# On The Persona-based Summarization of Domain-Specific Documents

1Ankan Mullick 1Sombit Bose 1Rounak Saha∗ 2Ayan Kumar Bhowmick
1Pawan Goyal 1 Niloy Ganguly 2Prasenjit Dey 2Ravi Kokku
{ankanm, sbcs.sombit<EMAIL_ADDRESS>
{pawang<EMAIL_ADDRESS>{ayan, prasenjit<EMAIL_ADDRESS>
1Computer Science and Engineering Department, IIT Kharagpur, India. 2Emergence
AI  Authors contributed equally

###### Abstract

In an ever-expanding world of domain-specific knowledge, the increasing
complexity of consuming, and storing information necessitates the generation
of summaries from large information repositories. However, every persona of a
domain has different requirements of information and hence their
summarization. For example, in the healthcare domain, a persona-based (such as
Doctor, Nurse, Patient etc.) approach is imperative to deliver targeted
medical information efficiently. Persona-based summarization of domain-
specific information by humans is a high cognitive load task and is generally
not preferred. The summaries generated by two different humans have high
variability and do not scale in cost and subject matter expertise as domains
and personas grow. Further, AI-generated summaries using generic Large
Language Models (LLMs) may not necessarily offer satisfactory accuracy for
different domains unless they have been specifically trained on domain-
specific data and can also be very expensive to use in day-to-day operations.
Our contribution in this paper is two-fold: 1) We present an approach to
efficiently fine-tune a domain-specific small foundation LLM using a
healthcare corpus and also show that we can effectively evaluate the
summarization quality using AI-based critiquing. 2) We further show that AI-
based critiquing has good concordance with Human-based critiquing of the
summaries. Hence, such AI-based pipelines to generate domain-specific persona-
based summaries can be easily scaled to other domains such as legal,
enterprise documents, education etc. in a very efficient and cost-effective
manner.

On The Persona-based Summarization of Domain-Specific Documents

1Ankan Mullick 1Sombit Bose††thanks:  Authors contributed equally 1Rounak
Saha∗ 2Ayan Kumar Bhowmick 1Pawan Goyal 1 Niloy Ganguly 2Prasenjit Dey 2Ravi
Kokku {ankanm, sbcs.sombit<EMAIL_ADDRESS>{pawang,
<EMAIL_ADDRESS>{ayan, prasenjit<EMAIL_ADDRESS>1Computer Science
and Engineering Department, IIT Kharagpur, India. 2Emergence AI

##  1 Introduction

In the rapidly expanding digital world, the exponential growth of domain-
specific knowledge has posed unprecedented challenges in efficiently storing
and consuming vast information repositories. With the increasing complexity of
managing such information, the need for generation of precise and specific
summaries becomes important. This need becomes particularly evident for
domain-specific data as there exist different personas within a domain who
have different information requirements which should be reflected in generated
summaries. For instance, if we consider the healthcare domain, there exist
diverse personas111http://tiny.cc/x1guwz ranging from healthcare professionals
like doctors and nurses to patients who require targeted information
customized to their specific roles and comprehension levels.

Traditional generic approaches to summarization have often relied on humans to
perform this high cognitive load task. However, as the volume and diversity of
information burgeon with growing number of domains and personas, human
generated persona-based summaries encounter limitations in scalability, cost-
effectiveness, and consistency. The inherent subjectivity and variability
among different human summarizers hinder the reliability and efficiency of
such an approach. Although there have been several approaches in prior
literature with focus on generic summaries through extractive and abstractive
methods Paulus et al. (2017); Erkan and Radev (2004) as well as goal-oriented
summaries Hayashi et al. (2021); Zhu et al. (2022) but none of them have
focused on persona-based summarization of domain-specific information.
Goldsack et al. (2023); Luo et al. (2022) focus on building layman-
summarization comprehensible to non-technical audiences but do not
differentiate the various technical summaries based on persona and they also
do not use LLMs as an alternative evaluator. Our work differs in the sense
that we develop a pipelined approach that generates persona-specific training
summaries (doctor, patient, normal person), fine-tune small-size LLMs on this
data, and use GPT-4 to efficiently evaluate summary quality.

