# Automated Evaluation for Student Argumentative Writing: A Survey

Xinyu Wang
Riiid Labs
<EMAIL_ADDRESS>
Yohan Lee
Riiid Labs
<EMAIL_ADDRESS>
Juneyoung Park
Riiid AI Research
<EMAIL_ADDRESS>

###### Abstract

This paper surveys and organizes research works in an under-studied area,
which we call automated evaluation for student argumentative writing. Unlike
traditional automated writing evaluation that focuses on holistic essay
scoring, this field is more specific: it focuses on evaluating argumentative
essays and offers specific feedback, including argumentation structures,
argument strength trait score, etc. The focused and detailed evaluation is
useful for helping students acquire important argumentation skill. In this
paper we organize existing works around tasks, data and methods. We further
experiment with BERT on representative datasets, aiming to provide up-to-date
baselines for this field.

##  1 Introduction

Automated writing evaluation uses computer programs to give evaluative
feedback to a piece of written work, which is often used in educational
settings (Parra et al., 2019). Most of the work on automated writing
evaluation focus on automated essay scoring (AES), which evaluates an essay’s
quality by assigning scores Shermis and Burstein (2013). It is a long-standing
research problem, with the first system proposed in 1966 by Page (1966). Since
then this area has been active and attracts a lot of research efforts.

A large portion of AES systems are developed for holistic scoring Ke and Ng
(2019), which outputs a single score to represent essay quality. This kind of
system is useful for summative assessment and can greatly reduce manual
grading efforts. However, holistic scoring is not sufficient for providing
instructional feedback to student learning because a low holistic score does
not provide enough information to the students about how to improve. To
address this issue, many research works tried to build trait-specific scoring
systems. These systems concern scoring particular quality dimensions of an
essay, such as grammar Burstein et al. (2004), word choice Mathias and
Bhattacharyya (2020), coherence Somasundaran et al. (2014), etc. Holistic
scoring systems, along with trait specific scoring systems, have already been
deployed to commercial settings successfully (Burstein et al. 2004, Rudner et
al. 2006).

However, most of the systems do not distinguish between essay types (e.g.
argumentative or narrative essay). It makes sense when the system is trying to
evaluate type-agnostic dimensions such as word choice of an essay, but the use
of such systems is limited when an user wants to know more about the in-depth
traits specific to an essay type, e.g. whether claims and counterclaims are
developed fairly in an argumentative essay. The evaluation of type specific
traits is important because these traits reflect important type-specific
skills — for example, whether claims and counterclaims are developed
thoroughly is indicative of critical thinking skills, which can be hard to
gauge through narrative essay writing.

In this paper we attempt to organize the research works regarding automated
evaluation specifically for student argumentative writing, which requires the
students to evaluate controversial claims, collect and judge evidence and
establish a position. The contributions of this paper are as follows. First,
we categorized and organized current research body regarding tasks and
datasets (section 2) and methods (section 3). Additionally, we experimented
with BERT Devlin et al. (2019) models and provided up-to-date baselines to the
community (section 4). Finally, we suggested directions (section 5) based on
missing elements in current research body which we hope to be filled in the
future.

Figure 1: An illustration of annotation scheme for S&G2014 and S&G2017a. This figure is adapted from Stab and Gurevych (2017a) Type | Example Tasks
---|---
AM | Argument Component Identification
| Argument Component Classification
| Argument Relation Identification
| Relation Labelling
CD | Opposing Argument Detection
| Valid Critique Detection
| Thesis Detection
QA | Sufficiency Recognition
| Argument Strength Scoring
Table 1: Example sub-tasks under the three main problems: argument mining
(AM), component detection (CD) and quality assessment (QA)

##  2 Tasks and Datasets

In this section, we introduce the tasks commonly studied in automated writing
evaluation for argumentative writing, together with their benchmark datasets.

Data | No. of Essays | Label Structure | Task Type
---|---|---|---
Stab and Gurevych 2014a | 90 | Tree | Argument mining
Stab and Gurevych 2017a | 402 | Tree | Argument mining
Stab and Gurevych 2016 | 402 | Binary value | Component Detection
Carlile et al. 2018 | 102 | Values on top of tree nodes | Quality Assessment
Stab and Gurevych 2017b | 402 | Binary value | Quality Assessment
Persing and Ng 2015 | 1000 | Score | Quality Assessment
Table 2:  Comparison of several popular public datasets

###  2.1 Argument Mining

Argument mining (AM) aims to identify and parse the argumentation structure of
a piece of text. Lawrence and Reed (2020) For argumentative essays,
argumentation structures may have variations but can typically be represented
as trees, as illustrated in Figure 1. The root of the tree is a major claim,
which expresses the author’s main point on the topic. Children of the major
claims are claims, which are controversial statements that either argue for or
against their parents. Claims could have children nodes, called premises (e.g.
$Premise1$ and $Premise3$ on Figure 1), which support or attack the
corresponding claims. Further, a premise (e.g. $Premise3$) can also be
supported or attacked by other premises (e.g. $Premise4$ and $Premise5$),
enriching the logical flow of the essay.

In order to generate an argumentation structure from text, argument mining
models typically decompose the process into four sub-tasks: (1) argument
component identification (ACI), which aims to identify the span of argument
components (e.g. text with squared brackets on Figure 1); (2) argument
component classification (ACC), which classifies argument components into
corresponding types (e.g. where a span is a Claim or Premise); 3) argument
relation identification (ARI), which aims at linking related argument
components on the argument structure; 4) relation labelling (RL), which
focuses on identifying relation type (e.g. for and support) between linked
components.

#### Datasets

Stab and Gurevych, 2014a (S&G2014) and Stab and Gurevych, 2017a (S&G2017a)
annotated the most widely adopted datasets regarding this direction. S&G2014
annotated 90 persuasive essays from essayforum, a site providing feedback for
students who wish to improve their writing. S&G2017a adopted similar
annotation scheme and annotated additional 402 essays.

Besides, Putra et al. (2021a) annotated 434 essays written by English to
Speakers of Other Languages (ESOL) college learners as quasi-trees. They
annotated each sentence as either argumentative component (ACs) or non-
argumentative component (non-AC). Annotators then identified relations between
ACs. Different from S&G2014, their relation labels include directed labels
(support, attack or detailing) as well as an undirected label(restatement).
They also reordered the sentences to make the essays better-structured.
Wambsganss et al. (2020) annotated 1000 student peer-reviews written in
German, indicating whether a text span is claim or premise or not, and the
relations between these argumentative components.

Alhindi and Ghosh (2021) annotated on a token level using BIO tagging schema
and this dataset can be used for ACI and ACC. Specifically, each token belongs
to one of the five classes: the begin token of a claim; a continuous token of
a claim; the begin token of a premise; a continuous token of a premise; non-
argumentative token.

###  2.2 Component Detection

Besides parsing the complex argumentative structures of essays as in argument
mining, a large portion of work in automated writing evaluation also considers
the simpler task of detecting whether an essay or a text span in an essay
contains a specific component. For example, Stab and Gurevych (2016) assigned
binary labels to essays depending on whether an essay has discussed the
opposing arguments to author’s own standpoint. Falakmasir et al. (2014)
labelled essays as whether or not containing a thesis and conclusion
statement. Beigman Klebanov et al. (2017) annotated essays written by college
students for criticizing a piece of argument, deciding if each sentence
contains a good critique or not. Ghosh et al. (2020) annotated on whether a
sentence contains a valid critique as well, but the essays they used are
written by middle school students.

Finally, we remark that component detection is different from the task of
argument component classification (ACC) discussed previously in Section 2.1.
Specifically, component detection does not require the text span to match the
component of interest. For example, Falakmasir et al. (2014) cares about
whether an essay contains a thesis and conclusion statement. In their case,
the length of text span is much longer than the component of interest.

###  2.3 Quality Assessment

Quality assessment concerns evaluating the quality of an argument. There are
many sub-tasks under this category due to the diversity of argumentation
quality theories.

Persing and Ng (2015) annotated 1000 essays over 10 prompts from International
Corpus of Learner English (ICLE) dataset Granger et al. (2009) on argument
strength. They defined argument strength as how well an essay makes an
argument for its thesis and convinces its readers. Horbach et al. (2017)
collected 2020 German essays written by prospective university students. Their
annotators evaluated the text regarding 41 aspects including quality of
argumentation. In addition, Stab and Gurevych (2017b) (S&G2017b) adopted
Relevance-Acceptability-Sufficiency criteria (Johnson and Blair, 2006) and
asked annotators to decide whether a piece of argument is sufficiently
supported or not.

Besides, Carlile et al. (2018) annotated detailed persuasiveness and
attributes values based on argument trees from S&G2014 and S&G2017a. They
defined an argument as a node in the argument tree along with all its
children. For each argument, they assigned an overall persuasive score, common
attribute values and type specific attribute values. For example, for
arguments using a major claim as the root node, annotations include persuasive
score, eloquence, specificity, evidence (common attributes) and persuasive
strategies (type-specific attribute).

##  3 Methods

All the methods to our knowledge are learning based and they can be divided
into supervised learning and unsupervised learning. Most of the work use
supervised learning, which can be further divided into feature-based
approaches and neural approaches. Next, we introduce these approaches in more
details.

###  3.1 Feature-based method

For feature-based approaches, off-the-shelf algorithms are typically used for
model training on hand-crafted input features. For example, support vector
machine (SVM), logistic regression and random forest are typically used for
tasks that can be framed as classification (Stab and Gurevych, 2014b, Stab and
Gurevych, 2017b, Stab and Gurevych, 2017a, Persing and Ng, 2016, Beigman
Klebanov et al., 2017, Wan et al., 2021). Linear regression and support vector
machine regression are often used for task that can be framed as regression
(Persing and Ng, 2015, Wachsmuth et al., 2016, Persing and Ng, 2015).

Next, we introduce more details of common features used in these methods.

Lexical features aim to capture word-level information and common lexical
features include n-gram and frequent words. They have been shown effective
(Stab and Gurevych, 2014b, Beigman Klebanov et al., 2017, Stab and Gurevych,
2017b). However, they do not perform as well when used in a cross-prompt
setting Beigman Klebanov et al. (2017) where prompts of testing essays are not
seen during training. This is intuitive as the actual wording of essays of
different prompts would differ significantly.

Syntactic features usually rely on parse trees. Common syntactic features
include number of sub-clauses in a parse tree, Boolean indicator of production
rules, part-of-speech tags, etc. Additionally, basic information such as tense
of verbs, presence of modal verbs can also serve as syntactic features. Stab
and Gurevych (2017a) showed that syntactic features are useful for identifying
the beginning of an argument component and Stab and Gurevych (2017b) suggested
that syntactic features are effective for recognizing insufficiently supported
arguments.

Structural features generally describe the position and frequency of a piece
of text. For example, they include position of a token, punctuation and an
argument component. They also include statistics such as number of tokens in
an argument component. Stab and Gurevych (2017a) reported effectiveness for
these features on both ACI and ACC tasks.

Embedding features are based on word vectors that represent words in a
continuous space and are supposed to capture more information than simple
n-grams. Stab and Gurevych (2017a) summed the word2vec Mikolov et al. (2013)
vectors for each token to represent a component. Putra et al. (2021b) used
BERT Devlin et al. (2019) to extract token embeddings in their work.

