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Alternating double t-values and T-values
Advances in Difference Equations volume 2020, Article number: 450 (2020)
Abstract
Recently, Hoffman (Commun. Number Theory Phys. 13:529–567, 2019), Kaneko and Tsumura (Tsukuba J. Math. (in press), 2020) introduced and systematically studied two variants of multiple zeta values of level two, i.e., multiple t-values and multiple T-values, respectively. In this paper, by the contour integration and residue theorem, we establish two general identities, which further reduce to the expressions of the alternating double t-values and T-values. Some examples are also provided.
1 Introduction and notations
For positive integers n and p, let \(H^{(p)}_{n}\) and \({\overline{H}}^{(p)}_{n}\) stand for the nth generalized harmonic number and the nth generalized alternating harmonic number defined by
respectively. If \(p>1\) (or resp. \(p>0\)), the generalized harmonic number \(H^{(p)}_{n}\) (or resp. \({\bar{H}}_{n}\)) converges to the (Riemann) zeta value \(\zeta (p)\) (or resp. alternating zeta value \({\bar{\zeta }}(p)\)):
When \(k=1\), \(H_{n}^{(1)}\equiv H_{n}\) (resp. \(\overline{H}_{n}^{(1)}\equiv \overline{H}_{n}\)) is the classical harmonic number (resp. the classical alternating harmonic number). The empty sums \(H^{(p)}_{0}\) and \({\bar{H}}^{(p)}_{0}\) are conventionally understood to be zero.
For positive integers \(p_{1},\ldots ,p_{k}\) with \(p_{1}>1\), the multiple zeta value (MZV for short) is defined by
The study of multiple zeta values began in the early 1990s with the works of Hoffman [4] and Zagier [16]. The study of multiple zeta values have attracted numerous research interests in the area in the last two decades. For detailed history and applications, please see the book of Zhao [17].
Let \(h_{n}^{(p)}\) be the nth odd harmonic number, which is defined for \(n\in \mathbb{N}_{0}\) and \(p\in \mathbb{N}\) by
If \(p>1\), the generalized harmonic number \(h^{(p)}_{n}\) converges to the t̃-value:
A twin sibling of the odd harmonic number is called alternating odd harmonic number, defined by
which was introduced in [14]. When taking the limit \(n\rightarrow \infty \) in above, we get the so-called alternating t̄-value
Note that from [3], for nonnegative integer k, we have the generating function of \({\bar{t}}(2k+1)\)
where \(E_{2k}\) is the Euler number. Thus, we compute
In a recent paper [5], Hoffman introduced and studied the more general multiple t-values
As the normalized version,
we call them multiple t̃-values.
Kaneko and Tsumura [6, 7] introduced and studied a new kind of multiple zeta values of level two:
which were called multiple T-values (MTVs). As the normalized version,
we call them multiple T̃-values.
In (1.1) and (1.6)–(1.9), we put a bar on the top of \(p_{j}\) (\(j=1,\ldots k\)) if there is a sign \((-1)^{n_{j}}\) appearing in the denominator on the right. These with one or more \(p_{j}\) barred are called the alternating MZVs, alternating multiple t-values, alternating multiple t̃-values, alternating multiple T-values, and alternating multiple T̃-values, respectively. For example,
In all of these definitions, we call k the “depth” and \(p_{1}+\cdots +p_{k}\) the “weight”.
The motivation for this paper arises from the results of Flajolet and Salvy’s paper [2] and Wang and Xu’s papers [12, 14]. In [2], Flajolet and Salvy used the methods of contour integration and residue theorem to determine the reducibility of some classical Euler sums. Similarly, in [12, 14], Wang and Xu used the contour integration and residue theorem to evaluate (alternating) Euler sums and Euler T-sums. There have been numerous contributions on the theory of Euler sums in the last two decades, for example, see [1, 8, 9, 11, 13, 15] and the references therein.
The main purpose of this paper is to study the four (alternating) double t-values
and the four (alternating) double T-values
by using the methods of contour integration and residue theorem.
2 Double t-values and T-values
In this section, we give explicit evaluations for some (alternating) double t-values and T-values. We will prove these results in Sect. 4.
Theorem 2.1
For positive integers p and \(q>1\),
where \(\zeta (1):=-2\log (2)\)and \(\widetilde{t}(1):=0\).
Remark 2.2
Note that formulas (2.1) and (2.4) can also be found in Xu and Wang [14].
Example 2.1
We have
Theorem 2.3
For positive integers p and \(q>1\),
where \(\zeta (1):=-2\log (2)\)and \(\widetilde{t}(1):=0\).
Remark 2.4
Note that the explicit evaluation of \(T(q,p)\) with odd weight was also proved by Kanenko and Tsumura [6, 7] by another method.
Example 2.2
We have
3 Notations and related expansions
In this section, we give some basic notations, definitions, and lemmas. Let \(A:=\{a_{k}\}\), \(-\infty < k < \infty \) be a sequence of complex numbers with \({a_{k}} = o ( {{k^{\alpha }}} )\) (\(\alpha < 1\)) if \(k\rightarrow \pm \infty \). For convenience, let \(A_{1}\) and \(A_{2}\) denote the constant sequence \(\{(1)^{k}\}\) and the alternating sequence \(\{(-1)^{k}\}\), respectively.
3.1 Notations and definitions
Now, we give three definitions.