One possible solution is to harness the power of generic large language models
(LLMs) such as GPT-4 to automate the generation of persona-based summaries as
such models have been used to generate data for other NLP tasks Sun et al.
(2023); Yu et al. (2023). ChatGPT 222https://chat.openai.com/ is also used in
educational data generation Kieser et al. (2023); Maddigan and Susnjak (2023).
However, AI-generated summaries using generic LLMs may not be guaranteed to
achieve optimal accuracy across different domains unless they are trained on
domain-specific data and they can also be very expensive to use for daily
repeated inferences. In this paper, we take a step towards introducing a two-
fold contribution aimed at overcoming these challenges of generating domain-
specific persona-based summaries. Firstly, we present an efficient approach
towards the training of domain-specific, small-sized Large Language Models
(LLMs) on a corpus related to healthcare domain. Though data distillation from
a stronger model for supervised fine-tuning is a standard method, the novelty
of our work lies in the fact that we effectively employ the approach in this
context to build a cost-optimised summarization framework catering to
different healthcare persona given the scarcity of domain-specific data. This
approach addresses the limitations of generic LLMs by aligning the trained
model specifically to the intricacies of summaries in the healthcare domain.
Moreover, we showcase the effectiveness of utilizing AI-based critiquing for
the evaluation of summarization quality, providing a more automated and
scalable solution.

Secondly, we demonstrate the strong agreement between AI-based and human-based
critiquing of generated summaries, establishing the reliability of our
proposed approach. This not only validates the effectiveness of domain-
specific small LLM-based models in generating accurate summaries but also
opens up avenues for scalability across diverse domains. The implications of
our findings extend beyond healthcare, as the proposed AI pipeline can be
seamlessly adapted to other domains, including legal, corporate documents,
education, and more, offering a versatile and cost-effective solution for
generating persona-based summaries.

##  2 Proposed Framework

We describe here our framework for training our small-size domain-specific
LLM, generation of data for finetuning and evaluation, and other model
baselines that we compare our model against.

###  2.1 Dataset

We create persona-based dataset (named ‘Persona-Data’) utilizing
GPT-4333https://openai.com/gpt-4 with specific prompts on 1455 articles from
the publicly available WebMD444https://www.webmd.com/ website. The mean ratio
between summary length and document length is $0.2:1$. After data generation
using GPT-4, we did a step by step validation of the generated summaries using
an automated approach followed by manual verification to filter out the bad
generations. Around $4.89\%$ of document-summary pairs were filtered out. We
provide the detailed filtering steps with criteria and removal pairs (in %) as
shown in Table 1.

Filtering Steps with Criteria (removed) % Step 1: Too many special characters
and other string (HTML tags and # ) 1.52 Step 2: Incomplete Summary (By
checking punctuations) 0.86 Step 3: Conflict identification - Very similar
summaries of different persona 1.12 Step 4: If the summary contains Medical
Terms or numbers not present in the document (using QuickUMLS -
https://github.com/Georgetown-IR-Lab/QuickUMLS/) 1.39 Overall summaries
filtered out 4.89

Table 1: Step-by-Step Data Filtering

These articles, related to healthcare, form the basis for creation of their
summaries related to three distinct persona: (a) Doctor: Summaries focus on
medical terminology, guidelines and provide detailed technical information
suitable for medical professionals. (b) Patient: Summaries are easily
understandable, addressing patient concerns without excessive technical
jargon, focusing on top-level information. (c) Normal Person: Summaries are
tailored for a general audience without medical background, presented in
simple language and engaging for laypersons while avoiding technical terms.
The dataset comprises 1091, 73 and 291 articles for training, validation, and
testing respectively. Additionally, we select 50 WebMD articles and generate
manual summaries for three personas (termed as Annotated-Data) using the
Prolific555https://www.prolific.com/ annotation platform and Doctors to
evaluate the GPT-4 generated summaries against human curated
summaries666Code/Dataset details are in https://github.com/ankan2/persona-
healthcare (Details are in Appendix F).

Model Rouge1 Rouge2 RougeL Meteor Bleu BERT-Prec BERT-Rec BERT-F1 Falcon 23.3
5.0 12.8 12.4 1.3 78.6 73.2 75.8 BART 23.6 8.1 15.0 11.3 0.8 83.8 74.9 79.1
Pegasus 27.8 8.3 16.2 14.3 2.0 80.7 75.0 77.8 T5-FT 42.6 15.6 23.7 30.2 11.4
84.2 80.7 82.4 FT5-FT 41.2 15.5 23.3 29.0 11.2 83.9 79.9 81.9 LED-B 33.3 9.3
17.4 17.3 3.3 80.7 76.7 78.6 LED-L 39.8 12.3 25.6 25.3 9.3 82.5 78.7 80.6
L-V-7b 14.2 4.0 8.5 6.7 0.6 72.0 65.7 68.7 L-V-13b 24.1 7.3 14.3 11.5 1.1 80.8
73.5 77.0 L-V-70b 25.1 7.2 13.7 16.2 3.7 65.7 64.6 65.1 L-F-7b 45.9 17.2 25.2
32.6 12.5 83.6 82.6 83.1 L-F-13b 53.7 24.3 33.8 38.3 18.4 87.4 85.3 86.3