Discourse features captures how sentences or clauses are connected together.
One kind of discourse features depend on discourse markers directly. The
markers, such as "therefore", suggest the relationship between current text
span and its adjacent text span. Another kind of discourse features use the
output of discourse parsers. For example, Beigman Klebanov et al. (2017)
parsed sentences into corresponding discourse roles, and then used these
discourse roles as features. Stab and Gurevych (2017a) reported usefulness of
discourse features on classifying argument components, indicating a
correlation between general discourse relation and argument component type.
Beigman Klebanov et al. (2017) found that discourse features remain useful in
cross-prompt settings, which is valuable as it’s not always possible to
collect a lot of data for a single prompt.

###  3.2 Neural method

As for neural approaches, neural architectures such as long short-term memory
(LSTM) networks and convolutional neural networks (CNN) are commonly adopted
(Eger et al., 2017, Alhindi and Ghosh, 2021, Putra et al., 2021b, Mim et al.,
2019a, Xue and Lynch, 2020,Stab and Gurevych, 2017b). Besides, Transformer
Vaswani et al. (2017) based architectures such as BERT Devlin et al. (2019)
are adopted recently (Ye and Teufel, 2021, Putra et al., 2021b, Ghosh et al.,
2020, Alhindi and Ghosh, 2021 Wang et al., 2020). We will describe more
details of that later on.

Use of pretrained models The use of pretrained language models has been
popular among natural language processing community. This is because state-of-
the-art models are pretrained on massive text corpus, allowing information
learned from a huge amount of text to be used for downstream tasks.

There are two ways to use pretrained models. First, it can be used as feature
extractor. For example, Putra et al. (2021b) has used BERT, a bi-directional
Transformer based architecture, to extract input embeddings, and then pass the
contextualized embeddings to downstream networks. Second, the pretrained
language models can be further fine-tuned for the task of interest. Ye and
Teufel (2021), Wang et al. (2020), Alhindi and Ghosh (2021) and Ghosh et al.
(2020) used this approach. In addition, other than fine-tuning on the task
data directly, Alhindi and Ghosh (2021) and Ghosh et al. (2020) experimented
with continued pre-training with a large unlabelled domain relevant corpus
first, and then fine-tuning with task data.

###  3.3 Address multiple sub-tasks

As mentioned, argument mining mostly concerns four sub-tasks and they are
often addressed together. The naive way to solve them simultaneously is to
model each sub-task separately in a pipeline fashion. This introduces at least
two issues: first, it does not enforce any constraints between different
tasks; second, the errors made early on in the pipeline could propagate.

One way to address these is to use integer linear programming (ILP) Roth and
Yih (2004) for joint inference. Stab and Gurevych (2017a) and Persing and Ng
(2016) adopted this approach and Persing and Ng (2016) further proposed an ILP
objective that directly optimizes F score.

Another way to address these issues is to build a joint model that can model
all these tasks at the same time. Eger et al. (2017) proposed two different
frameworks for joint modelling the full argument mining task and has been
influential. Figure 2 showed an illustration of these two formulations. They
first framed the task as sequence tagging. For example, they used S&G2017a
data for experiments and in this case, each token’s label space would include
1) whether this token is a begin or continue of an argument component or it is
non-argumentative); 2) the type of the component to which the token belongs;
3) distance between the corresponding component and the component it relates
to; 4) the relation type between the two related components. This way they can
use off-the-shelf taggers to solve all four sub-tasks at once. They also
framed the task as dependency parsing. When framed as dependency parsing, the
text is represented as directed trees where each token has a labelled head so
that argument component relation information can be encoded. They further
labelled these edges with tokens’ component types and relation types so that
argument trees can be converted to quasi-dependency trees. Finally, they
adapted a joint neural model designed for entity detection and relation
extraction Miwa and Bansal (2016). In this case, they modelled argument
components as entities and argument relations as semantic relations.

Figure 2: Illustration of the formulations described in 3.3. On the top is an
annotated sentence. The second chunk represents sequence tagging formulation
and the bottom is dependency parsing formulation. In the figure, B means the
begin of a component, I means the continue of a component, C stands for Claim,
P stands for Premise, O stands for non-argumentative, Supp stands for Support
and $\emptyset$ means not filled. Figure is adapted from Eger et al. (2017)

###  3.4 Unsupervised learning

Aside from the supervised learning work described above, Persing and Ng (2020)
proposed an unsupervised method for argument mining. The key for their work is
to use heuristics for bootstrapping a small set of labels and then train the
model in a self-training fashion.

##  4 Experiments

The use of Transformer-based models has been dominating in other natural
language processing tasks but is still in its infant stage in this field.
Therefore, we experiment with vanilla BERT models on three representative
datasets, hoping to facilitate research to that end. The three datasets cover
argumentation structure parsing and argument quality assessment. Specifically,
we used S&G2017a dataset, which is the benchmark for parsing essay
argumentation structure. Besides, we used S&G2017b dataset, which assess
argument quality from a logic aspect Wachsmuth et al. (2017) by annotating
whether a piece of argument is sufficiently supported. We also used P&N2015
dataset which assess argument quality from a rhetoric aspect Wachsmuth et al.
(2017) by assigning an overall argument strength score to each essay.
Statistics for these datasets can be found in Table 3, 4, 5 and 6.

###  4.1 Implementation details

| All | Per Essay
---|---|---
Token | 147,271 | 366.3
Sentence | 7,116 | 17.7
Paragraph | 1,833 | 4.6
Essay | 402 | 1
Table 3: Data statistics for the whole S&G2017a dataset.

S&G2017a For S&G2017a dataset, we followed Eger et al. (2017) and used 286
essays for training, 36 essays for validation and 80 essays for testing on
paragraph level. For model building, we used Huggingface Wolf et al. (2020)
library’s cased base BERT model. We added one shared dropout layer and three
linear heads for predicting labels mentioned in 3.3. The loss is computed by
summing up the cross entropy loss on each head. Note that BERT’s tokenizer can
tokenize each word into multiple sub-tokens but we want to predict only one
set of tags for each word. To address this, we only use the first sub-token
for training. For model training, we used AdamW optimizer Loshchilov and
Hutter (2017), learning rate 3e-5 and Cosine Annealing Warm Restart scheduler
Loshchilov and Hutter (2016) implemented by PyTorch Lightning library Falcon
et al. (2019). We also monitored validation loss for early stopping. We only
tuned dropout rate and patience used for early stopping minimally and we set
patience to 5 and dropout rate to 0.5.

| All | Per Argument
---|---|---
Token | 97,370 | 94.6
Sentence | 4,593 | 4.5
Argument | 1,029 | 1
| Number | Percentage
Sufficient | 681 | 66.2%
Insufficient | 348 | 33.8%
Table 4: Data statistics for S&G2017b dataset. It includes the size of the
corpus and class distributions.

S&G2017b For S&G2017b dataset, we used 823 arguments for training, 103
arguments for validation and 103 arguments for testing. We used the same BERT
model and added a dropout layer and a linear layer. We used binary cross
entropy loss as objective function. For model hyperparameters, we set patience
to 5, dropout rate to 0.3 and kept everything else the same as S&G2017a.

| All | Per Essay
---|---|---
Token | 649,549 | 649.5
Sentence | 31,589 | 31.6
Paragraph | 7,537 | 7.5
Essay | 1,000 | 1
Table 5: Data statistics for P&N2015 dataset. Score | 1.0 | 1.5 | 2.0 | 2.5 | 3.0 | 3.5 | 4.5
---|---|---|---|---|---|---|---
Essays | 2 | 21 | 116 | 342 | 372 | 132 | 15
Table 6: Score distribution for P&N2015 dataset.

P&N2015 For P&N2015 dataset, we used 800 essays for training, 100 essays for
validation and 100 essays for testing. Additionally, we ran cross-prompt
experiments by randomly selecting 247 essays from prompt 1 for training, 31
essays for validation, and rest of prompt 1 essays as well as essays from
other prompts for testing. We also used one dropout layer and one linear layer
on top of the base BERT model. For objective function, we used mean square
loss. Besides, we set patience to 5 and dropout rate to 0.1 and kept
everything else the same as S&G2017a.

###  4.2 Metrics and comparison methods

S&G2017a Similar to most work, we followed Persing and Ng (2016) for
evaluation. They defined F1 score as F1 = $\frac{2TP}{2TP+FP+FN}$, where TP
stands for true positive, FP stands for false positive and FN stands for false
negative. They also define ’level $\alpha$ matching’ Eger et al. (2017): for
$\alpha$% level match, the predicted component span and ground truth share at
least $\alpha$% of their tokens. For comparison methods, we chose the LSMT-ER
model from Eger et al. (2017), which is a common baseline. In addition, we
compared with the BiPAM model from Ye and Teufel (2021), which is a BERT-
enhanced biaffine dependency parser Dozat and Manning (2018).

S&G2017b We used macro F1 and accuracy scores for evaluation, following Stab
and Gurevych (2017b). As for comparison methods, we chose the best performing
CNN model in their work and human upper bound.

P&N2015 We used mean absolute error (MAE) and mean square error (MSE) for
evaluation. We compared the baseline with a model developed by Wachsmuth et
al. (2016), which is the best performing model to our knowledge.

###  4.3 Results and discussion

S&G2017a The experiment results for this argument mining task are shown in
Table 7. The vanilla BERT did not outperform state-of-the-art methods but are
close. The BiPAM model, which is a BERT-enhanced dependency parser, performs
the best. These demonstrate the power of Transformer-based models for the
argument mining task. Additionally, LSTM-ER has 1) designed separate module
for handling components and relations; 2) encoded explicit syntax information
through syntactic parser. BiPAM has carefully designed the argument relation
representation so that it can benefit from state-of-the-art dependency
parsers. These considerations can be used for further unlocking the potential
of Transformer-based models.

| C-F1 | R-F1
---|---|---
| 100% | 50% | 100% | 50%
LSTM-ER | 70.8 | 77.2 | 45.5 | 50.1
BiPAM | 72.9 | N/A | 45.9 | N/A
BERT+linear | 69.3 | 76.7 | 43.7 | 47.6
Table 7: Performance of LSTM-ER, BiPAM and vanilla BERT on S&G2017a dataset.
The models are all trained on paragraph level and we report both 100% level
match and 50% level match results. C-F1 stands for argument component F1 and
R-F1 stands for argument relation F1. Best scores are in bold.

S&G2017b The experiment results for sufficiency recognition are shown in Table
8. We can see that the vanilla BERT model outperforms the previous best
performing CNN model by a large margin. This is expected as the BERT model has
been pretrained on a huge amount of text. It is surprising that the vanilla
BERT model already achieved near-human performance. Therefore, we foresee that
Transformer-based model can equal or even surpass human upper bound on
sufficiency recognition in the near future.

| Accuracy | Macro F1
---|---|---
Human | 0.911±.022 | 0.887±.026
CNN | 0.843±.025 | 0.827±.027
BERT+linear | 0.882±.018 | 0.869±.012
Table 8: Performance of CNN, vanillaBERT and human upper bound on S&G2017b.
Best scores are in bold.

