Poly-Cauchy numbers with level 2

Komatsu, Takao; Pita-Ruiz, Claudio

doi:10.1080/10652469.2019.1710745

Cited by 9 publications

(9 citation statements)

References 12 publications

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“…Poly-Bernoulli numbers with level 2 can be expressed explicitly in terms of the Stirling numbers of the second kind with level 2. It is a natural extension of the expression in (10). Proof.…”

Section: Explicit Formulae and Recurrence Relationsmentioning

confidence: 96%

“…When s = 2, the Stirling numbers of the first kind with level 2 ( [8]) are related with the central factorial numbers of the first kind t(n, k) ( [1]) as n k 2 = t(2n, 2k). Notice that the original Stirling numbers of the first kind and the Stirling numbers of the first kind with level 2 are used to express poly-Cauchy numbers ( [5]) and poly-Cauchy numbers with level 2 ( [8,10]), respectively.…”

Section: Introductionmentioning

confidence: 99%

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Stirling numbers with level $2$ and poly-Bernoulli numbers with level $2$

Komatsu

2021

Preprint

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In this paper, we introduce poly-Bernoulli numbers with level 2, related to the Stirling numbers of the second kind with level 2, and study several properties of poly-Bernoulli numbers with level 2 from their expressions, relations, and congruences. Poly-Bernoulli numbers with level 2 have strong connections with poly-Cauchy numbers with level 2. In a special case, we can determine the denominators of Bernoulli numbers with level 2 by showing a von Staudt-Clausen like theorem.

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Section: Explicit Formulae and Recurrence Relationsmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Stirling numbers with level $2$ and poly-Bernoulli numbers with level $2$

Komatsu

2021

Preprint

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“…Using the definition above, we can extend the Theorem 4.2 using the Stirling numbers of the first kind with higher level, defined in [11,12]. Moreover, some applications of the Stirling numbers of higher level in special polynomials can be found in [10,9].…”

Section: Modular S-stirling Numbersmentioning

confidence: 98%

“…We also give a relationship with the Stirling numbers with higher level. This last sequence was recently studied in the context of special polynomials [10].…”

Section: Introductionmentioning

confidence: 99%

New modular symmetric function and its applications: Modular $s$-Stirling numbers

Bazeniar¹,

Ahmia²,

Ramírez³

et al. 2021

Preprint

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In this paper, we consider a generalization of the Stirling number sequence of both kinds by using a specialization of a new family of symmetric functions. We give combinatorial interpretations for this symmetric functions by means of weighted lattice path and tilings. We also present some new convolutions involving the complete and elementary symmetric functions. Additionally, we introduce different families of set partitions to give combinatorial interpretations for the modular s-Stirling numbers.

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“…Recently in [19,20], combinatorial properties of the numbers [[ n k ]] s are studied, also some relations with other sequences such as poly-Cauchy numbers with higher level, Bernoulli polynomials, and central factorial numbers are discussed. In [18,21], the poly-Cauchy numbers with level 2, and with higher level s are introduced and studied as extensions of the original poly-Cauchy numbers.…”

Section: Introductionmentioning

confidence: 99%

Laboratory Characterization of Asphalt Mixtures Containing Steel Slag

Shiha¹,

Gabr²,

El-Badawy³

2020

Bulletin of the Faculty of Engineering. Mansoura University

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The use of industrial by-products needs better understanding of their characteristics and behavior when subjected to traffic loading and environmental conditions. This paper presents a comprehensive laboratory characterization testing of asphalt mixtures containing steel slag. Electric Arc Furnace Steel Slag (EAFS) products were chosen as one of the most significant types of non-hazardous metallurgical waste, which were sourced from a steel factory in Egypt. The routine tests were conducted on EAFS materials to examine the physical, mechanical, and chemical properties, which were compared with the virgin aggregate (limestone). Blends of limestone aggregates/EAFS with percentages of 100/0, 80/20, 60/40, and 0/100% were selected to be employed as binder course layer. Asphalt mixtures were designed by Marshall method to find the optimum binder content. Test results showed that as the EAFS percentage in the mix increased, both the stability and density of the mixes increased. In addition, the flow and voids in mineral aggregates (VMA) were found to decrease with an increase in the EAFS content.

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Poly-Cauchy numbers with level 2

Cited by 9 publications

References 12 publications

Stirling numbers with level $2$ and poly-Bernoulli numbers with level $2$

Stirling numbers with level $2$ and poly-Bernoulli numbers with level $2$

New modular symmetric function and its applications: Modular $s$-Stirling numbers

Laboratory Characterization of Asphalt Mixtures Containing Steel Slag

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