Table 2: Traditional Metrics’ based evaluation results (all are in %)

Model Rouge1 Rouge2 RougeL Meteor Bleu BERT-Prec BERT-Rec BERT-F1 Doctor 53.9
24.7 34.0 37.0 18.7 88.0 85.1 86.5 Patient 53.5 24.2 33.5 36.7 18.3 87.2 84.8
86.0 Nor-Per 53.6 23.9 33.9 41.0 18.1 86.9 86.1 86.5 Average 53.7 24.3 33.8
38.3 18.4 87.4 85.3 86.3

Table 3: Traditional Metrics’ based evaluation on different persona using
Llama2-13b Finetuning model (in %)

###  2.2 Model Architecture

Our training process consists of employing small foundation LLMs such as
Llama2 and finetuning such models on the training set of the WebMD data
described above. Specifically, we use supervised fine-tuning approach Li et
al. (2023) on the pre-trained vanilla model versions of Llama2.

Llama2777https://ai.meta.com/llama/: We perform supervised fine-tuning (SFT)
on the pretrained vanilla Llama2-7b and Llama2-13b models using the training
data that consists of prompt-completion pairs where the prompt comprises a
WebMD article in the training set and the instruction to generate a summary
based on a persona (doctor/patient/normal person) and the completion is the
corresponding persona-based summary generated using GPT-4. We use a parameter
efficient finetuning approach i.e. Quantized Low-Rank Adaptation (QLoRA)
Dettmers et al. (2023) to optimize the training process. After training, the
finetuned Llama2-7b and Llama2-13b models (referred to as L-F-7b and L-F-13b
respectively) acquire the ability to generate a persona-based summary for a
given medical article depending on the persona specified in the prompt.

Baselines: For comparison with our finetuned models on the persona-based
summary generation task, we use different state-of-the-art models as baselines
such as Falcon 7b-instruction tuned model Penedo et al. (2023), BART-large
Lewis et al. (2019), instruction-tuned Pegasus Zhang et al. (2020) and
Longformer Beltagy et al. (2020) Base (LED-B), Large
(LED-L)888https://huggingface.co/allenai/led-base-16384. Besides these, we
also use finetuned versions of T5-Large Raffel et al. (2020) (T5-FT) and
Flan-T5-Large Chung et al. (2022) (FT5-FT) on our training data as baselines.
Further, we also compare the performance with the different vanilla Llama2
model variants (7b, 13b, 70b referred to as L-V-7b, L-V-13b and L-V-70b
respectively).

##  3 Evaluation and Results

We evaluate the performance of our finetuned models (L-F-7b and L-F-13b) in
terms of generating high quality persona-based summaries for medical articles
and compare against the baseline models.

Evaluation metrics: Our evaluation relies on two different approaches:
(i) Traditional \- Here we use traditional metrics such as Rouge [1, 2 and L]
Lin (2004), Meteor Banerjee and Lavie (2005), Bleu Papineni et al. (2002),
BERTScore Zhang et al. (2019) [Precision (BERT-Prec), Recall (BERT-Rec) and
F1-score (BERT-F1)] to assess the quality of generated summaries.
(ii) GPT-4 critique \- Here we use the GPT-4 LLM as a critic to evaluate the
quality of the model generated summaries against the gold standard GPT-4
generated summary (Section 2) from different dimensions. Specifically, we
provide suitable critique based prompts to GPT-4 where we evaluate the
summaries based on a set of five predefined criterias (termed as GPT-4
criteria) defined below:
Criteria 1: Relevance (Rel): The extent to which the generated persona-based
summary is relevant to the intended persona (doctor/patient/normal person)
given the document.
Criteria 2: Coverage (Cov): The extent to which the generated persona-based
summary correctly covers important key points described in the gold standard
persona-based summary of the document.
Criteria 3: Impurity (Imp): The extent to which the persona-based summary does
not contain information specific to all other possible personas
$\\{persona\\_set$ \- persona$\\}$.
Criteria 4: Quality (Qlt): The extent to which the persona-based summary is of
overall good quality from the perspective of the intended persona.
Criteria 5: Goodness (Gds): Extending from 4, we manually verify the goodness
of the summary.
(Details of these criteria with prompts are provided in Appendix D).