P&N2015 The experiment results for argument strength scoring are shown in
Table 9. On this dataset, the vanilla BERT model did not outperform previous
methods. There are two possible reasons for explaining it. First, the average
token in an single essay exceed the maximum length(512) supported by vanilla
BERT model. This results in part of the essay being truncated. From the data
statistics, we can know that at least 25% of the essay is being truncated,
resulting in large information loss. Second, both Persing and Ng (2015) and
Wachsmuth et al. (2016) encoded argument structure information explicitly by
crafting a list of argument structure related features. Mim et al. (2019b)
incorporated paragraph’s argument function information by pretraining on
large-scale essays with shuffled paragraphs. At the same time, BERT model was
not pretrained on relevant task and might fail at capturing the overall
argument structure of an essay.

| MAE | MSE
---|---|---
Persing2015 | 0.392 | 0.244
Wachsmuth2016 | 0.378 | 0.226
Mim2019 | N/A | 0.231
BERT+linear | 0.394 | 0.250
Table 9: Performance of best models from Persing and Ng (2015), Wachsmuth et al. (2016) and Mim et al. (2019b), and vanillaBERT on P&N2015.  Test Prompt | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
---|---|---|---|---|---|---|---|---|---|---
MAE | 0.421 | 0.399 | 0.485 | 0.443 | 0.471 | 0.389 | 0.371 | 0.369 | 0.398 | 0.439
MSE | 0.237 | 0.268 | 0.349 | 0.286 | 0.403 | 0.225 | 0.217 | 0.246 | 0.251 | 0.283
Table 10: Results of cross-prompt experiments on P&N2015 dataset

To further gauge the generalization ability of Transformer-based models, we
ran the vanilla BERT in a cross-prompt setting and the results are shown in
10. Being able to generalize across prompt is valuable in practice because it
is expensive to collect data for each new prompt. Recall that the model is
trained and validated on essays of prompt 1. From 3, we can see that the MSE
and MAE remains similar across prompts except for prompt 3 and prompt 5. We
took a look at these two prompts and found that they are much more abstract
and provide less concrete context than the other prompts. 111Due to licensing
of the ICLE dataset, we cannot provide the detailed information here. We
hypothesize it to be the performance drop on prompt 3 and prompt 5 and thus
believe that BERT can generalize reasonably well.

Figure 3: Line chart based on Table 10

Overall, the vanilla BERT models perform comparably to previous methods across
datasets. We believe this can be a good baseline model for future research in
the community.

##  5 Conclusion and Future Directions

In this paper we have organize existing works around tasks, data and methods
regarding automated evaluation for student argumentative writing. We provided
a baseline to the community by experimenting with BERT models on benchmark
datasets. Now the question would be: what is important for the future? Towards
this end, we identified several directions we deem interesting.

First of all, there is a lack of emphasis on language diversity in current
datasets. This concerns two aspect: language of the essays and language
backgrounds of the authors. For language of the essays, most datasets are
annotated essays written in English. Horbach et al. (2017) and Wambsganss et
al. (2020) annotated essays written in German and these two datasets are the
only ones not written in English to our knowledge. Intuitively speaking,
characteristics such as overall argumentation structure would vary across
essays written in different languages. Besides, annotators for essays written
in different languages may have different criteria when assessing the rhetoric
quality of an essay. As for authors’ language backgrounds, current datasets
either do not take into account authors’ language levels or use essays written
by proficient writers (e.g. college-level writing). 222Although P&N2015
dataset annotated English learner’s texts, most of the authors’ native
languages belong to Indo-European languages and have received at least 6 or
more years’ English education Putra et al. (2021a) and Alhindi and Ghosh
(2021) are the only accessible datasets that target authors whose interested
language skills are not proficient yet, to our knowledge. Specifically, Putra
et al. (2021a) annotated essays written by English learners from various Asian
countries. They discarded already well-written essays and only reserved the
ones with intermediate quality. Alhindi and Ghosh (2021) annotated
argumentative essays by middle school students and found these essays less
structured and thus more challenging. We believe collecting more diverse
datasets would be valuable because it will not only expand the impact of
argumentative writing support systems, but also pose more challenging research
problems.

Second, adoption of Transformer-based models is still in the infant stage
despite some recent works have used Transformer-based architectures. As
described in 4, we have built the simplest form of BERT models and
demonstrated that they have comparable performance to previous state-of-the-
art methods. In addition, Ghosh et al. (2020) and Alhindi and Ghosh (2021)
showed that performance of BERT can be greatly improved by continued pre-
training on unlabelled essay datasets or architectural design that takes
account of data characteristics. Therefore, we believe there is still huge
potential for Transformer-based models.

Third, there is no research regarding generalization in this field. We are
aware of relevant research works in the AES area (Jin et al. 2018, Cao et al.
2020, Ridley et al. 2021) but not for systems that care about argument-related
attributes. Not being able to generalize across prompts remains a major
bottleneck in AES Woods et al. (2017) and we believe this can also be a main
obstacle for deploying automated evaluation systems specifically for
argumentative essays. This is because it is costly to collect data for every
prompt. At the same time, learners usually need to practice writing over a
large amount of prompts and get feedback before seeing significant
improvement. This echoes our promotion for Transformer-based models as we have
shown in 4.3 a vanilla BERT model can generalize well across prompts within
one dataset. Besides, Transformer-based models have shown to be effective on
unseen domains in other NLP tasks (Houlsby et al. 2019, Han and Eisenstein
2019). Overall, we believe it is critical to build generalizeable systems and
hope to see more research addressing this issue in the future.

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Integral Representations of Three Novel Multiple Zeta Functions

for Barnes Type: A Probabilistic Approach

Gwo Dong Lin a and Chin-Yuan Hu b

a Academia Sinica, Taiwan

b National Changhua University of Education, Taiwan

Abstract. Integral representation is one of the powerful tools for studying
analytic continuation of the zeta functions. It is known that Hurwitz zeta
function generalizes the famous Riemann zeta function which plays an important
role in analytic number theory. They both have several multiple versions in
the literature. In this paper, we introduce three novel multiple zeta
functions for Barnes type and study their integral representations through
hyperbolic probability distributions given by Pitman and Yor (2003, Canad. J.
Math., 55, 292–330). The analytically continued properties of the three
multiple zeta functions are also investigated. Surprisingly, two of them,
unlike the previous results, can extend analytically to entire functions in
the whole complex plane.

Key words and phrases: Integral representation, Analytical continuation,
Barnes zeta function, Hyperbolic functions, Hurwitz zeta function, Riemann
zeta function.
2020 Mathematics Subject Classifications: Primary 11M06, 11M32, 60E05.
Running title: Integral representations of three novel Barnes multiple zeta
functions
Postal addresses:
Gwo Dong Lin, Institute of Statistical Science, Academia Sinica, No. 128,
Section 2, Academia Road, Nankang District, Taipei 11529, Taiwan (ROC).
(E-mail<EMAIL_ADDRESS>
Chin-Yuan Hu, National Changhua University of Education, No. 1, Jinde Road,
Changhua 50007, Taiwan (ROC). (E-mail<EMAIL_ADDRESS>

1\. Introduction

Consider the Hurwitz zeta function

$\displaystyle\zeta(s;b)=\sum_{n=0}^{\infty}\frac{1}{(n+b)^{s}},$ (1)

where the parameter $b>0$ and the complex variable $s\in\mathbb{C}$ has the
real part $\rm{Re}(s)>1.$ When $b=1$, $\zeta(s;b)$ reduces to the famous
Riemann zeta function

$\displaystyle\zeta(s)\equiv\zeta(s;1)=\sum_{n=1}^{\infty}\frac{1}{n^{s}},\quad{\rm
Re}(s)>1,$ (2)

which plays an important role in analytic number theory (see, e.g., Apostal
1976). It is known that each of both zeta functions in (1) and (2) can extend
to a meromorphic function in the whole complex plane $\mathbb{C}$ and has a
simple pole at $s=1$ where it has residue 1.

As for multidimensional generalizations of (1) and (2), there are various
multiple ($r$-tuple or $r$-fold) zeta functions proposed in the literature
(see, e.g., the survey paper by Matsumoto 2010, the thesis by Onozuka 2014,
and the references therein), where $r\geq 1$ is an integer. We recall a few of
the results in the following:
(i) Euler–Zagier multiple (Riemann) zeta function (Euler 1776, Zagier 1994,
Hoffman 1992, Zhao 2000, Matsumoto 2002a, Mehta et al. 2016)

$\zeta(s_{1},s_{2},\ldots,s_{r})=\sum_{0<n_{1}<n_{2}<\cdots<n_{r}}\frac{1}{n_{1}^{s_{1}}n_{2}^{s_{2}}\cdots
n_{r}^{s_{r}}},\quad\sum_{k=j}^{r}{\rm Re}(s_{k})>r-j+1,\ \ 1\leq j\leq r;$

(ii) Multiple Hurwitz zeta function (Kamano 2006, Murty and Sinha 2006,
Kelliher and Masri 2008, Mehta and Viswanadham 2017)

$\zeta(s_{1},s_{2},\ldots,s_{r};b)=\sum_{0\leq
n_{1}<n_{2}<\cdots<n_{r}}\prod_{j=1}^{r}\frac{1}{(n_{j}+b)^{s_{j}}},\quad
b>0,\quad\sum_{k=j}^{r}{\rm Re}(s_{k})>r-j+1,\ 1\leq j\leq r,$

or, in general,

$\zeta(s_{1},\ldots,s_{r};b_{1},\ldots,b_{r})=\sum_{0\leq
n_{1}<n_{2}<\cdots<n_{r}}\prod_{j=1}^{r}\frac{1}{(n_{j}+b_{j})^{s_{j}}},\ \
b_{j}>0,\ \ \sum_{k=j}^{r}{\rm Re}(s_{k})>r-j+1,\ \ 1\leq j\leq r;$

(iii) Barnes multiple zeta function (of one variable) (Barnes 1901, 1904, Choi
and Quine 1992, Ruijsenaars 2000, Matsumoto 2002a, 2010)

$\zeta_{B}(s)\equiv\zeta_{B}(s;a_{1},\ldots,a_{r};b)=\sum_{n_{1},n_{2},\ldots,n_{r}=0}^{\infty}\frac{1}{(a_{1}n_{1}+a_{2}n_{2}+\cdots+a_{r}n_{r}+b)^{s}},\quad{\rm
Re}(s)>r,$

where $a_{1},\ldots,a_{r}$ and $b$ are complex parameters satisfying some
suitable conditions (e.g., all of them have positive real parts).

For further generalizations of the Barnes multiple zeta function
$\zeta_{B}(s)$, see Shintani (1977, 1978) and Mellin (1902, 1904) for the case
of one variable, and Matsumoto (2002a, 2002b, 2003a, 2003b, 2005) for the case
of several variables. The analytically continued properties of the above-
mentioned multiple zeta functions have been investigated. For example, the
Barnes multiple zeta function $\zeta_{B}(s)$ extends to a meromorphic function
in $s\in\mathbb{C}$ and has simple poles only at $s=1,2,...,r$ (see Barnes
1904, Ruijsenaars 2000 or Matsumoto 2010). The main tools of analytic
continuation of the zeta functions include integral representation,
Mellin–Barnes integral formula, Euler–Maclaurin summation, translation formula
(identity), generalized function, etc. For the recent development on the
multiple zeta functions, see the monograph by Zhao (2016).

