Yamamoto's interpolation of finite multiple zeta and zeta-star values

Murahara, Hideki; Ono, Masataka

doi:10.48550/arxiv.1908.09307

Cited by 4 publications

(3 citation statements)

References 35 publications

(32 reference statements)

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“…A very important variant of the MHS (see [25,27,30]) often referred to as multiple zeta star values MZSV or multiple harmonic star series MHSS (or simply multiple zeta values) is defined by:…”

Section: Introduction and Notationmentioning

confidence: 99%

A generalization of multiple zeta values. Part 1: Recurrent sums

Haddad¹

2022

NNTDM

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Multiple zeta star values have become a central concept in number theory with a wide variety of applications. In this article, we propose a generalization, which we will refer to as recurrent sums, where the reciprocals are replaced by arbitrary sequences. We introduce a toolbox of formulas for the manipulation of such sums. We begin by developing variation formulas that allow the variation of a recurrent sum of order $m$ to be expressed in terms of lower order recurrent sums. We then proceed to derive theorems (which we will call inversion formulas) which show how to interchange the order of summation in a multitude of ways. Later, we introduce a set of new partition identities in order to then prove a reduction theorem which permits the expression of a recurrent sum in terms of a combination of non-recurrent sums. Finally, we use these theorems to derive new results for multiple zeta star values and recurrent sums of powers.

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Section: Introduction and Notationmentioning

confidence: 99%

A generalization of multiple zeta values. Part 1: Recurrent sums

Haddad¹

2022

NNTDM

View full text Add to dashboard Cite

show abstract

“…A very important variant of the MHS (see [27,25,30]) often referred to as multiple zeta star values MZSV or multiple harmonic star series MHSS (or simply multiple zeta values) is defined by:…”

Section: Introduction and Notationmentioning

confidence: 99%

Recurrent Sums and Partition Identities

Haddad

2021

Preprint

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Sums of the form;N 1 where the a (k);N k 's are same or distinct sequences appear quite often in mathematics. We will refer to them as recurrent sums. In this paper, we introduce a variety of formulas to help manipulate and work with this type of sums. We begin by developing variation formulas that allow the variation of a recurrent sum of order m to be expressed in terms of lower order recurrent sums. We then proceed to derive theorems (which we will call inversion formulas) which show how to interchange the order of summation in a multitude of ways. Later, we introduce a set of new partition identities in order to then prove a reduction theorem which permits the expression of a recurrent sum in terms of a combination of non-recurrent sums. Finally, we apply this reduction theorem to a recurrent form of two famous types of sums: The p-series and the sum of powers.

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“…In 2018, Ce Xu [10] illustrated the connection between similar integrals involving logarithms and a particular recurrent sum (the multiple zeta star function or multiple harmonic star sum). The multiple harmonic star sum (MHSS) [11,12,13] is defined as follows:…”

Section: Introductionmentioning

confidence: 99%

Repeated Integration and Explicit Formula for the $n$-th Integral of $x^m (\ln x)^{m'}$

Haddad

2021

Preprint

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Repeated integration is a major topic of multivariable calculus. In this article, we study repeated integration. In particular, we study repeated integrals and recurrent integrals. For each of these integrals, we develop reduction formulae for both the definite as well as indefinite form. These reduction formulae express these repetitive integrals in terms of single integrals. We also derive a generalization of the fundamental theorem of calculus that expresses a definite integral in terms of an indefinite integral for repeated and recurrent integrals. From the recurrent integral formulae, we derive some partition identities. Then we provide an explicit formula for the n-th integral of x m (ln x) m ′ in terms of a shifted multiple harmonic star sum. Additionally, we use this integral to derive new expressions for the harmonic sum and repeated harmonic sum.

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Yamamoto's interpolation of finite multiple zeta and zeta-star values

Cited by 4 publications

References 35 publications

A generalization of multiple zeta values. Part 1: Recurrent sums

A generalization of multiple zeta values. Part 1: Recurrent sums

Recurrent Sums and Partition Identities

Repeated Integration and Explicit Formula for the $n$-th Integral of $x^m (\ln x)^{m'}$

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