Results: We provide a comparison of our finetuned models against the baselines
in terms of both traditional metrics as well as GPT-4 criteria in Tables 2 and
4 respectively (all values are in $\%$) on the WebMD test set of size $873$
(prompt specific to three different persona each for $291$ articles). Table 2
infers that both our finetuned models (L-F-7b and L-F-13b) achieve superior
performance compared to the baseline methods in terms of traditional metrics.
In fact, finetuned Llama2-13b (L-F-13b) outperforms the baselines in terms of
all the traditional metrics, demonstrating the superiority of our finetuning
approach which helps to adapt the model to the healthcare domain and perform
better on specific applications such as persona-based summarization. Similar
observation holds true when we compare the values of the GPT-4 critique based
criteria shown in Table 4. Here we also compare the quality of the gold
standard GPT-4 generated summaries in terms of the GPT-4 critique based
criteria. We find that the finetuned Llama2-13b model (L-F-13b) can generate
summaries pretty close in quality to the gold standard, while being much
faster in terms of training and inference time as well as cost-effective and
cheaper in terms of memory requirement.

Model Rel Cov Imp Qlt Gds Falcon 56.3 45.4 82.0 50.6 46.8 BART 65.4 42.6 84.8
49.7 25.8 Pegasus 47.7 33.0 74.0 36.1 11.5 T5-FT 72.4 70.1 84.9 67.6 78.1
FT5-FT 72.2 70.2 88.0 68.3 80.3 LED-B 36.1 19.2 79.3 31.3 17.6 LED-L 69.1 56.2
82.3 59.3 56.6 L-V-7b 19.1 18.4 41.3 16.6 15.2 L-V-13b 32.1 29.1 73.1 28.5
23.4 L-V-70b 49.7 45.1 78.0 46.4 47.8 L-F-7b 75.8 58.7 85.8 63.8 58.6 L-F-13b
93.5 90.1 91.7 88.5 99.1 GPT-4 98.2 96.3 98.6 98.5 99.7

Table 4: GPT-4 Critique evaluation results (in %)

Framework: We use 80GB A100 GPU, 210MHz clock cycle along with NLTK/SpaCy
python packages for all experiments. For 6 epochs, Llama2-13b takes 20 hrs for
finetuning and 3 hrs for inference, Llama2-7b takes 8 hrs for finetuning and
2.5 hrs for inference (Details in Appendix E.3).

Ablation study: Here we investigate the quality of persona-based summaries
generated by different variations of our best performing finetuned Llama2-13b
(L-F-13b) model on WebMD test set:

(A) Performance specific to different persona: Table 3 and 5 shows the
performance in terms of standard evaluation metrics and outcomes of GPT-4
critique based criteria for each of the three persona [Doctor, Patient and
Normal Person (Nor-Per)] for the best performing Llama2-13b Finetuned model
(L-F-13b). We observe that the model performs uniformly across the three
persona which confirms that our finetuned model generalizes well across
multiple persona, generating distinct persona-based summaries for the same
medical article.

Persona Rel Cov Imp Qlt Gds Doctor 90.0 89.1 91.0 86.2 98.8 Patient 94.4 90.4
92.2 88.5 99.0 Nor-Per 93.2 91.0 91.8 87.7 99.3 Average 93.5 90.1 91.7 88.5
99.1

Table 5: GPT-4 critique on different persona using
Llama2-13b Finetuning model (in %)

(B) Validation of GPT-4 generated gold standard summaries: To verify the
robustness of the GPT-4 generated summaries for the WebMD articles and to
mitigate the GPT-4 introduced inherent bias in generated summaries, we perform
different types of human annotation experiments:

(i) Persona-based Summary of GPT-4: We randomly select $50$ different WebMD
articles and provide three different persona-based (doctor, patient, normal
person) summaries (without their actual labels) to three different doctors
with domain knowledge expertise along with a good working proficiency in
English and ask them to identify the intended persona, i.e. - which summary
belongs to which specific persona. Initial human labeling is done by two
doctors and any annotation discrepancy is checked and resolved by the third
doctor after discussing with others. The inter-annotator agreement is found to
be $0.91$. On comparing with actual persona labels, we found that human labels
have $86.67\%$ accuracy for correctly identifying the actual persona which
shows the reliability of GPT-4 generated persona-based summaries.