In this paper, we will introduce the following three variants (a) – (c) of the
Barnes multiple zeta function, mimicking the Shintani double zeta functions of
the form (Shintani 1975, Kim et al. 2022)

$\zeta_{1}(s_{1},s_{2})=\sum_{n_{1},n_{2}=1}^{\infty}\frac{A(4n_{1},n_{2})}{n_{1}^{s_{1}}n_{2}^{s_{2}}},\quad\zeta_{2}(s_{1},s_{2})=\sum_{n_{1},n_{2}=1}^{\infty}\frac{A(n_{1},-n_{2})}{n_{1}^{s_{1}}(4n_{2})^{s_{2}}},$

where $A(n_{1},n_{2})$ denotes the number of distinct solutions to the
quadratic congruence equation $x^{2}=n_{2}\ ({\rm mod}\ n_{1}).$

Let $r\geq 1$ be an integer and for convenience let
$\alpha_{1},\alpha_{2},\ldots,\alpha_{r},$ $a_{1},a_{2},\ldots,a_{r},b$ be
positive real numbers. Denote
${\bf\alpha}=(\alpha_{1},\alpha_{2},\ldots,\alpha_{r}),$ ${\bf
a}=(a_{1},a_{2},\ldots,a_{r})$ and
$\beta=\alpha_{1}+\alpha_{2}+\cdots+\alpha_{r}.$ As usual, we adopt the
notations $0!=1,$ and for real $\gamma,$

${\gamma\choose 0}=1,\quad{\gamma\choose
n}=\frac{\gamma(\gamma-1)\cdots(\gamma-n+1)}{n!}\quad{\rm if}\
n\in\mathbb{N}\equiv\\{1,2,\dots\\}.$

Then, we define
(a) the multiple hyperbolic sinh-moment function

$\displaystyle S_{{\bf\alpha},{\bf
a}}(s;b)=\sum_{n_{1},n_{2},\ldots,n_{r}=0}^{\infty}\frac{{n_{1}+\alpha_{1}-1\choose
n_{1}}\cdots{n_{r}+\alpha_{r}-1\choose
n_{r}}}{(a_{1}n_{1}+\cdots+a_{r}n_{r}+b)^{s}},\quad{\rm Re}(s)>\beta;$ (3)

(b) the multiple hyperbolic cosh-moment function

$\displaystyle C_{{\bf\alpha},{\bf
a}}(s;b)=\sum_{n_{1},n_{2},\ldots,n_{r}=0}^{\infty}\frac{{-\alpha_{1}\choose
n_{1}}\cdots{-\alpha_{r}\choose
n_{r}}}{(a_{1}n_{1}+\cdots+a_{r}n_{r}+b)^{s}},\quad{\rm Re}(s)>\beta;$ (4)

(c) the multiple hyperbolic tanh-moment function

$\displaystyle T_{{\bf\alpha},{\bf
a}}(s;b)=\sum_{n_{1},n_{2},\ldots,n_{r}=0}^{\infty}E\Big{\\{}\frac{{-\alpha_{1}\choose
n_{1}}\cdots{-\alpha_{r}\choose
n_{r}}}{(a_{1}(n_{1}+V_{1})+\cdots+a_{r}(n_{r}+V_{r})+b)^{s}}\Big{\\}},\quad{\rm
Re}(s)>\beta,$ (5)

where $\alpha_{1},\ldots,\alpha_{r}$ are integers,
$V_{j}=U_{j1}+\cdots+U_{j\alpha_{j}},\ 1\leq j\leq r,$ and
$U_{j1},\ldots,U_{j\alpha_{j}},\ 1\leq j\leq r,$ are independently and
identically distributed (i.i.d.) uniform random variables on $(0,1).$

It is seen that when all $\alpha_{j}=1,\ 1\leq j\leq r,$ the function
$S_{{\bf\alpha},{\bf a}}(s;b)$ in (3) above reduces to the Barnes multiple
zeta function $\zeta_{B}(s)$ (see Corollary 1 below). We will provide the
integral representations of the above three novel multiple zeta functions
through the hyperbolic probability distributions given by Pitman and Yor
(2003), and investigate their analytically continued properties including the
set of singularities. Surprisingly, unlike the previous results, the functions
$C_{{\bf\alpha},{\bf a}}(s;b)$ and $T_{{\bf\alpha},{\bf a}}(s;b)$ in (4) and
(5) above can extend analytically to entire functions in $s\in\mathbb{C}.$ The
main results are stated in Section 3\. Section 2 provides the needed lemmas.
Finally, the proofs of the main results are given in Section 4.

2\. Lemmas
We need the following preparatory lemmas. Lemma 1 presents the basic
properties of the gamma function (see, e.g., Apostal 1976, Chapter 12). Lemma
2 is about Liouville’s extension of Dirichlet integral theorem (see, e.g.,
Sharma 2005, Chapter 2, or Gibson 1931, p. 492). The crucial Lemma 3 modifies
slightly Cheng and Hu’s (2022) Lemma 2.1. The hyperbolic random variables in
Lemma 4 are introduced and investigated by Pitman and Yor (2003). On the other
hand, for the derivations of the hyperbolic density functions in Lemma 5, see
Cheng and Hu (2022) as well as Pitman and Yor (2003).
Lemma 1 (Gamma function). Let $\Gamma(s)$ be the gamma function defined by

$\Gamma(s)=\int_{0}^{\infty}x^{s-1}e^{-x}dx,\quad{\rm Re}(s)>0.$

Then the function $\Gamma(s)$ can extend to a meromorphic function in the
whole complex plane $\mathbb{C}$ and has simple poles at the points
$s=0,-1,-2,\ldots,$ with residue $(-1)^{n}/n!$ at $s=-n,$ where
$n=0,1,2,\ldots.$ The gamma function satisfies the equation
$\Gamma(s+1)=s\,\Gamma(s).$ It has no zeros, and hence the reciprocal gamma
function, $1/\Gamma(s)$, is an entire function in $s\in\mathbb{C}$ with zeros
at $s=0,-1,-2,\ldots.$ Moreover, if $x>0$ is not an integer, then

$\Gamma(-x)=\frac{(-1)^{[x]+1}\Gamma([x]+1-x)}{x(x-1)\cdots(x-[x])},$

where $[x]$ denotes the greatest integer less than or equal to $x.$
Lemma 2 (Liouville–Dirichlet integral). Let $m\geq 2$ be an integer and let
$f$ be a Lebesgue measurable function on $[0,1].$ Then we have

$\displaystyle\idotsint\limits_{A}f(t_{1}+\cdots+t_{m-1})\,t_{1}^{s_{1}-1}\cdots
t_{m-1}^{s_{m-1}-1}dt_{1}\cdots dt_{m-1}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}=\frac{\Gamma(s_{1})\cdots\Gamma(s_{m-1})}{\Gamma(s_{1}+\cdots+s_{m-1})}\int_{0}^{1}f(t)\,t^{s_{1}+\cdots+s_{m-1}-1}dt,$

provided that the resulting single integral on the right exists. Here, the
parameters $s_{j}>0,$ $1\leq j\leq m-1,$ and the region of multiple integral

$A=\\{(t_{1},\ldots,t_{m-1}):t_{j}\geq 0,\,1\leq j\leq
m-1,\,t_{1}+\cdots+t_{m-1}\leq 1\\}.$

In particular, if $f(t)=(1-t)^{s_{m}-1},$ where $t\in[0,1]$ and $s_{m}>0,$
then the following formula holds:

$\displaystyle\idotsint\limits_{A}t_{1}^{s_{1}-1}\cdots
t_{m-1}^{s_{m-1}-1}(1-t_{1}-\cdots-t_{m-1})^{s_{m}-1}dt_{1}\cdots dt_{m-1}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=\frac{\Gamma(s_{1})\cdots\Gamma(s_{m})}{\Gamma(s_{1}+\cdots+s_{m})},\quad
s_{j}>0,\ 1\leq j\leq m.$

Lemma 3 (Integral transform of a distribution). Let $X$ be a random variable
with distribution $F_{X}$ on $\mathbb{R},$ denoted $X\sim F_{X},$ and let $X$
have finite moments of all orders. For any nonzero real $c,$ define the
complex-valued function

$h_{X}(s;c)\equiv\int_{\mathbb{R}}\frac{1}{(c+ix)^{s}}dF_{X}(x)=E\big{\\{}(c+iX)^{-s}\big{\\}},\quad
s\in\mathbb{C},$

where $(c+ix)^{s}=\exp({s{\rm Log}(c+ix)}).$ Then $h_{X}(s;c)$ is an entire
function in $s\in\mathbb{C}.$
Proof. We sketch the proof below. For each $d>0,$ define the closed square

${S}(d)=\\{s=\sigma+it:\ |\sigma|\leq d,\ |t|\leq
d,\quad\sigma,t\in\mathbb{R}\\}.$

Denote the integer $n_{0}=[d]+1.$ Then there exists a constant
$C=C(c,d,n_{0},F)$ such that

$\int_{\mathbb{R}}\Big{|}\frac{1}{(c+ix)^{s}}\Big{|}dF_{X}(x)\leq C\quad{\rm
for\ all}\ s\in S(d).$

This implies that the integral transformation $h_{X}(s;c)$ is absolutely
convergent and converges uniformly in each compact subset of $\mathbb{C}.$
Moreover, $h_{X}(s,c)$ is a continuous function in $s\in\mathbb{C}.$ Let
$\triangle$ be any closed triangle in $\mathbb{C}$ and recall that for each
fixed real $x,$ $(c+ix)^{-s}$ is an entire function in $s.$ Then we have, by
Cauchy’s theorem,

$\int_{\partial{\triangle}}h_{X}(s,c)ds=\int_{\partial{\triangle}}\int_{\mathbb{R}}\frac{1}{(c+ix)^{s}}dF_{X}(x)ds=\int_{\mathbb{R}}\int_{\partial{\triangle}}\frac{1}{(c+ix)^{s}}dsdF_{X}(x)=0,$

where $\partial{\triangle}$ is the boundary of $\triangle.$ Finally, Morera’s
theorem completes the proof.
Remark 1. Note that for $s=-n,$ $n\in\mathbb{N},$ the function value of
$h_{X}(s;c)$ becomes a linear combination of moments of $X\sim F_{X}.$ Thus,
for simplicity, we might call the integral transformation $h_{X}(s;c)$ a
moment function of the distribution $F_{X}.$
Lemma 4 (Hyperbolic random variables). For $t>0,$ let
$\hat{S}_{t},\hat{C}_{t},\hat{T}_{t}$ be three random variables with
characteristic functions