Definition 3.1
With A defined above, we define the parametric digamma function \(\varPhi ( { - s;A} )\) by
Definition 3.2
For nonnegative integers \(j\geq 1\) and n, we define
with \(D^{(A)}(1):=\overrightarrow{{F}_{0}}^{(A)}(1)\). Clearly, \(D^{(A_{1})}(1):=-2\log (2)\) and \(D^{(A_{2})}(1):=-\log (2)+\frac{\pi }{2}\).
Remark 3.1
It should be emphasized that many notations in Definition 3.2 were introduced in the reference [12].
Obviously,
Clearly, if we let \(A=A_{1}\) or \(A_{2}\) in Definition 3.2, elementary calculations yield
Definition 3.3
([12, Def. 1.2])
Define the cotangent function with sequence A by
It is clear that if we let \(A=A_{1}\) and \(A_{2}\) in (3.2), respectively, then it becomes
3.2 Several identities among Φ-functions
Proposition 3.2
Let \(p\geq 1\)and n be nonnegative integers, if \(|s-n+1/2|<1\), then
Proposition 3.3
Let \(p\geq 1\)and n be nonnegative integers, if \(|s-n|<1\)with \(s\neq n\), then
If we set \(n=0\), then for any \(|s|<1\) with \(s\neq 0\),
Proposition 3.4
Let p and n be positive integers, if \(|s+n-1/2|<1\), then
The method of the proofs of identities (3.3)–(3.6) is completely similar to that in [12, Theorems 2.1–2.3]. Thus, we omit it.
3.3 Lemmas
We define \(\tan (s;A):=\cot (\pi /2-s;A)\). It is clear that \(\tan (s;A_{1})=\tan (s)\) and \(\tan (s;A_{2})=\sec (s)\).
Lemma 3.5
([12, Thm. 2.3])
With \(\cot (\pi s;A)\)defined above, if \(|s-n|<1\)with \(s\neq n\) (\(n\in \mathbb{Z}\)), then
where \(\sigma _{n}\)is defined by the symbol of n, namely
Hence, an elementary calculation yields
for \(|s-n+1/2|<1\) with \(s\neq n-1/2\) (\(n\in \mathbb{Z}\)),
Let \(B:=\{b_{k}\}\), \(-\infty < k < \infty \) be a sequence of complex numbers with \({b_{k}} = o ( {{k^{\beta }}} )\) (\(\beta < 1\)) if \(k\rightarrow \pm \infty \). Define a kernel function \(\xi ( s )\) by the two requirements: 1. \(\xi ( s )\) is meromorphic in the whole complex plane. 2. \(\xi ( s )\) satisfies \(\xi ( s )=o(s)\) over an infinite collection of circles \(\vert s \vert = {\rho _{k}}\) with \({\rho _{k}} \to \infty \). Applying these two conditions of kernel function \(\xi ( s )\), Flajolet and Salvy showed the following residue theorem.
Lemma 3.6
([2])
Let \(\xi ( s )\)be a kernel function, and let \(r(s)\)be a rational function which is \(O(s^{-2})\)at infinity. Then
where S is the set of poles of \(r(s)\)and O is the set of poles of \(\xi ( s )\)that are not poles of \(r(s)\). Here \(\operatorname{Res} { [ {r ( s )},s = \alpha ]} \)denotes the residue of \(r(s)\)at \(s= \alpha \).
4 Two general theorems
In this section, we prove two general theorems which will be used to obtain the explicit evaluations of (alternating) double t-values and (alternating) double T-values.
Theorem 4.1
For positive integers p and \(q>1\),
where
Proof
Apply the kernel function
to the base function \(r(s)=s^{-q}\). Namely, we need to compute the residue of the function
Clearly, \(f_{1}(s;A,B)\) only has poles at \(s=0,\pm (n-1/2)\) and n (n is a positive integer). With the help of identities (3.3)–(3.6), we deduce the following residues:
and (4.2)–(4.4). Applying Lemma 3.6 yields the desired result. □
Proof of Theorem 2.1
Setting \(A,B\in \{A_{1},A_{2}\}\) in Theorem 4.1 yields the four desired evaluations. □
Theorem 4.2
For positive integers p and \(q>1\),
where
and
Proof
Apply the kernel function
to the base function \(r(s)=(s+1/2)^{-q}\). Namely, we need to compute the residues of the function
Clearly, \({f_{2}}( {s;A,B})\) only has poles at 0, n and \(\pm (n-1/2)\) (n is a positive integer). With the help of identities (3.3)–(3.6), these residues are
Then summing these four contributions and using Lemma 3.6, we may easily deduce the desired evaluation. □
Proof of Theorem 2.3
Setting \(A,B\in \{A_{1},A_{2}\}\) in Theorem 4.2 yields the four desired evaluations. □
It is possible that closed form representations of some other similar infinite series can be proved using techniques of the present paper.
Remark 4.3
It should be emphasized that Xu [12] defined another parametric digamma function \(\varPsi (-s;A)\). Very recently, Wang and Xu [10] used the parametric digamma function \(\varPsi (-s;A)\) to define several new kernel functions. Then they used the methods of contour integration and residue theorem to prove two general theorems (using the two theorems, they obtained Theorems 2.1 and 2.3), which are similar to Theorems 4.1 and 4.2. Moreover, they also showed many other types of results.
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Acknowledgements
We thank the anonymous referee for suggestions which led to improvements in the exposition.
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Quan, J. Alternating double t-values and T-values. Adv Differ Equ 2020, 450 (2020). https://doi.org/10.1186/s13662-020-02917-1
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DOI: https://doi.org/10.1186/s13662-020-02917-1