(ii) Content Quality Check: We ask human annotators (doctors) to annotate
summaries on the basis of whether the persona-based summary is relevant while
correctly covering appropriate key points based on information need of
different persona and its overall usefulness. 96% of the GPT-4 generated
summaries for different personas are found to be useful by human annotators
(doctors).

(iii) GPT-4 Generated and Ground Truth Summary Check: Both doctors with domain
expertise and GPT-4 evaluate 50 document-summary pairs in terms of whether the
persona-based summary is relevant and correctly covers appropriate key points
based on information need of different persona, each for GPT-4 generated
summaries and ground truth summaries generated by annotators. We obtain an
inter-annotator agreement of $0.893$ which signifies strong consensus between
human and GPT-4 based evaluation. We separately test our best fine-tuned
(Llama2-13b-FT) model on the human-generated summaries (50 articles) and
obtain the following scores for different metrics - Traditional: R1-52.9,
R2-24.1, RL-33.2, Meteor-38.7, Bleu-18.0, Bert-P-87.7, Bert-R-84.9,
Bert-F1-86.3; GPT-4 criteria: Rel-91.2, Cov-90.8, Imp-90.4, Qlt-88.7,
Gds-98.5. This shows that there is strong alignment between results of on
human generated and GPT-4 generated summaries, signifying the high quality of
GPT-4 generated summaries.

(C) Validation of finetuned model generated summaries: To further investigate
the reliability of our finetuned model generated summaries, we choose $50$
different WebMD articles and provide persona-based summaries for each persona
(generated by GPT-4 in ground truth v/s Llama2-13b finetuned model generated)
to two doctors to annotate: (i) whether finetuned generated summary is better,
(ii) Both are Good, (iii) Ground Truth/GPT-4 summary is better and (iv) Both
are bad. We find that for - 20% cases Llama2-13b finetuned model summaries are
better (i), for 50% cases both finetuned and ground truth generated summaries
are good (ii) and rest 30% cases ground truth generated summaries are better
(iii) and no instance is found where both performs bad (iv).

(D) Different LLM Evaluators: We evaluate the fine-tuned model generated
summaries in the test set with Gemini model999https://gemini.google.com/
keeping the same prompts and criteria as used earlier for GPT-4 and the obtain
values of the same LLM based criteria are: Rel - 95.2, Cov - 92.4, Imp - 87.6,
Qlt - 90.7, Gds - 99.4. Thus, Gemini scores are also aligned with GPT-4 scores
with a correlation coefficient of 0.808 (Gemini provides higher scores for all
criteria except Criteria 3 - ‘Imp’). This verifies that the GPT-4 based
evaluation is impartial and robust.

(E) Llama2-13b performance on Other data: We test our best performing
Llama2-13b finetuned model on healthcare domain articles of OASUM Yang et al.
(2022) dataset which is publicly available. We select OASUM articles with
aspects related to healthcare [Death, Diagnosis, Differential Diagnosis and
Diagnosis Classification] and obtain $234$ such documents. We perform GPT-
critique based evaluation and observe that $82.77\%$ of the summaries are
labeled as good which signifies the robustness of our model in terms of
generating high quality summaries.

##  4 Conclusion

In this paper, we propose a framework for the efficient training of a small
foundation LLM on AI-generated datasets to obtain high quality domain-specific
persona-based summaries. Our focus is on training a finetuned version of
Llama2 on a corpus related to healthcare domain such that the trained model
captures the intricacies of persona-based summaries in healthcare domain. We
also demonstrate the effectiveness of using AI-based critiquing for the
evaluation of the model generated summaries, providing a more automated and
scalable solution. Our experiments also reveal the superior quality of
persona-based summaries generated by our finetuned model compared to
contemporary baselines. Further, AI-based critiquing of the summaries show
high inter-annotator agreement with human-based critiquing methods, further
confirming the effectiveness of our proposed approach. We plan to extend our
work in generating accurate persona-based summaries for documents in other
domains such as legal, enterprises, education and more, which is the focus of
our future work.

## Acknowledgements

The work was supported in part by a research grant from IIT KHARAGPUR AI4ICPS
I HUB FOUNDATION.

