Alternating double t-values and T-values

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.


Introduction and notations
When k = 1, H (1) n ≡ H n (resp. H (1) n ≡ H n ) is the classical harmonic number (resp. the classical alternating harmonic number). The empty sums H (1.1) 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 (p) n be the nth odd harmonic number, which is defined for n ∈ N 0 and p ∈ N by If p > 1, the generalized harmonic number h (p) n converges to the t-value: A twin sibling of the odd harmonic number is called alternating odd harmonic number, defined bȳ which was introduced in [14]. When taking the limit n → ∞ in above, we get the so-called alternatingt-valuē (1.5) Note that from [3], for nonnegative integer k, we have the generating function oft(2k + 1) where E 2k is the Euler number. Thus, we computē In a recent paper [5], Hoffman introduced and studied the more general multiple tvalues As the normalized version, t(p 1 , p 2 , . . . , p k ) := 2 p 1 +···+p k t(p 1 , p 2 , . . . , p k ), (1.7) we call them multiple t-values.
In all of these definitions, we call k the "depth" and p 1 + · · · + 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 T(q, p), T(q,p), T(q,p), T(q, p)
by using the methods of contour integration and residue theorem.

Double t-values and T-values
In this section, we give explicit evaluations for some (alternating) double t-values and Tvalues. We will prove these results in Sect. 4.
Remark 2.2 Note that formulas (2.1) and (2.4) can also be found in Xu and Wang [14].

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.

Notations and related expansions
In this section, we give some basic notations, definitions, and lemmas. Let A := {a k },

Notations and definitions
Now, we give three definitions.

Definition 3.1 With A defined above, we define the parametric digamma function
Definition 3.2 For nonnegative integers j ≥ 1 and n, we define 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 It is clear that if we let A = A 1 and A 2 in (3.2), respectively, then it becomes cot(πs; A 1 ) = cot(πs) and cot(πs; A 2 ) = csc(πs).

Proposition 3.3
Let p ≥ 1 and n be nonnegative integers, if |s -n| < 1 with s = n, then If we set n = 0, then for any |s| < 1 with s = 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.

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. 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 ∈ {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 Ψ (-s; A). Very recently, Wang and Xu [10] used the parametric digamma function Ψ (-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.