Generalized rational zeta series for ζ(2n) and ζ(2n+1)

Orr, Derek

doi:10.1080/10652469.2017.1393809

Cited by 5 publications

(6 citation statements)

References 13 publications

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“…In this paper, building upon previous work from [19] we derive elementary and direct proofs for both H(a, b) (Theorem 1.1) and T (a, b) (Theorem 1.2) using the same approach with identities explored in [23]. Our proofs are significantly simpler than those given in [21,29] and rely on the Taylor series of integer powers of arcsin.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

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Elementary proofs of Zagier's formula for multiple zeta values and its odd variant

Li¹,

Lupu²,

Orr³

2022

Preprint

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In this paper, we give elementary proofs of Zagier's formula for multiple zeta values involving Hoffman elements and its odd variant due to Murakami. Zagier's formula was a key ingredient in the proof of Hoffman's conjecture. Moreover, using the same approach, we prove Murakami's formula for multiple t-values. This formula is essential in proving a Brown type result which asserts that each multiple zeta value is a Q-linear combination of multiple t-values of the same weight involving 2's and 3's.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…Re s > 0 is the Dirichlet beta function. As we have already highlighted in [20,23], we can express the cotangent integral in terms of Clausen functions.…”

Section: Preliminariesmentioning

confidence: 99%

Elementary proofs of Zagier's formula for multiple zeta values and its odd variant

Li¹,

Lupu²,

Orr³

2022

Preprint

Self Cite

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“…where (6) and (8) have been used again. Note this formula is the same as in other papers as well (see [6], [15]). Again, taking more partial derivatives, we will have (23) ∂ p F ∂m p (n,0,π/2) = 2 p π/2 0 x n log p (sin(x)) dx = π 2 n π 2(n + 1)…”

Section: Lsmentioning

confidence: 87%

“…where ( 6) and ( 8) have been used again. Note this formula is the same as in other papers as well (see [6], [15]). Again, taking more partial derivatives, we will have…”

Section: Lsmentioning

confidence: 87%

Generalized Log-sine integrals and Bell polynomials

Orr

2019

Journal of Computational and Applied Mathematics

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In this paper, we investigate the integral of x n log m (sin(x)) for natural numbers m and n. In doing so, we recover some well-known results and remark on some relations to the log-sine integral Ls (n) n+m+1 (θ). Later, we use properties of Bell polynomials to find a closed expression for the derivative of the central binomial and shifted central binomial coefficients in terms of polygamma functions and harmonic numbers.π 3, k ∈ N.Here, we will focus on a similar integral, F (n, m, z) = z 0x n sin 2m (x) dx.Further, we can define

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“…(2n + 2r + 2s + 3)2 2n . Now, we state the following Theorem 2.5 (Orr, [19]). For p ∈ N, and |z| < 1, πz 0…”

Section: Andmentioning

confidence: 97%

Another look at Zagier's formula for multiple zeta values involving Hoffman elements

Lupu¹

2020

Preprint

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In this paper, we give an elementary account into Zagier's formula for multiple zeta values involving Hoffman elements. Our approach allows us to obtain direct proof in a special case via rational zeta series involving the coefficient ζ(2n). This formula plays an important role in proving Hoffman's conjecture which asserts that every multiple zeta value of weight k can be expressed as a Q-linear combinations of multiple zeta values of the same weight involving 2's and 3's. Also, using a similar hypergeometric argument via rational zeta series, we produce a new Zagier-type formula for the multiple special Hurwitz zeta values.

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Generalized rational zeta series for ζ(2n) and ζ(2n+1)

Abstract: 9) Cl 1 (θ) = − log 2 sin θ 2 , |θ| < 2π.Writing others out,

Cited by 5 publications

References 13 publications

Elementary proofs of Zagier's formula for multiple zeta values and its odd variant

Elementary proofs of Zagier's formula for multiple zeta values and its odd variant

Generalized Log-sine integrals and Bell polynomials

Another look at Zagier's formula for multiple zeta values involving Hoffman elements

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