Extension of the four Euler sums being linear with parameters and series involving the zeta functions

Sofo, Anthony; Choi, Junesang

doi:10.1016/j.jmaa.2022.126370

Cited by 7 publications

(13 citation statements)

References 45 publications

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“…In the vast mathematical literature, many researchers have undertaken research on Euler, Euler-type sums, and other versions of these sums using a variety of techniques (see, e.g., [2], [3], [4], [6], [7], [9], [11], [12], [16], [18], [19], [20], [21], [23] and the references therein).…”

Section: Et Seq])mentioning

confidence: 99%

“…Example 2. Setting p = 1 and m = 0 in (3.31) yields an interesting sum which involves harmonic numbers and binomial coefficients: Extended parametric harmonic sums involving H qk (q ∈ N) were investigated in [18]. In fact, Shallit and Zikan [17] revealed that a relatively traditional (more than two centuries old) theorem of Christian Goldbach (1690-1764), which was given in a letter dated 1729 from Goldbach to Daniel Bernoulli (1700-1782):…”

Section: Particular Cases and Remarksmentioning

confidence: 99%

“…The research topic of series involving zeta functions such as (4.10), (4.11), and (4.12) has been popularly investigated by many researchers who have presented closed form expressions of a variety of series involving zeta functions and given applications (see, e.g., [22,Chapter 3] and the references cited therein; for recent ones, see also [1], [2], [8], [10], [18]).…”

Section: Particular Cases and Remarksmentioning

confidence: 99%

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Four types of variant Euler harmonic sums

Batır¹,

Choi²

2023

Preprint

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We aim to investigate the four types of variant Euler harmonic sums. Also, as corollaries, we provide particular examples of our core findings, some of whose further instances are evaluated in terms of basic and well-known functions as well as certain mathematical constants. We explore relevant linkages between our results and those of other previously established studies. An examination of a specific case of one result shows a relationship to series involving zeta functions, which is also a popular area of research.

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Section: Et Seq])mentioning

confidence: 99%

Section: Particular Cases and Remarksmentioning

confidence: 99%

Section: Particular Cases and Remarksmentioning

confidence: 99%

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Four types of variant Euler harmonic sums

Batır¹,

Choi²

2023

Preprint

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“…Different Euler sums with parameters have been studied (consult, for instance, [18,21,22,28,38,39]). Very recently, Sofo and Choi [40] extended the four parameterized linear Euler sums in (33) as follows:…”

Section: (Z)mentioning

confidence: 99%

Applications of Euler Sums and Series Involving the Zeta Functions

Choi,

Sofo

2023

Symmetry

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A very recent article delved into and expanded the four parametric linear Euler sums, revealing that two well-established subjects—Euler sums and series involving the zeta functions—display particular correlations. In this study, we present several closed forms of series involving zeta functions by using formulas for series associated with the zeta functions detailed in the aforementioned paper. Another closed form of series involving Riemann zeta functions is provided by utilizing a known identity for a series of rational functions in the series index, expressed in terms of Gamma functions. Furthermore, we demonstrate a myriad of applications and relationships of series involving the zeta functions and the extended parametric linear Euler sums. These include connections with Wallis’s infinite product formula for π, Mathieu series, Mellin transforms, determinants of Laplacians, certain integrals expressed in terms of Euler sums, representations and evaluations of some integrals, and certain parametric Euler sum identities. The use of Mathematica for various approximation values and certain integral formulas is elaborated upon. Symmetry naturally occurs in Euler sums.

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“…was identified by Flajolet and Salvy [6], in the case a = 0, b = 0 and q = 1. The case a ∈ Z ≥1 , b ∈ Z ≥1 and q = 1 was examined by Alzer and Choi [1], and finally the case a ∈ Z ≥1 , b ∈ Z ≥1 and q ∈ Z ≥1 was examined by Sofo and Choi [13].…”

Section: Introductionmentioning

confidence: 99%

Families of skew linear harmonic Euler sums involving some parameters

Sofo

2024

Cubo (Temuco, Online)

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In this study we investigate a family of skew linear harmonic Euler sums involving some free parameters. Our analysis involves using the properties of the polylogarithm function, commonly referred to as the Bose-Einstein integral. A reciprocity property is utilized to highlight an explicit representation for a particular skew harmonic linear Euler sum. A number of examples are also given which highlight the theorems. This work generalizes some results in the published literature and introduces some new results.

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Extension of the four Euler sums being linear with parameters and series involving the zeta functions

Cited by 7 publications

References 45 publications

Four types of variant Euler harmonic sums

Four types of variant Euler harmonic sums

Applications of Euler Sums and Series Involving the Zeta Functions

Families of skew linear harmonic Euler sums involving some parameters

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