Abstract
We identify a partition-theoretic generalization of Riemann zeta function and the equally positive integer-indexed harmonic sums at infinity, to obtain the generating function and the integral representations of the latter. The special cases coincide with zeta values at positive integer arguments.
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1 Background
The harmonic sum of indices \(a_{1},\ldots ,a_{k}\in \mathbb {R}\backslash \left\{ 0\right\} \) is defined as (see [1, eq. 4, pp. 1])
which is naturally connected to the Riemann zeta function, by noting that \(N=\infty \), \(k=1\) and \(a_{1}>0\) gives \(S_{a_{1}}\left( \infty \right) =\zeta \left( a_{1}\right) \). A variety of the study can be found in the literature. For instance, Hoffman [4] established the connection between harmonic sums and multiple zeta values. We especially focus on the equally positively indexed harmonic sums, given by the case \(a_{1}=\cdots =a_{k}=a>0\)
and also the equally positive integer-indexed harmonic sums (EPIIHS), namely \(a=m\in \mathbb {Z}_{>0}\). If \(N=\infty \), we additionally assume \(m\in \mathbb {Z}_{>1}\) for convergence.
Recently, Schneider [6] studied the generalized q-Pochhammer symbol and obtained [6, pp. 3]
where
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\(X\subseteq \mathbb {\mathbb {Z}}_{>0}\) and \(f:\mathbb {\mathbb {Z}}_{>0}\longrightarrow \mathbb {C}\) such that if \(n\not \in X\) then \(f\left( n\right) =0\);
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\(\mathcal {P}_{X}\) is the set of partitions into elements of X;
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\(\lambda \vdash n\) means \(\lambda \) is a partition of n, the size \(|\lambda |\) is the sum of the parts of \(\lambda \), i.e., the number n being partitioned, and \(\lambda _{i}\in \lambda \) means \(\lambda _{i}\in \mathbb {Z}_{>0}\) is a part of partition \(\lambda \).
Further define \(l\left( \lambda \right) :=k\), \(n_{\lambda }:=\lambda _{1}\cdots \lambda _{k}\) and denote \(\mathcal {P}:=\mathcal {P}_{\mathbb {Z}_{>0}}\). Noting \(\lambda _{1}\ge \cdots \ge \lambda _{k}\ge 1\), a partition-theoretic generalization of Riemann zeta function [6, eq. 11, pp. 4] is defined and identified as
which leads to the generating function and the integral representation of \(S_{\varvec{m}_{k}}\left( \infty \right) \), presented in the next section.
2 Main results
We first apply (1.2) to the case \(X=\left\{ 1,2,\ldots ,N\right\} \) and \(f\left( n\right) :=\frac{t^{a}}{n^{a}}\), obtaining
which, by further letting \(q\rightarrow 1\), yields the following generating function.
Theorem 1
The generating function of \(S_{\varvec{a}_{k}}\left( N\right) \) is given by
Remark 2
The special case for \(a=1\) is [2, eq. 9, pp. 1272]
involving the beta function B, defined by
where the integral representation holds for \(\text {Re}\left( x\right) \), \(\text {Re}\left( y\right) >0\).
Corollary 3
For \(m\in \mathbb {Z}_{>1}\), denote \(\xi _{m}:=\exp \left( \frac{2\pi \text {i}}{m}\right) \) with \(\text {i}^{2}=-1\). Then,
Proof
Then, apply the limit (see [7, pp. 254, ex. 5]) \(\Gamma \left( z\right) =\underset{N\rightarrow \infty }{\lim }N^{z}B\left( N,z\right) \) to \(z_{j}=1-\xi _{m}^{j}t\), \(j=0,\ldots ,m-1\), by noting \(\xi _{m}^{0}+\cdots +\xi _{m}^{m-1}=0\), to complete the proof. \(\square \)
Remark 4
An alternative proof can be given by letting \(N=\infty \) in (2.1) and applying [3, Thm. 1.1, pp. 547].
Remark 5
For general \(a>0\), we failed to obtain a closed form of \(\overset{\infty }{\underset{n=1}{\prod }}\frac{n^{a}}{n^{a}-t^{a}}.\)
Example 6
When \(m=2\), we apply (2.4) to get
From the integral representation (2.3), we obtain (also see Remark 7)
Then it follows, by comparing coefficients of t,
In particular, \(k=1\) yields
Remark 7
We may interchange the integral and the sum of the series in (2.5), by restricting t to a closed compact set, e.g., \(\left[ -\frac{1}{2},\frac{1}{2}\right] \), satisfying \(\text {Re}\left( 1-t\right) \), \(\text {Re}\left( 1+t\right) >0\) as that in (2.3), in order to guarantee uniform convergence of the integral representation. (Similar discussion is omitted for the multiple beta function, defined next.)
Definition 8
The multiple beta function [5, Ch. 49] is defined as
where \(\Omega _{m}=\left\{ \left( x_{1},\ldots ,x_{m}\right) \in \mathbb {R}_{>0}^{m}:x_{1}+\cdots +x_{m-1}<1,\ x_{1}+\cdots +x_{m}=1\right\} \) and the integral representation requires \(\text {Re}\left( \alpha _{1}\right) ,\ldots ,\text {Re}\left( \alpha _{m}\right) >0\).
Following the same idea as that in Example 6, we first have, from (2.4),
Then, apply the integral representation (2.6), expand the integrand as a power series in t, and compare coefficients of t, to obtain the following integral representation.
Theorem 9
For all \(m,\,k\in \mathbb {Z}_{>0}\) with \(m\ge 2\),
Corollary 10
In particular, the case \(k=1\) implies for integer \(m\in \mathbb {Z}_{>1}\) that
or alternatively
Acknowledgements
The author would like to thank Dr. Jakob Ablinger for his help on harmonic sums; Prof. Johannes Blümlein for his handwritten notes on the proof of (2.2); and especially his mentors, Prof. Peter Paule and Prof. Carsten Schneider, for their valuable suggestions. This work is supported by the Austrian Science Fund (FWF) Grant SFB F50 (F5006-N15 and F5009-N15).
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Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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Jiu, L. Integral representations of equally positive integer-indexed harmonic sums at infinity. Res. number theory 3, 10 (2017). https://doi.org/10.1007/s40993-017-0074-x
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DOI: https://doi.org/10.1007/s40993-017-0074-x