$E(e^{i\theta\hat{S}_{t}})=\Big{(}\frac{\theta}{\sinh\theta}\Big{)}^{t},\quad
E(e^{i\theta\hat{C}_{t}})=\Big{(}\frac{1}{\cosh\theta}\Big{)}^{t},\quad
E(e^{i\theta\hat{T}_{t}})=\Big{(}\frac{\tanh\theta}{\theta}\Big{)}^{t},\quad\theta\in\mathbb{R},$

respectively. Then the hyperbolic random variables
$\hat{S}_{t},\hat{C}_{t},\hat{T}_{t}$ are infinitely divisible and have finite
moments of all orders.
For $t>0,$ let $\hat{S}_{t}\sim F_{S,t},\,\hat{C}_{t}\sim
F_{C,t},\,\hat{T}_{t}\sim F_{T,t},$ where $F_{S,t},\,F_{C,t},\,F_{T,t}$ are
called hyperbolic $\sinh$-, $\cosh$-, $\tanh$-distributions, respectively.
Lemma 5 (Hyperbolic distributions). For $t>0,$ $F_{S,t},\,F_{C,t},\,F_{T,t}$
are absolutely continuous symmetric distributions $($about zero$)$ and have
probability density functions

$F_{S,t}^{\prime}(x)=f_{S,t}(x)=\frac{2^{t}}{\pi}\int_{0}^{1}\frac{\cos(x\ln
y)y^{t-1}(\ln y)^{t}}{(y^{2}-1)^{t}}dy,\quad x\in\mathbb{R},$
$F_{C,t}^{\prime}(x)=f_{C,t}(x)=\frac{2^{t-2}}{\pi}Be((t+ix)/2,(t-ix)/2),\quad
x\in\mathbb{R},$
$F_{T,t}^{\prime}(x)=f_{T,t}(x)=\frac{1}{\pi}\int_{0}^{1}\frac{\cos(x\ln
y)(y^{2}-1)^{t}}{(\ln y)^{t}(y^{2}+1)^{2}}\cdot\frac{1}{y}dy,\quad
x\in\mathbb{R},$

respectively, where $Be$ is the Beta function

$Be(s_{1},s_{2})=\int_{0}^{1}t^{s_{1}-1}(1-t)^{s_{2}-1}dt=\frac{\Gamma(s_{1})\Gamma(s_{2})}{\Gamma(s_{1}+s_{2})},\quad{\rm
Re}(s_{1})>0,\ \ {\rm Re}(s_{2})>0.$

In particular, for $t=1,2,$ we have the even density functions:

$f_{S,1}(x)=\frac{\pi}{4\cosh^{2}(\pi x/2)},\quad
f_{S,2}(x)=\frac{\frac{\pi}{2}(\frac{\pi}{2}x\coth(\frac{\pi}{2}x)-1)}{\sinh^{2}(\pi
x/2)},\quad x\in\mathbb{R};$ $f_{C,1}(x)=\frac{1}{2\cosh(\pi x/2)},\quad
f_{C,2}(x)=\frac{x}{2\sinh(\pi x/2)},\quad x\in\mathbb{R};$
$f_{T,1}(x)=\frac{1}{\pi}\log\coth(\frac{\pi}{4}|x|),\quad
f_{T,2}(x)=\int_{|x|}^{\infty}\frac{y(y-|x|)}{2\sinh(\pi y/2)}dy,\quad
x\in\mathbb{R}.$

3\. Main results

Throughout the section we assume that $r\geq 1$ is an integer and that
$\alpha_{1},\ldots,\alpha_{r}$,$a_{1},\ldots,a_{r}$ and $b$ are positive real
numbers satisfying $c\equiv
b-\frac{1}{2}(a_{1}\alpha_{1}+a_{2}\alpha_{2}+\cdots+a_{r}\alpha_{r})\neq 0.$
Denote ${\bf\alpha}=(\alpha_{1},\ldots,\alpha_{r}),\,{\bf
a}=(a_{1},\ldots,a_{r}),$ $\beta=\alpha_{1}+\alpha_{2}+\cdots+\alpha_{r}$ and
$c_{r}=(a_{1}^{\alpha_{1}}\cdots a_{r}^{\alpha_{r}})^{-1}.$ Moreover, the
notation “$\,\stackrel{{\scriptstyle d}}{{=}}\,$” means equality in
distribution. Then we have the following results, which as mentioned below
generalize some of the previous results given in Cheng and Hu (2022).
Theorem 1. Assume that $X_{j}\stackrel{{\scriptstyle
d}}{{=}}\frac{1}{2}\hat{S}_{\alpha_{j}},j=1,2,\ldots,r,$ are independent
random variables, where $\hat{S}_{\alpha_{j}}$ are hyperbolic sinh random
variables defined in Lemma $4$. Then the multiple hyperbolic sinh-moment
function $S_{{\bf\alpha},{\bf a}}(s;b)$ in $(3)$ has an integral
representation of the form:

$\displaystyle S_{{\bf\alpha},{\bf
a}}(s;b)=c_{r}\frac{\Gamma(s-\beta)}{\Gamma(s)}\\!\int_{\mathbb{R}}\frac{1}{(c+iy)^{s-\beta}}dF_{Y_{r}}(y)=c_{r}\frac{\Gamma(s-\beta)}{\Gamma(s)}E\big{\\{}(c+iY_{r})^{-s+\beta}\big{\\}},\
{\rm Re}(s)>\beta,$ (6)

where $Y_{r}=a_{1}X_{1}+a_{2}X_{2}+\cdots+a_{r}X_{r}.$ Moreover, the following
statements hold true.
$(a)$ The zeta function $S_{{\bf\alpha},{\bf a}}(s;b)$ in $(6)$ can extend to
a meromorphic function in $s\in\mathbb{C}$ and all the poles are simple.
$(b)$ If the parameter $\beta$ is a positive integer, then the function
$S_{{\bf\alpha},{\bf a}}(s;b)$ in $(6)$ is an analytic function in
$s\in\mathbb{C}$ except for $s=1,2,\ldots,\beta.$ Precisely,

$S_{{\bf\alpha},{\bf
a}}(s;b)=\frac{c_{r}}{(s-1)\cdots(s-\beta)}\int_{\mathbb{R}}\frac{1}{(c+iy)^{s-\beta}}dF_{Y_{r}}(y),\quad
s\neq 1,2,\ldots,\beta.$

$(c)$ If the parameter $\beta>0$ is not an integer, then the zeta function
$S_{{\bf\alpha},{\bf a}}(s;b)$ in $(6)$ has trivial zeros at
$s=0,-1,-2,\ldots,$ and we have the following two cases for poles.
$(i)$ For non-negative integer $n<\beta,$ the point $\beta-n$ is a simple pole
of $S_{{\bf\alpha},{\bf a}}(s;b)$ with residue

$\lim_{s\to\beta-n}(s-\beta+n)S_{{\bf\alpha},{\bf
a}}(s;b)=\frac{(-1)^{n}c_{r}E\big{(}(c+iY_{r})^{n}\big{)}}{n!\Gamma(\beta-n)}.$

$(ii)$ For positive integer $n>\beta,$ the point $\beta-n$ is a simple pole of
$S_{{\bf\alpha},{\bf a}}(s;b)$ with residue

$\lim_{s\to\beta-n}(s-\beta+n)S_{{\bf\alpha},{\bf
a}}(s;b)=\frac{(-1)^{n}c_{r}E\big{(}(c+iY_{r})^{n}\big{)}}{n!\,d_{\beta,n}},$

where

$d_{\beta,n}=\frac{(-1)^{[n-\beta]+1}\Gamma(\beta-n+1+[n-\beta])}{(n-\beta)(n-\beta-1)\cdots(n-\beta-[n-\beta])}.$

If $\alpha_{1}=\alpha_{2}=\cdots=\alpha_{r}=1,$ then $\beta=r$ and the
multiple hyperbolic sinh-moment function $S_{{\bf\alpha},{\bf a}}(s;b)$ in (3)
reduces to the Barnes multiple zeta function $\zeta_{B}(s).$ Thus, we have the
following.
Corollary 1. The Barnes multiple zeta function $\zeta_{B}(s)$ has the integral
representation

$\displaystyle\zeta_{B}(s)=\frac{c_{r}}{(s-1)\cdots(s-r)}\int_{\mathbb{R}}\frac{1}{(c+iy)^{s-r}}dF_{Y_{r}}(y),\quad{\rm
Re}(s)>r,$ (7)

where $c_{r}=(a_{1}\cdots a_{r})^{-1},$
$c=b-\frac{1}{2}(a_{1}+\cdots+a_{r})\neq 0,$
$Y_{r}=a_{1}X_{1}+\cdots+a_{r}X_{r}$ and $X_{1},\ldots,X_{r}$ are $r$
independent random variables, each $X_{j}\stackrel{{\scriptstyle
d}}{{=}}\hat{S}_{1}/2$ having the density function

$\displaystyle f_{1}(x)=\frac{\pi}{2\cosh^{2}(\pi x)},\quad x\in\mathbb{R}.$
(8)

Moreover, $\zeta_{B}(s)$ in $(7)$ extends to a meromorphic function in
$s\in\mathbb{C}$ and has simple poles at $s=1,2,\ldots,r.$
If, further, $r=1$ and $\alpha_{1}=a_{1}=1,$ then the multiple hyperbolic
sinh-moment function $S_{{\bf\alpha},{\bf a}}(s;b)$ reduces to the Hurwitz
zeta function $\zeta(s;b).$ Thus, we have the following.
Corollary 2 (Cheng and Hu 2022). The Hurwitz zeta function $\zeta(s;b)$ has
the integral representation of the form

$\displaystyle\zeta(s;b)=\frac{1}{s-1}\int_{\mathbb{R}}\frac{1}{(c+iy)^{s-1}}dF_{Y_{1}}(y),\quad{\rm
Re}(s)>1,$ (9)

where $c=b-\frac{1}{2}\neq 0,$ $Y_{1}=X_{1}$ and $X_{1}\stackrel{{\scriptstyle
d}}{{=}}\hat{S}_{1}/2$ has the density function in $(8).$ Moreover,
$\zeta(s;b)$ in $(9)$ extends to a meromorphic function in $s\in\mathbb{C}$
and has a simple pole at $s=1.$
If $r=2$ and $\alpha_{1}=\alpha_{2}=1,a_{1}=a_{2}=1,$ then $\beta=2$ and the
multiple hyperbolic sinh-moment function $S_{{\bf\alpha},{\bf a}}(s;b)$ in (3)
reduces to

$\displaystyle\zeta_{2}(s;b)=\sum_{n_{1},n_{2}=0}^{\infty}\frac{1}{(n_{1}+n_{2}+b)^{s}}=\sum_{n=1}^{\infty}\frac{n}{(n+b-1)^{s}},\quad{\rm
Re}(s)>2.$ (10)

Corollary 3 (Cheng and Hu 2022). The function $\zeta_{2}(s;b)$ in $(10)$ has
the integral representation

$\displaystyle\zeta_{2}(s;b)=\frac{1}{(s-1)(s-2)}\int_{\mathbb{R}}\frac{1}{(c+iy)^{s-2}}dF_{Y_{2}}(y),\quad{\rm
Re}(s)>2,$ (11)

where $c=b-1\neq 0$ and $Y_{2}=X_{1}+X_{2}\stackrel{{\scriptstyle
d}}{{=}}\hat{S}_{2}/2$ has the density function