## Limitation and Discussion

There are a few limitations in our works - (i) Not all LLMs are useful
(similar to GPT-4) to generate personalized contents properly - like GPT2,
GPT3.5, Llama2-vanilla models do not perform very well mostly due to
hallucination and not covering important informations. (ii) We only explore
the data from healthcare domain, but we plan to extend our work to other
domains such as legal, corporate and education among others. (iii) Our
experimental dataset is only English in heathcare domain, we wish to extend
the work in multilingual setup, specifically for low-resource settings in
diverse domains. (iv) Prompt command is very important. Unless, we
specifically mention with explanation in Prompt about different Persona
(Doctor/Patient/Normal Person), GPT-4 does not perform well to generate
appropriate summary.

## Ethical Concerns

We use the publicly available content of the WebMD platform for non-commercial
and academic purpose only without violating any ethical concerns. The dataset
neither reveals any personal sensitive information of the patients nor any
toxic statement. Consent has been taken from all annotators including doctors.
For experiments, we use publicly available free frameworks - Llama2, Falcon,
BART, Pegasus, T5-FT, Flan-T5 (FT5-FT), LED-Base, LED-Large, Llama2-7b, 13b
and 70b - vanilla and finetune.

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## Appendix

##  Appendix A Related Work

In this section, we conduct a survey of state-of-the-art literature on closely
related topics.

Summarization: We perform a survey of various summarization techniques, that
encompasses generic approaches such as abstractive Chopra et al. (2016); See
et al. (2017); Paulus et al. (2017) and extractive summarization like TextRank
Mihalcea and Tarau (2004) and LexRank Erkan and Radev (2004). Subsequent
research has explored summarization in specific directions, including aspect-
based summaries Hayashi et al. (2021); Coavoux et al. (2019); Mukherjee et al.
(2020); Akhtar et al. (2017) and query-focused summaries Vig et al. (2021);
Zhu et al. (2022); Xu and Lapata (2020), but none focuses on domain-specific
persona-based summarization.

Persona Concept: Goldsack et al. (2023); Luo et al. (2022) focus on building
lay-summarization for comprehensible to non-technical audiences but do not
have the differentiating factor of persona concept to distinguish the various
technical summaries. based on persona and no usage of LLMs as an alternative
metrics to evaluate the summaries. Our work differs from the fact that we show
a pipelined approach - data generation, persona-specific (doctor vs patient vs
normal person) summarization with key points and GPT-4 based evaluation to
save time and assist different persona to augment their knowledge on it to
make conclusions more efficiently. The concept of Persona also different from
the notion of intent Mullick et al. (2022d, 2023, b); Mullick (2023b, a);
Mullick et al. (2022a), entity-relation Mullick et al. (2024); Guha et al.
(2021); Mullick et al. (2022c) or opinion/fact Mullick et al. (2017b, 2016,
2018a, 2018b, 2019, a) idea. Our work differs from the fact that we show a
pipelined approach - data generation, ‘persona concept’ specific (doctor vs
patient vs normal person) summarization with key points and GPT-4 based
evaluation to save time and assist different persona to augment their
knowledge on it to make conclusions more efficiently.

Data Generation using LLMs: Large Language Models (LLMs), such as the GPT
family, have been utilized to generate training datasets for NLP tasks,
addressing data scarcity issues in a cost effective manner Yu et al. (2023);
Meng et al. (2022); Ye et al. (2022). ChatGPT 101010https://chat.openai.com/
aids in educational data augmentation and data visualization Kieser et al.
(2023); Maddigan and Susnjak (2023); however, in healthcare domain, GPT-3 and
GPT-4 are employed for generating medical dialogue summarization data
Chintagunta et al. (2021) and radiology reports Sun et al. (2023). However, no
prior work focuses on benchmark domain-specific persona-based summary data
generation with appropriate human validation.

LLMs as Evaluation Metrics: The rise of LLMs presents a potential, cost-
effective alternative for evaluating various NLP tasks. Existing efforts
include a taxonomy of LLM-based NLG evaluation methods Gao et al. (2024), the
development of ‘ChatEval’ for assessing response quality Chan et al. (2023),
and proposed guidelines for LLM-based evaluation Zhou et al. (2023). While
some studies explore LLM-based assessments with human alignment Liu et al.
(2023), there is a lack of work utilizing LLMs to evaluate solution
architectures comprehensively with a critique scoring system.

Our paper takes a significant step towards addressing the shortcomings of
prior literature in terms of persona-based summary data generation followed by
human valiadation as well as performing LLM based critic evaluation.

