Generalized harmonic numbers via poly-Bernoulli polynomials

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

doi:10.48550/arxiv.2008.00284

Cited by 3 publications

(4 citation statements)

References 20 publications

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“…j+1 and H (x+1) j+1 as a polynomial in x, complementing the formula given in [9,Equation (19)]. Furthermore, we have shown the relationship between the (exponential) complete Bell polynomials and the r-Stirling numbers of the first kind, and we have derived some identities involving the Bernoulli numbers and polynomials, the r-Stirling numbers of the first kind, the Stirling numbers of both kinds, and the harmonic numbers.…”

Section: Discussionsupporting

confidence: 58%

A note on the hyper-sums of powers of integers, hyperharmonic polynomials and r-Stirling numbers of the first kind

Cereceda¹

2021

Preprint

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Recently, Kargın et al. (arXiv:2008.00284 [math.NT]) obtained (among many other things) the following formula for the hyper-sums of powers of integerswhereIn this note we point out that a formula equivalent to the preceding one was already established in a different form, namely, a form in which m+n+1 i+n+1 n+1 is given explicitly as a polynomial in n of degree m − i. We find out the connection between this polynomial and the so-called r-Stirling polynomials of the first kind. Furthermore, we determine the hyperharmonic polynomials and their successive derivatives in terms of the r-Stirling polynomials of the first kind, and show the relationship between the (exponential) complete Bell polynomials and the r-Stirling numbers of the first kind. Finally, we derive some identities involving the Bernoulli numbers and polynomials, the r-Stirling numbers of the first kind, the Stirling numbers of both kinds, and the harmonic numbers.

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Section: Discussionsupporting

confidence: 58%

A note on the hyper-sums of powers of integers, hyperharmonic polynomials and r-Stirling numbers of the first kind

Cereceda¹

2021

Preprint

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“…that is yielded from the orthogonal relation in (7) and (8). More details concerning the r-Stirling numbers, which are used in this paper, are mentioned in the next section.…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, similar formulas with positive index have been studied in [7,13]. Surprisingly, relations include the harmonic numbers H n := n ℓ=1 1/ℓ and their generalization.…”

Section: Introductionmentioning

confidence: 99%

Some recurrence relations of poly-Cauchy numbers

Komatsu¹

2019

J. Nonlinear Sci. Appl.

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Poly-Cauchy numbers c (k) n (n 0, k 1) have explicit expressions in terms of the Stirling numbers of the first kind. When the index is negative, there exists a different expression. However, the sequence {c (−k) n } n 0 seem quite irregular for a fixed integer k 2. In this paper we establish a certain kind of recurrence relations among the sequence {c (−k) n } n 0 , analyzing the structure of poly-Cauchy numbers. We also study those of poly-Cauchy numbers of the second kind, poly-Euler numbers, and poly-Euler numbers of the second kind. Some different proofs are given. As applications, some leaping relations are shown.

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“…k (x) are the poly-Bernoulli polynomials [1], and H (p,r) n are the so-called generalized hyperharmonic numbers H (p,r) n which are defined recursively by [17,25] , and H (p,1) n = n k=1 1/k p . Furthermore, for p = 1, the generalized hyperharmonic numbers H (1,r) k reduce to the standard hyperharmonic numbers defined in (30).…”

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confidence: 99%

Daehee, hyperharmonic, and power sums polynomials

Cereceda¹

2021

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In this paper we consider the Daehee numbers and polynomials of the first and second kind, and give several explicit representations for them. In particular, we express the Daehee polynomials as the derivative of a generalized binomial coefficient. This is done by performing the Stirling transform of the power sum polynomial S k (x) associated with the sum of kth powers of the first n positive integersFurthermore, we show the relationship between the Daehee polynomials and the hyperharmonic polynomials. This allows us to also express the hyperharmonic polynomials as the derivative of a generalized binomial coefficient. In addition, we reformulate and generalize a number of identities involving the Stirling, Bernoulli, and harmonic numbers, in terms of the Daehee and hyperharmonic polynomials. Finally, in the light of a recent result of Kargın et al., we discuss a further extension of the obtained identities in terms of the r-Stirling numbers.

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Generalized harmonic numbers via poly-Bernoulli polynomials

Cited by 3 publications

References 20 publications

A note on the hyper-sums of powers of integers, hyperharmonic polynomials and r-Stirling numbers of the first kind

A note on the hyper-sums of powers of integers, hyperharmonic polynomials and r-Stirling numbers of the first kind

Some recurrence relations of poly-Cauchy numbers

Daehee, hyperharmonic, and power sums polynomials

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