$f_{2}(y)=\frac{\pi(\pi y\coth(\pi y)-1)}{\sinh^{2}(\pi y)},\quad
y\in\mathbb{R}.$

Moreover, $\zeta_{2}(s;b)$ in $(11)$ extends to a meromorphic function in
$s\in\mathbb{C}$ and has simple poles at $s=1,2.$

Theorem 2. Assume that $X_{j}\stackrel{{\scriptstyle
d}}{{=}}\frac{1}{2}\hat{C}_{\alpha_{j}},j=1,2,\ldots,r,$ are independent
random variables, where $\hat{C}_{\alpha_{j}}$ are hyperbolic cosh random
variables defined in Lemma $4.$ Then the multiple hyperbolic cosh-moment
function $C_{{\bf\alpha},{\bf a}}(s;b)$ in $(4)$ has an integral
representation of the form:

$\displaystyle C_{{\bf\alpha},{\bf
a}}(s;b)=2^{-\beta}\int_{\mathbb{R}}\frac{1}{(c+iz)^{s}}dF_{Z_{r}}(z)=2^{-\beta}E\big{\\{}(c+iZ_{r})^{-s}\big{\\}},\quad{\rm
Re}(s)>\beta,$ (12)

where $Z_{r}=a_{1}X_{1}+a_{2}X_{2}+\cdots+a_{r}X_{r}.$ Moreover, the zeta
function $C_{{\bf\alpha},{\bf a}}(s;b)$ in $(12)$ can extend analytically to
an entire function in $s\in\mathbb{C}.$
If $\alpha_{1}=\alpha_{2}=\cdots=\alpha_{r}=1,$ then $\beta=r$ and the
multiple hyperbolic cosh-moment function $C_{{\bf\alpha},{\bf a}}(s;b)$ in (4)
reduces to the following zeta function

$\displaystyle\zeta_{BC}(s)=\sum_{n_{1},\ldots,n_{r}=0}^{\infty}\frac{(-1)^{n_{1}+\cdots+n_{r}}}{(a_{1}n_{1}+\cdots+a_{r}n_{r}+b)^{s}},\quad{\rm
Re}(s)>r$ (13)

(compare with the zeta function $\zeta_{B}(s)$). Thus, we have the following.
Corollary 4. The zeta function $\zeta_{BC}(s)$ in $(13)$ has the integral
representation of the form

$\displaystyle\zeta_{BC}(s)=2^{-r}\int_{\mathbb{R}}\frac{1}{(c+iz)^{s}}dF_{Z_{r}}(z),\quad{\rm
Re}(s)>r,$ (14)

where $c_{r}=(a_{1}\cdots a_{r})^{-1},$
$c=b-\frac{1}{2}(a_{1}+\cdots+a_{r})\neq 0,$
$Z_{r}=a_{1}X_{1}+a_{2}X_{2}+\cdots+a_{r}X_{r}$ and $X_{1},\ldots,X_{r}$ are
$r$ independent random variables, each $X_{j}\stackrel{{\scriptstyle
d}}{{=}}\hat{C}_{1}/2$ having the density function

$\displaystyle g_{1}(x)=\frac{1}{\cosh(\pi x)},\quad x\in\mathbb{R}.$ (15)

Moreover, the zeta function $\zeta_{BC}(s)$ in $(14)$ extends to an entire
function in $s\in\mathbb{C}.$
If, further, $r=1$ and $a_{1}=\alpha_{1}=1,$ then the zeta function
$\zeta_{BC}(s)$ in (13) reduces to

$\displaystyle\zeta_{C}(s)=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n+b)^{s}},\quad{\rm
Re}(s)>1.$ (16)

Thus, we have the following.
Corollary 5 (Cheng and Hu 2022). The zeta function $\zeta_{C}(s)$ in $(16)$
has the integral representation

$\displaystyle\zeta_{C}(s)=2^{-1}\int_{\mathbb{R}}\frac{1}{(c+iz)^{s}}dF_{Z_{1}}(z),\quad{\rm
Re}(s)>1,$ (17)

where $c=b-\frac{1}{2}\neq 0$ and $Z_{1}=X_{1}$ has the density function
$g_{1}$ in $(15).$ Moreover, the zeta function $\zeta_{C}(s)$ in $(17)$
extends to an entire function in $s\in\mathbb{C}.$
If $r=2$ and $\alpha_{1}=\alpha_{2}=1,a_{1}=a_{2}=1,$ then $\beta=2$ and the
multiple hyperbolic cosh-moment function $C_{{\bf\alpha},{\bf a}}(s;b)$ in (4)
reduces to

$\displaystyle\zeta_{C_{2}}(s)=\sum_{n_{1},n_{2}=0}^{\infty}\frac{(-1)^{n_{1}+n_{2}}}{(n_{1}+n_{2}+b)^{s}}=\sum_{n=0}^{\infty}\frac{(-1)^{n}(n+1)}{(n+b)^{s}},\quad{\rm
Re}(s)>2.$ (18)

Then we have the following.
Corollary 6 (Cheng and Hu 2022). The zeta function $\zeta_{C_{2}}(s)$ in
$(18)$ has the integral representation of the form

$\displaystyle\zeta_{C_{2}}(s)=\frac{1}{4}\int_{\mathbb{R}}\frac{1}{(c+iz)^{s}}dF_{Z_{2}}(z),\quad{\rm
Re}(s)>2,$ (19)

where $c=b-1\neq 0$ and $Z_{2}=X_{1}+X_{2}\stackrel{{\scriptstyle
d}}{{=}}\hat{C}_{2}/2$ has the density function

$g_{2}(z)=\frac{2z}{\sinh(\pi z)},\quad z\in\mathbb{R}.$

Moreover, the zeta function $\zeta_{C_{2}}(s)$ in $(19)$ extends to an entire
function in $s\in\mathbb{C}.$
Theorem 3. Assume that $X_{j}\stackrel{{\scriptstyle
d}}{{=}}\frac{1}{2}\hat{T}_{\alpha_{j}},j=1,2,\ldots,r,$ are independent
random variables, where $\alpha_{1},\ldots,\alpha_{r}$ are positive integers
and $\hat{T}_{\alpha_{j}}$ are hyperbolic tanh random variables defined in
Lemma $4.$ Then the multiple hyperbolic tanh-moment function
$T_{{\bf\alpha},{\bf a}}(s;b)$ in $(5)$ has an integral representation of the
form:

$\displaystyle T_{{\bf\alpha},{\bf
a}}(s;b)=2^{-\beta}\int_{\mathbb{R}}\frac{1}{(b+iw)^{s}}dF_{W_{r}}(w)=2^{-\beta}E\\{(b+iW_{r})^{-s}\\},\quad{\rm
Re}(s)>\beta,$ (20)

where $W_{r}=a_{1}X_{1}+a_{2}X_{2}+\cdots+a_{r}X_{r}.$ Moreover, the zeta
function $T_{{\bf\alpha},{\bf a}}(s;b)$ in $(20)$ can extend analytically to
an entire function in $s\in\mathbb{C}.$
If $\alpha_{1}=\alpha_{2}=\cdots=\alpha_{r}=1,$ then $\beta=r$ and the
multiple hyperbolic tanh-moment function $T_{{\bf\alpha},{\bf a}}(s;b)$ in (5)
reduces to

$\displaystyle\zeta_{T,{\bf
a}}(s;b)=\sum_{n_{1},n_{2},\ldots,n_{r}=0}^{\infty}E\Big{\\{}\frac{(-1)^{n_{1}+\cdots+n_{r}}}{(a_{1}(n_{1}+V_{1})+\cdots+a_{r}(n_{r}+V_{r})+b)^{s}}\Big{\\}},\quad{\rm
Re}(s)>r,$ (21)

where $V_{j}=U_{j1}+\cdots+U_{j\alpha_{j}},\ 1\leq j\leq r,$ and
$U_{j\ell_{j}},\ 1\leq j\leq r,\ 1\leq\ell_{j}\leq\alpha_{j},$ are i.i.d.
uniform random variables on $(0,1).$ Thus, we have the following.
Corollary 7. The multiple hyperbolic tanh-moment function $\zeta_{T,{\bf
a}}(s;b)$ in $(21)$ has the integral representation of the form

$\displaystyle\zeta_{T,{\bf
a}}(s;b)=\frac{1}{2^{r}}\int_{\mathbb{R}}\frac{1}{(b+iw)^{s}}dF_{W_{r}}(w),\quad{\rm
Re}(s)>r,$ (22)

where $W_{r}=a_{1}X_{1}+\cdots+a_{r}X_{r}$ and $X_{1},\ldots,X_{r}$ are i.i.d.
random variables, each $X_{j}\stackrel{{\scriptstyle d}}{{=}}\hat{T}_{1}/2$
having the density function

$\displaystyle h_{1}(x)=\frac{2}{\pi}\log\coth(\frac{\pi}{2}|x|),\quad
x\in\mathbb{R}.$ (23)

Moreover, $\zeta_{T,{\bf a}}(s;b)$ in $(22)$ extends to an entire function in
$s\in\mathbb{C}.$
If, further, $r=1$ and $\alpha_{1}=a_{1}=1,$ then the multiple hyperbolic
tanh-moment function $\zeta_{T,{\bf a}}(s;b)$ in (21) reduces to

$\displaystyle\zeta_{T}(s;b)=\sum_{n=0}^{\infty}E\Big{\\{}\frac{(-1)^{n}}{(n+b+U)^{s}}\Big{\\}},\quad{\rm
Re}(s)>1,$ (24)

where $U$ has the uniform distribution in $(0,1).$
Corollary 8 (Cheng and Hu 2022). The multiple hyperbolic tanh-moment function
in $(24)$ has the integral representation of the form

$\displaystyle\zeta_{T}(s;b)=\frac{1}{2}\int_{\mathbb{R}}\frac{1}{(b+iw)^{s}}dF_{W_{1}}(w),\quad{\rm
Re}(s)>1,$ (25)

where $W_{1}=X_{1}\stackrel{{\scriptstyle d}}{{=}}\hat{T}_{1}/2$ has the
density function in $(23).$ Moreover, $\zeta_{T}(s;b)$ in $(25)$ extends to an
entire function in $s\in\mathbb{C}.$
If $r=2$ and $\alpha_{1}=\alpha_{2}=1,a_{1}=a_{2}=1,$ then $\beta=2$ and the
multiple hyperbolic tanh-moment function $\zeta_{T,{\bf a}}(s;b)$ in (21)
reduces to

$\displaystyle\zeta_{T_{2}}(s;b)=\sum_{n=0}^{\infty}E\Big{\\{}\frac{(-1)^{n}(n+1)}{(n+b+U_{1}+U_{2})^{s}}\Big{\\}},\quad{\rm
Re}(s)>2,$ (26)

where $U_{1},U_{2}$ are i.i.d. uniform random variables on $(0,1).$
Corollary 9 (Cheng and Hu 2022). The zeta function $\zeta_{T_{2}}(s;b)$ in
$(26)$ has the integral representation

