Hypergeometric series and the Riemann zeta function

Chu, Wenchang

doi:10.4064/aa-82-2-103-118

Cited by 62 publications

(45 citation statements)

References 5 publications

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“…Since then many interesting finite and infinite sums involving generalized harmonic numbers have been evaluated by many authors by using different techniques. For example Chu [11]:…”

Section: Introductionmentioning

confidence: 99%

On some combinatorial identities and harmonic sums

Batır

2017

Int. J. Number Theory

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For any $m,n\in\mathbb{N}$ we first give new proofs for the following well known combinatorial identities \begin{equation*} S_n(m)=\sum\limits_{k=1}^n\binom{n}{k}\frac{(-1)^{k-1}}{k^m}=\sum\limits_{n\geq r_1\geq r_2\geq...\geq r_m\geq 1}\frac{1}{r_1r_2\cdots r_m} \end{equation*} and $$ \sum\limits_{k=1}^n(-1)^{n-k}\binom{n}{k}k^n = n!, $$ and then we produce the generating function and an integral representation for $S_n(m)$. Using them we evaluate many interesting finite and infinite harmonic sums in closed form. For example, we show that $$ \zeta(3)=\frac{1}{9}\sum\limits_{n=1}^\infty\frac{H_n^3+3H_nH_n^{(2)}+2H_n^{(3)}}{2^n}, $$ and $$ \zeta(5)=\frac{2}{45}\sum\limits_{n=1}^{\infty}\frac{H_n^4+6H_n^2H_n^{(2)}+8H_nH_n^{(3)}+3\left(H_n^{(2)}\right)^2+6H_n^{(4)}}{n2^n}, $$ where $H_n^{(i)}$ are generalized harmonic numbers defined below.Comment: to appear in Int. J. Number Theory, 201

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“…Since then many interesting finite and infinite sums involving generalized harmonic numbers have been evaluated by many authors by using different techniques. For example Chu [11]:…”

Section: Introductionmentioning

confidence: 99%

On some combinatorial identities and harmonic sums

Batır

2017

Int. J. Number Theory

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“…Thus, the complete reducibility of ζ(m + 2, {1} n ) is a simple consequence of the instance (6.5) of Gauss's 2 F 1 hypergeometric summation theorem [1], [3], [62]. Wenchang Chu [19] has elaborated on this idea, applying additional hypergeometric summation theorems to evaluate a wide variety of depth-2 sums, including nonlinear (cf. [31]) sums.…”

Section: Theorem 66 Let |P| ≥ 1 the Double Generating Function Equmentioning

confidence: 99%

Special values of multiple polylogarithms

Borwein

Bradley

Broadhurst

et al. 2000

Trans. Amer. Math. Soc.

356

390

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Abstract. Historically, the polylogarithm has attracted specialists and nonspecialists alike with its lovely evaluations. Much the same can be said for Euler sums (or multiple harmonic sums), which, within the past decade, have arisen in combinatorics, knot theory and high-energy physics. More recently, we have been forced to consider multidimensional extensions encompassing the classical polylogarithm, Euler sums, and the Riemann zeta function. Here, we provide a general framework within which previously isolated results can now be properly understood. Applying the theory developed herein, we prove several previously conjectured evaluations, including an intriguing conjecture of Don Zagier.

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“…Instead for λεθ = 0, Theorem 1 will lead to several infinite series identities involving both harmonic numbers and the Riemann zeta function. For further identities of finite and infinite series of similar type, the reader can refer to [5][6][7][8][12][13][14] and [4,9,11,17], respectively. In addition, it is pointed out by an anonymous referee that the expansion of hypergeometric sum expressions, as carried out in this paper, has applications in particle physics (cf.…”

Section: Reformulation Of Dixon's Classical Identitymentioning

confidence: 99%

Dixon’s F23(1)-series and identities involving harmonic numbers and the Riemann zeta function

Chen¹,

Chu²

2010

Discrete Mathematics

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a b s t r a c tDixon's classical summation theorem on 3 F 2 (1)-series is reformulated as an equation of formal power series in an appropriate variable x. Then by extracting the coefficients of x m , we establish a general formula involving harmonic numbers and the Riemann zeta function. Several interesting identities are exemplified as consequences, including one of the hardest challenging identities conjectured by Weideman (2003).

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Hypergeometric series and the Riemann zeta function

Cited by 62 publications

References 5 publications

On some combinatorial identities and harmonic sums

On some combinatorial identities and harmonic sums

Special values of multiple polylogarithms

Dixon’s F23(1)-series and identities involving harmonic numbers and the Riemann zeta function

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