##  Appendix B Dataset Requirement

Access to robust, comprehensive datasets is crucial for training NLP models to
understand and generate persona-specific content, emphasizing the challenges
of data scarcity and summary-making capabilities. Manual annotation of large
healthcare datasets is both costly and time-intensive, demanding domain
expertise and meticulous attention to detail. Consider the example where $n$
individuals generate summaries for $m$ persona, resulting in $n\times m$
distinct summary generations for a single article — it is a highly expensive
and time consuming process. Moreover, acquiring suitable human resources for
labeling is challenging, as people are often hesitant to undertake the tedious
and difficult task of summary generation, even with a standard payment
agreement.

We provide healthcare data to the Prolific annotation platform with specific
criteria: annotators with a PhD / Graduate degree, a medical background,
approval rates of 90%-100%, and expertise in medicine. However, out of 157,341
potential annotators, only 189 meet these criteria, and even among them, there
is a high rejection rate of 71.43% for selecting documents for manual summary
generation, despite offering more than standard payment. Further, eligible
annotators tend to heavily rely on ChatGPT and similar automated approaches,
leading to the need for extensive re-evaluations and revisions, along with
issues related to annotator rejection, even to assess a limited number of
documents. Hence, obtaining a high quality dataset remains a formidable
challenge.

##  Appendix C GPT-4 Bias

It is acknowledged that using summaries generated exclusively by GPT-4 could
introduce biases inherent in its summarization capabilities, it may also be
noted that alternatives, such as human evaluation, also carry their own
biases. Despite the potential for bias, leveraging GPT-4 for summarization may
still be a pragmatic choice, especially in scenarios where access to diverse
datasets or sophisticated validation methods is limited. However, in this
work, we remain vigilant, recognizing the limitations inherent in both
automated and human-generated summaries, and take proactive steps such as
human intervention to validate and contextualise the results to mitigate
biases to the best extent possible within the given constraints.

##  Appendix D Prompts

We use prompting in three stages - data generation, finetune-inference and
critique. There are two kinds of prompts - system prompt and user prompt.

###  D.1 Data Generation

system : You are an AI assistant who are to generate a summary of a medical
document specific to a certain persona which can be doctor, patient, normal
person. The summary of a medical document should be generated from the
perspective of the respective persona.
user : Summarize the medical document given below from the perspective of a
{persona} [doctor/patient/normal person] and return the summary only. The
medical document is as follows: Document: {document}

###  D.2 Finetune and Inference prompt

user prompt \- Summarize the medical document given below from the perspective
of a persona:
$\\#\\#\\#$ Document: document

###  D.3 Critique

system: You are an AI assistant who is to evaluate the summary of a medical
document specific to a certain persona which can be doctor, patient or a
normal person. A doctor requires a detailed and technical summary about the
medical document. Patients require a layman’s summary about the medical
document, with information about things like causes, effects, treatment etc.
that may be helpful to them. A normal person has no medical knowledge and
requires a generic summary about the medical document. You need to return a
score between 0 and 1 reflecting the quality of the generated summary based on
some criteria.
user: You are given a medical document and the corresponding summary of the
document generated from the perspective of a {persona} predicted by a language
model as follows.
Document: {document}
Ground truth summary : {label summary}
Summary from the perspective of a {persona} [doctor/patient/normal person]:
{model generated summary}
Evaluate the above persona based summary for the document in terms of each of
the following criteria and return only a score between 0 and 1 without any
explanation:

  * •

The extent to which the generated summary is relevant to the specific persona
{persona}[doctor/patient/normal person] based summary of the document.

  * •

The extent to which the generated persona-based summary correctly covers all
the important key points described in the persona
{persona}[doctor/patient/normal person] based summary of the document.

  * •

The extent to which the summary does not contain information specific to all
other possible personas {persona_set - persona}[doctor/patient/normal person]
based summary.

  * •

Rate the summary from the point of view of the persona – whether the summary
is good, average, or bad. A good summary effectively captures the essential
points, presenting them clearly and concisely. It maintains accuracy,
encourages reader engagement, and serves as a compelling introduction to the
content. An average summary conveys the main points but may lack some clarity
or detail, presenting a decent overview without standing out in terms of
conciseness or precision. It provides a basic understanding but not from a
more refined focused summary and fails to accurately convey the main points,
containing inaccuracies or misinterpretations. It is either overly verbose or
lacks coherence, making it difficult for the reader to grasp the core
information effectively.