$\displaystyle\zeta_{T_{2}}(s;b)=\frac{1}{4}\int_{\mathbb{R}}\frac{1}{(b+iw)^{s}}dF_{W_{2}}(w),\quad{\rm
Re}(s)>2,$ (27)

where $W_{2}=X_{1}+X_{2}\stackrel{{\scriptstyle d}}{{=}}\hat{T}_{2}/2$ has the
density function

$h_{2}(w)=\int_{2|w|}^{\infty}\frac{y(y-2|w|)}{\sinh(\pi y/2)}dy,\quad
w\in\mathbb{R}.$

Moreover, $\zeta_{T_{2}}(s;b)$ in $(27)$ extends to an entire function in
$s\in\mathbb{C}.$
Remark 2. In addition to the above single variable cases, it is also possible
to consider the following cases of several variables, where ${\bf
s}=(s_{1},s_{2},\ldots,s_{r})\in\mathbb{C}^{r}:$

$S^{*}_{{\bf\alpha},{\bf a}}({\bf s};{\bf
b})=\sum_{n_{1},n_{2},\ldots,n_{r}=0}^{\infty}\prod_{j=1}^{r}\frac{{n_{j}+\alpha_{j}-1\choose
n_{j}}}{(a_{1}n_{1}+\cdots+a_{r}n_{r}+b_{j})^{s_{j}}},\quad{\rm
Re}(s_{j})>\alpha_{j},\ 1\leq j\leq r,$

and

$C^{*}_{{\bf\alpha},{\bf a}}({\bf s};{\bf
b})=\sum_{n_{1},n_{2},\ldots,n_{r}=0}^{\infty}\prod_{j=1}^{r}\frac{{-\alpha_{j}\choose
n_{j}}}{(a_{1}n_{1}+\cdots+a_{r}n_{r}+b_{j})^{s_{j}}},\quad{\rm
Re}(s_{j})>\alpha_{j},\ 1\leq j\leq r,$

in which the parameters $\alpha_{j},a_{j},\ b_{j}>0,\ 1\leq j\leq r,$ and
${\bf\alpha}=(\alpha_{1},\ldots,\alpha_{r}),\,{\bf a}=(a_{1},\ldots,a_{r}),$
${\bf b}=(b_{1},\ldots,b_{r}).$ In these cases, Lemma 2 applies. The results
are too complicated to be presented here.

4\. Proofs of the main results

It suffices to prove Theorems 1, 2 and 3, because the corollaries can be
derived easily from the theorems.

Proof of Theorem 1. We first claim that the multiple series on the RHS of (3)
converges absolutely provided ${\rm Re}(s)>\beta.$ To see this, note that by
Stirling’s formula, or more generally,
$\Gamma(x)\approx\sqrt{2\pi}x^{x-\frac{1}{2}}e^{-x}$ as $x\to\infty,$ there
exist some constants $c_{\alpha_{j}}>0,\ 1\leq j\leq r,$ such that

${n_{j}+\alpha_{j}-1\choose
n_{j}}=\frac{\Gamma(n_{j}+\alpha_{j})}{\Gamma(n_{j}+1)\Gamma(\alpha_{j})}\approx
c_{\alpha_{j}}n_{j}^{\alpha_{j}-1}\ \ {\rm as}\ n_{j}\to\infty,\ 1\leq j\leq
r.$

On the other hand, for fixed $s\in\mathbb{C}$ with ${\rm Re}(s)>\beta,$ take
$\varepsilon=\varepsilon(s)>0$ such that ${\rm Re}(s)=\beta+r\varepsilon,$
then

$\displaystyle(a_{1}n_{1}+\cdots+a_{r}n_{r}+b)^{\rm Re(s)}$ $\displaystyle=$
$\displaystyle(a_{1}n_{1}+\cdots+a_{r}n_{r}+b)^{\beta+r\varepsilon}$
$\displaystyle>$
$\displaystyle\prod_{j=1}^{r}a_{j}^{\alpha_{j}+\varepsilon}n_{j}^{\alpha_{j}+\varepsilon}.$

Finally, apply the fact that each series
$\sum_{n_{j}=1}^{\infty}{n_{j}^{-(1+\varepsilon)}}$ converges.

Multiplying both sides of the multiple hyperbolic sinh-moment function
$S_{{\bf\alpha},{\bf a}}(s;b)$ in (3) by $\Gamma(s),$ we have

$\displaystyle\Gamma(s)S_{{\bf\alpha},{\bf
a}}(s;b)=\sum_{n_{1},n_{2},\ldots,n_{r}=0}^{\infty}$
$\displaystyle{n_{1}+\alpha_{1}-1\choose
n_{1}}\cdots{n_{r}+\alpha_{r}-1\choose n_{r}}$ (28) $\displaystyle\cdot\ \
\frac{\Gamma(s)}{(a_{1}n_{1}+\cdots+a_{r}n_{r}+b)^{s}},\quad{\rm
Re}(s)>\beta.$

Recall the gamma identity

$\displaystyle\frac{\Gamma(s)}{a^{s}}=\int_{0}^{\infty}x^{s-1}e^{-ax}dx,\quad{\rm
Re}(s)>0,\quad a>0,$ (29)

and the binomial series $(1+x)^{-r}=\sum_{k=0}^{\infty}{-r\choose
k}x^{k},\quad|x|<1,$ or, equivalently,

$\displaystyle(1-x)^{-r}=\sum_{k=0}^{\infty}{-r\choose
k}(-x)^{k}=\sum_{k=0}^{\infty}{k+r-1\choose k}x^{k},\quad|x|<1.$ (30)

Now, by (29), (30), the above claim and Fubini’s theorem, we can rewrite (28)
as follows:

$\displaystyle\Gamma(s)S_{{\bf\alpha},{\bf a}}(s;b)$ $\displaystyle=$
$\displaystyle\sum_{n_{1},n_{2},\ldots,n_{r}=0}^{\infty}{n_{1}+\alpha_{1}-1\choose
n_{1}}\cdots{n_{r}+\alpha_{r}-1\choose n_{r}}$ (31)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\cdot\int_{0}^{\infty}x^{s-1}e^{-(a_{1}n_{1}+\cdots+a_{r}n_{r}+b)x}dx$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}x^{s-1}e^{-bx}\Big{\\{}\sum_{n_{1},n_{2},\ldots,n_{r}=0}^{\infty}{n_{1}+\alpha_{1}-1\choose
n_{1}}\cdots{n_{r}+\alpha_{r}-1\choose n_{r}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\cdot
e^{-(a_{1}n_{1}+\cdots+a_{r}n_{r})x}\Big{\\}}dx$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}x^{s-1}e^{-bx}\prod_{j=1}^{r}(1-e^{-a_{j}x})^{-\alpha_{j}}dx,\quad{\rm
Re}(s)>\beta.$

Next, consider the product term of the integrand in (31) and write, using
Lemma 4,

$\displaystyle\prod_{j=1}^{r}(1-e^{-a_{j}x})^{-\alpha_{j}}$ $\displaystyle=$
$\displaystyle\prod_{j=1}^{r}\Big{\\{}\frac{e^{a_{j}x/2}a_{j}x/2}{a_{j}x\sinh(a_{j}x/2)}\Big{\\}}^{\alpha_{j}}=\prod_{j=1}^{r}\frac{e^{a_{j}\alpha_{j}x/2}}{a_{j}^{\alpha_{j}}x^{\alpha_{j}}}E\big{(}e^{ia_{j}xX_{j}}\big{)}$
(32) $\displaystyle=$ $\displaystyle\frac{x^{-\beta}}{a_{1}^{\alpha_{1}}\cdots
a_{r}^{\alpha_{r}}}E\big{(}e^{(\lambda+iY_{r})x}\big{)},\quad x>0,$

where $\lambda=\frac{1}{2}(a_{1}\alpha_{1}+\cdots+a_{r}\alpha_{r}).$

Inserting (32) in (31) yields, by (29) and Fubini’s theorem again,

$\displaystyle\Gamma(s)S_{{\bf\alpha},{\bf a}}(s;b)$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\frac{x^{s-\beta-1}}{a_{1}^{\alpha_{1}}\cdots
a_{r}^{\alpha_{r}}}E\big{(}e^{-(b-\lambda-iY_{r})x}\big{)}dx$ (33)
$\displaystyle=$ $\displaystyle\frac{1}{a_{1}^{\alpha_{1}}\cdots
a_{r}^{\alpha_{r}}}\int_{0}^{\infty}\int_{-\infty}^{\infty}x^{s-\beta-1}e^{-(b-\lambda-
iy)x}dF_{Y_{r}}(y)dx$ $\displaystyle=$
$\displaystyle\frac{1}{a_{1}^{\alpha_{1}}\cdots
a_{r}^{\alpha_{r}}}\int_{-\infty}^{\infty}\int_{0}^{\infty}x^{s-\beta-1}e^{-(b-\lambda-
iy)x}dxdF_{Y_{r}}(y)$ $\displaystyle=$
$\displaystyle\frac{1}{a_{1}^{\alpha_{1}}\cdots
a_{r}^{\alpha_{r}}}\int_{-\infty}^{\infty}\frac{\Gamma(s-\beta)}{(b-\lambda-
iy)^{s-\beta}}dF_{Y_{r}}(y)$ $\displaystyle=$
$\displaystyle\frac{\Gamma(s-\beta)}{a_{1}^{\alpha_{1}}\cdots
a_{r}^{\alpha_{r}}}E\big{(}(c-iY_{r})^{-s+\beta}\big{)}$ $\displaystyle=$
$\displaystyle
c_{r}\Gamma(s-\beta)E\big{(}(c+iY_{r})^{-s+\beta}\big{)},\quad{\rm
Re}(s)>\beta,$

where we apply the symmetric property of the hyperbolic sinh-distribution.
This proves the integral representation in (6) by dividing both sides of (33)
by $\Gamma(s).$

Recall that the integral transformation $h_{Y_{r}}(s)\equiv
E\big{(}(c+iY_{r})^{-s+\beta}\big{)}$ is an entire function in
$s\in\mathbb{C}$ due to Lemma 3. Therefore, the zeta function
$S_{{\bf\alpha},{\bf a}}(s;b)$ in (6) can extend analytically to a meromorphic
function in $s\in\mathbb{C}$ and all the poles are simple due to Lemma 1. This
proves part (a). Parts (b) and (c) follow from part (a) and Lemma 1 again
immediately. The proof of Theorem 1 is completed.
Proof of Theorem 2. Note first that there is a relationship between
$C_{{\bf\alpha},{\bf a}}(s;b)$ and $S_{{\bf\alpha},{\bf a}}(s;b);$
specifically, we can rewrite

$C_{{\bf\alpha},{\bf
a}}(s;b)=\sum_{n_{1},n_{2},\ldots,n_{r}=0}^{\infty}(-1)^{\sum_{j=1}^{r}n_{j}}\frac{{n_{1}+\alpha_{1}-1\choose
n_{1}}\cdots{n_{r}+\alpha_{r}-1\choose
n_{r}}}{(a_{1}n_{1}+\cdots+a_{r}n_{r}+b)^{s}},$

so that the multiple series on the RHS of (4) also converges absolutely
provided ${\rm Re}(s)>\beta.$ Then proceeding along the same lines in the
proof of Theorem 1, we can prove the theorem as follows. Multiplying both
sides of the multiple hyperbolic sinh-moment function $C_{{\bf\alpha},{\bf
a}}(s;b)$ in (4) by $\Gamma(s),$ we have

