Explicit formulas of alternating multiple zeta star values $\zeta^\star({\bar 1}, \{1\}_{m-1}, {\bar 1})$ and $\zeta^\star(2, \{1\}_{m-1}, {\bar 1})$

1.   Introduction

We begin with some basic notation. Let ${\bf s}: = (s_1, \ldots, s_k)\in {\mathbb{N}}^k$, the (alternating) multiple harmonic (star) sums are defined by

where $s_j$ stands for non-zero integer, and

We may compactly indicate the presence of an alternating sign. When ${\rm sgn}(s_j) = -1$, by placing a bar over the corresponding integer exponent $s_j$. For example,

Moreover, for convenience, by ${\left\{ {{s_1}, \ldots, {s_j}} \right\}_d}$ we denote the sequence of depth $dj$ with $d$ repetitions of ${\left\{ {{s_1}, \ldots, {s_j}} \right\}}$. For example,

The sums of types (1.1) and (1.2) (one of more the $s_j$ barred) are called the alternating multiple harmonic sums and alternating multiple harmonic star sums, respectively. Conventionally, we call ${\rm dep}(\bf{s}) = k$ the depth and $w\equiv{\rm wt}(\bf{s}): = s_1+\cdots+s_k$ the weight.

Obviously, the limit cases of alternating multiple harmonic (star) sums give rise to alternating multiple zeta (star) values, for example,

The systematic study of multiple zeta values began in the early 1990s with the works of Hoffman ^[2] and Zagier ^[6]. After that it has been attracted a lot of research on them in the last three decades (see, for example, the book of Zhao ^[7]).

The main purpose of the present paper is to obtain the explicit formulas of alternating multiple zeta star values $\zeta^\star({\bar 1}, \{1\}_{m-1}, {\bar 1})$ and $\zeta^\star(2, \{1\}_{m-1}, {\bar 1})$.

2.   Main results

In this section we first give a lemma, which will be useful in the development of the main results of this paper.

Lemma 2.1. (see ^[3,4]) For $n, m\in {\mathbb{N}}$ and $x\in[-1, 1)$, the following relation holds:

where the parametric multiple harmonic star sum $\zeta^\star_n(s_1, \cdots, s_{k-1}, s_k; x)$ is defined by

and $\zeta^\star_n(\emptyset; x): = x^n$.

We note the fact that $\int_0^x = \int_0^1+\int_1^x$ and $\int_0^1 {{t^{n - 1}}{{\ln }^m}\left({1 - t} \right)} dt = m!\frac{{{{\left({ - 1} \right)}^m}}}{n}\zeta^\star_n\left(\{1\}_m\right)$, then (2.1) can be rewritten as

From (2.2), by a direct calculation, we can find that

Theorem 2.2. For positive integers $k, m$ and real $x\in [-1, 1)$,

where ${\mathrm{Li}}_{{{s_1}, {s_2}, \cdots, {s_k}}}(x)$ is multiple polylogarithm function defined by

Proof. Multiplying (2.3) by $\frac{\zeta_{n-1}(\{1\}_{k-1})}{n^2}$, then summing both sides of it over $n$ yields

From [3,Eq (2.24)], an elementary calculation shows that

From ^[1,3], we have the duality relation $\zeta(m+1, \{1\}_{k-1}) = \zeta(k+1, \{1\}_{m-1})$. Hence, letting $m = 1$ yields $\zeta(2, \{1\}_{k-1}) = \zeta(k+1)$. Thus, the formula (2.4) holds.

Similarly, we can also give the following results of alternating zeta type values:

Theorem 2.3. For positive integer $m$ and $x\in[-1, 1]$,

Proof. We note that the formula (2.3) can be rewritten as

Multiplying (2.8) by $\frac{(-1)^n}{n}$ and summing with respect to $n$, we obtain

In ^[5], Xu proved the result

Combining (2.9) with (2.10), by an elementary calculation, we have the result (2.7).

Corollary 2.4. For a positive integer $m$,

where (^[1])

Proof. Letting $k = 1, x = -1$ in (2.4) and $x = -1$ in (2.7) and noting the facts that (according to the definitions of alternating multiple zeta star values)

yield the desired results.

Hence, from Corollary 2.4, we obtain the explicit evaluations of alternating multiple zeta values $\zeta^\star({\bar 1}, \{1\}_{m-1}, {\bar 1})$ and $\zeta^\star(2, \{1\}_{m-1}, {\bar 1})$.

3.   Conclusions

In this paper, we use the integrals of logarithmic function to establish explicit formulas of the two (alternating) multiple zeta type values involving multiple harmonic star sum and parametric multiple harmonic star sum

in terms of zeta values and multiple polylogarithms. Further, by applying these formulas obtained, we obtain the explicit evaluations of alternating multiple zeta values $\zeta^\star({\bar 1}, \{1\}_{m-1}, {\bar 1})$ and $\zeta^\star(2, \{1\}_{m-1}, {\bar 1})$.

It is possible that some similar sums involving (parametric) multiple harmonic star sum can be computed by the methods and techniques given in this paper. For example, we can get the following theorem:

Theorem 3.1. For positive integers $m, k$ and $x\in[-1, 1)$,

Proof. The proof of Theorem 3.1 is similar as the proof of Theorem 2.3. Multiplying (2.3) by $\frac{\zeta_n({\bar 1}, \{1\}_{k-1})}{n}(-1)^n$ and summing with respect to $n$, and applying the identity

we have

Hence, we may easily deduce the desired result by a direct calculation.

In particular, setting $x = 0$ in (3.1) yields

Acknowledgments

We thank the anonymous referee for suggestions which led to improvements in the exposition. The author is supported by the National Natural Science Foundation of China (Grant No. 12101008), the Natural Science Foundation of Anhui Province (Grant No. 2108085QA01) and the University Natural Science Research Project of Anhui Province (Grant No. KJ2020A0057).

Conflict of interest

The author declares no conflict of interest in this paper.