integer), and k j k−j , respectively. Moreover, we provide a pair of formulas for the Bernoulli polynomials B k (x) involving the numbers k j x .

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“…For further investigations concerning with generalized harmonic numbers, the readers may consult with [1,2,4,7,10,11,27,28] and references cited therein.…”

Section: Introductionmentioning

confidence: 99%