$\displaystyle\Gamma(s)C_{{\bf\alpha},{\bf
a}}(s;b)=\\!\\!\sum_{n_{1},n_{2},\ldots,n_{r}=0}^{\infty}{-\alpha_{1}\choose
n_{1}}\cdots{-\alpha_{r}\choose
n_{r}}\frac{\Gamma(s)}{(a_{1}n_{1}+\cdots+a_{r}n_{r}+b)^{s}},\ \ {\rm
Re}(s)>\beta.$ (34)

As before, using the gamma identity and the binomial series, we can rewrite
(34) as follows:

$\displaystyle\Gamma(s)C_{{\bf\alpha},{\bf a}}(s;b)$ $\displaystyle=$
$\displaystyle\sum_{n_{1},n_{2},\ldots,n_{r}=0}^{\infty}{-\alpha_{1}\choose
n_{1}}\cdots{-\alpha_{r}\choose
n_{r}}\int_{0}^{\infty}x^{s-1}e^{-(a_{1}n_{1}+\cdots+a_{r}n_{r}+b)x}dx$ (35)
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}x^{s-1}e^{-bx}\Big{\\{}\sum_{n_{1},n_{2},\ldots,n_{r}=0}^{\infty}{-\alpha_{1}\choose
n_{1}}\cdots{-\alpha_{r}\choose
n_{r}}e^{-(a_{1}n_{1}+\cdots+a_{r}n_{r})x}\Big{\\}}dx$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}x^{s-1}e^{-bx}\prod_{j=1}^{r}(1+e^{-a_{j}x})^{-\alpha_{j}}dx,\quad{\rm
Re}(s)>\beta.$

Next, consider the product term of the integrand in (35) and write, using
Lemma 4,

$\displaystyle\prod_{j=1}^{r}(1+e^{-a_{j}x})^{-\alpha_{j}}$ $\displaystyle=$
$\displaystyle\prod_{j=1}^{r}\Big{\\{}\frac{e^{a_{j}x/2}}{2\cosh(a_{j}x/2)}\Big{\\}}^{\alpha_{j}}=\prod_{j=1}^{r}\frac{{e^{a_{j}\alpha_{j}x/2}}}{2^{\alpha_{j}}}E\big{(}e^{ia_{j}xX_{j}}\big{)}$
(36) $\displaystyle=$
$\displaystyle\frac{1}{2^{\beta}}E\big{(}e^{(\lambda+iZ_{r})x}\big{)},\quad
x>0,$

where $\lambda=\frac{1}{2}(a_{1}\alpha_{1}+\cdots+a_{r}\alpha_{r}).$

Inserting (36) in (35) yields, by (29) and Fubini’s theorem again,

$\displaystyle\Gamma(s)C_{{\bf\alpha},{\bf
a}}(s;b)=\int_{0}^{\infty}\frac{x^{s-1}}{2^{\beta}}E\big{(}e^{-(b-\lambda-
iZ_{r})x}\big{)}dx$ (37) $\displaystyle=$
$\displaystyle\frac{1}{2^{\beta}}\int_{0}^{\infty}\int_{-\infty}^{\infty}x^{s-1}e^{-(b-\lambda-
iz)x}dF_{Z_{r}}(z)dx$ $\displaystyle=$
$\displaystyle\frac{1}{2^{\beta}}\int_{-\infty}^{\infty}\int_{0}^{\infty}x^{s-1}e^{-(b-\lambda-
iz)x}dxdF_{Z_{r}}(z)=\frac{1}{2^{\beta}}\int_{-\infty}^{\infty}\frac{\Gamma(s)}{(b-\lambda-
iz)^{s}}dF_{Z_{r}}(z)$ $\displaystyle=$
$\displaystyle\frac{\Gamma(s)}{2^{\beta}}E\big{(}(c-iZ_{r})^{-s}\big{)}=\frac{1}{2^{\beta}}\Gamma(s)E\big{(}(c+iZ_{r})^{-s}\big{)},\quad{\rm
Re}(s)>\beta,$

where we apply the symmetric property of the hyperbolic cosh-distribution. By
cancelling the common term $\Gamma(s)$ in both sides of (37), we proves the
integral representation in (9).

Recall again that the integral transformation $h_{Z_{r}}(s)\equiv
E\big{(}(c+iZ_{r})^{-s}\big{)}$ is an entire function in $s\in\mathbb{C}$ due
to Lemma 3. Therefore, the zeta function $C_{{\bf\alpha},{\bf a}}(s;b)$ in
(12) can extend analytically to an entire function in $s\in\mathbb{C}.$ This
completes the proof of Theorem 2.
Proof of Theorem 3. As in the proof of Theorem 2, the multiple series on the
RHS of (5) converges absolutely provided ${\rm Re}(s)>\beta,$ by noting that

$a_{1}(n_{1}+v_{1})+\cdots+a_{r}(n_{r}+v_{r})+b>a_{1}n_{1}+\cdots+a_{r}n_{r}+b\
\ \ {\rm for}\ \ v_{j}>0,\ j=1,2,\dots,r.$

Multiplying both sides of the function $T_{{\bf\alpha},{\bf a}}(s;b)$ in (5)
by $\Gamma(s),$ we have

$\displaystyle\Gamma(s)T_{{\bf\alpha},{\bf a}}(s;b)$ $\displaystyle=$
$\displaystyle\sum_{n_{1},n_{2},\ldots,n_{r}=0}^{\infty}{-\alpha_{1}\choose
n_{1}}\cdots{-\alpha_{r}\choose
n_{r}}E\Big{\\{}\frac{\Gamma(s)}{(a_{1}(n_{1}+V_{1})+\cdots+a_{r}(n_{r}+V_{r})+b)^{s}}\Big{\\}},$
(38)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\quad{\rm
Re}(s)>\beta.$

Then, by (29) and Fubini’s theorem, we can rewrite the expectation

$\displaystyle
E\Big{\\{}\frac{\Gamma(s)}{(a_{1}(n_{1}+V_{1})+\cdots+a_{r}(n_{r}+V_{r})+b)^{s}}\Big{\\}}$
(39) $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\cdots\int_{0}^{\infty}\Gamma(s)\big{(}a_{1}(n_{1}+v_{1})+\cdots+a_{r}(n_{r}+v_{r})+b\big{)}^{-s}dF_{V_{1}}(v_{1})\cdots
dF_{V_{r}}(v_{r})$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\cdots\int_{0}^{\infty}\Big{(}\int_{0}^{\infty}x^{s-1}e^{-(a_{1}(n_{1}+v_{1})+\cdots+a_{r}(n_{r}+v_{r})+b)x}dx\Big{)}dF_{V_{1}}(v_{1})\cdots
dF_{V_{r}}(v_{r})$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}x^{s-1}e^{-(a_{1}n_{1}+\cdots
a_{r}n_{r}+b)x}\Big{(}\int_{0}^{\infty}\cdots\int_{0}^{\infty}\prod_{j=1}^{r}e^{-a_{j}xv_{j}}dF_{V_{1}}(v_{1})\cdots
dF_{V_{r}}(v_{r})\Big{)}dx$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}x^{s-1}e^{-(a_{1}n_{1}+\cdots
a_{r}n_{r}+b)x}\prod_{j=1}^{r}E\big{(}e^{-a_{j}xV_{j}}\big{)}dx$
$\displaystyle=$ $\displaystyle\int_{0}^{\infty}x^{s-1}e^{-(a_{1}n_{1}+\cdots
a_{r}n_{r}+b)x}\prod_{j=1}^{r}\\{\frac{1-e^{-a_{j}x}}{a_{j}x}\\}^{\alpha_{j}}dx,\quad\
{\rm Re}(s)>\beta,$

where we use the fact that for uniform random variable $U$ on $(0,1),$

$E\big{(}e^{-xU}\big{)}=\frac{1-e^{-x}}{x}\quad\ {\rm for}\ \ x>0.$

Next, recall that

$\displaystyle E\big{(}e^{i\theta
T_{\alpha_{j}}}\big{)}=\big{(}\frac{\tanh\theta}{\theta}\big{)}^{\alpha_{j}}\quad{\rm
for}\ \ \theta\in\mathbb{R}$ (40)

and that

$\displaystyle\sum_{n_{1},n_{2},\ldots,n_{r}=0}^{\infty}{-\alpha_{1}\choose
n_{1}}\cdots{-\alpha_{r}\choose n_{r}}e^{-(a_{1}n_{1}+\cdots
a_{r}n_{r}+b)x}=e^{-bx}\prod_{j=1}^{r}\big{(}1+e^{-a_{j}x}\big{)}^{-\alpha_{j}}.$
(41)

Then it follows from (38)–(41) and the definition of the random variable
$W_{r}$ that

$\displaystyle\Gamma(s)T_{{\bf\alpha},{\bf
a}}(s;b)=\int_{0}^{\infty}x^{s-1}e^{-bx}\prod_{j=1}^{r}\\{\frac{1}{a_{j}x}\\}^{\alpha_{j}}\prod_{j=1}^{r}\\{\frac{1-e^{-a_{j}x}}{1+e^{-a_{j}x}}\\}^{\alpha_{j}}dx$
(42) $\displaystyle=$ $\displaystyle
2^{-\beta}\int_{0}^{\infty}x^{s-1}e^{-bx}\prod_{j=1}^{r}\\{\frac{1}{a_{j}x/2}\\}^{\alpha_{j}}\prod_{j=1}^{r}\\{\tanh(a_{j}x/2)\\}^{\alpha_{j}}dx$
$\displaystyle=$ $\displaystyle
2^{-\beta}\int_{0}^{\infty}x^{s-1}e^{-bx}\prod_{j=1}^{r}E\\{e^{ia_{j}xX_{j}}\\}dx=2^{-\beta}\int_{0}^{\infty}x^{s-1}e^{-bx}E(e^{ixW_{r}})dx$
$\displaystyle=$ $\displaystyle
2^{-\beta}\int_{0}^{\infty}x^{s-1}e^{-bx}\int_{-\infty}^{\infty}e^{ixw}dF_{W_{r}}(w)dx=2^{-\beta}\int_{-\infty}^{\infty}\int_{0}^{\infty}x^{s-1}e^{-(b-iw)x}dxdF_{W_{r}}(w)$
$\displaystyle=$ $\displaystyle
2^{-\beta}\int_{-\infty}^{\infty}\frac{\Gamma(s)}{(b-iw)^{s}}dF_{W_{r}}(w)=2^{-\beta}\Gamma(s)E\\{(b-iW_{r})^{-s}\\}$
$\displaystyle=$ $\displaystyle
2^{-\beta}\Gamma(s)E\\{(b+iW_{r})^{-s}\\},\quad{\rm Re}(s)>\beta.$

Cancelling $\Gamma(s)$ in both sides of (42), we obtain the integral
representation in (20). Finally, Lemma 3 completes the proof of the theorem.

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