  * •

Calculated summary from the point of view of the persona [Good/Bad/Average]
[Calculated from 4 with the help of manual annotation]

##  Appendix E Experiments

###  E.1 Varying Training Size Dataset

To understand the effect of training data size on the performance, we vary the
WebMD training data for the Llama2-13b model - taking $10$-shot ($k$-shot
settings where $k$=$10$), 10%, 40% and 70% of the initial training data, and
finetune the Llama2-13b model with same parameter and hyper-parameter settings
and the five criterias of GPT-4 Critique outcome (in %) are shown in Fig 1. We
see that with increasing the dataset size, the performance of Llama2-13b
improves in terms of GPT-4 critique and traditional metrics. Even at 40% of
the dataset, the model is able to achieve a very good performance. It shows
the effectiveness of the Llama2-13b model. It also infers that even with very
little amount of data in $10$-shot Llama2-13b can able to generate appropriate
persona and aspect based summary.

Figure 1: Different Training Data Sizes

###  E.2 Tuning generation parameters during model inference:

We investigate the impact of tuning the max-new-token and temperature
generation parameters on the performance of our finetuned model during
inference. The variation in performance in terms of the five GPT-4 critique
based criteria are shown in Figures 2(a) and 2(b) respectively. We observe
that our model performs the best for a temperature of $0$ and performance
degrades significantly as we increase the temperature beyond $0.4$. Similarly,
the best model performance is achieved for a max-new-token size of $350$. We
have used NLTK, Spacy, openai (version=0.28), huggingface_hub, torch and
transformers python packages for all experiment.

(a) w.r.t. max-new-token

(b) w.r.t. temperature

Figure 2: Variations of Llama2-13b-Finetune (in %)

###  E.3 Time and GPU

We experiment on 80GB A100 GPU with GPU clock cycle 210 MHz. The finetuning
and inference time of our finetuned models are in Table 6.

Model | Finetune Time | Inference Time
---|---|---
Llama2-7b | 8 hrs | 2hrs 30 mins
Llama2-13b | 20 hrs | 2hrs 50 mins
Table 6: Model Training Time [using 80GB A100 GPU]

##  Appendix F Human Annotations

Annotation Guidelines for Comparative Rating: We provide instructions with
explanations of different persona and ask to identify which summary belongs to
which persona as shown in Fig 5. We also provide the link of document along
with distinct summaries of GPT-4 and Llama2-13b finetune model for comprison
as shown in Fig 6. The instructions are the following -
“You are given a summary of a medical document specific to the perspective of
a certain group of people (doctor, patient and normal person).
A doctor requires a detailed and technical summary about the medical document.
A patient requires a layman summary about the medical document, with
information about things like causes, effects, treatment etc. that may be
helpful to him. A patient only requires a top level view of the extensive
medical details and not so much medical details like a doctor.
A normal person has no medical knowledge and requires a generic summary about
the medical document and does not require extensive medical details.”

Annotation Guidelines to Prolific and Doctors

For annotator selection, we have several criterias. Annotator selection
includes specific criteria such as ‘Degree subject’ in Health and welfare,
‘Highest education level completed’ as Doctorate degree or Graduate degree,
‘Fluent languages’ in English, ‘Approval Rate’ of 90–100, ‘Subject’ in
Medicine, and ‘Employment Sector’ as Doctor. Further annotations are conducted
by graduate doctors (details in Section - Human Evaluation Part).

Following is the annotation guideline to ‘Prolific’ annotation platform -

Objective: Generate Personified Summaries by Prolific

Introduction: In this study, you are tasked with generating a summary tailored
to three different personas: a Doctor, a Patient, and a Normal Person. You
will be provided with a document link containing a Source Document (SD) –
which can be a medical research document or a general article related to
health from https://www.webmd.com/. . Additionally, a Persona (P) will be
present.

Your Task: Read the Source Document (and Persona) and craft three summaries,
each targeted towards one of the personas mentioned. Use your understanding
and perspective to tailor the information in a way that is most relevant and
comprehensible to each persona.

Summary Persona:

Doctor Persona: Craft a summary that focuses on medical terminology,
guidelines, and provides information suitable for a medical professional.
Emphasize technical accuracy and relevance to medical practice. A doctor
requires a detailed and technical summary about the medical document.

Patient Persona: Generate a summary with a patient-centric approach, avoiding
excessive technical jargon. Ensure that the information is clear, easily
understandable, and addresses concerns that a patient might have. A patient
requires a non-technical summary about the medical document, with information
about things like causes, effects, treatment etc. that may be helpful to him.
A patient only requires a top level view of the extensive medical details and
not so much medical details like a doctor.

Normal Person Persona: Tailor a summary for a general audience without a
medical background. Use simple language, avoid technical terms, and present
the information in a way that is accessible and engaging to a layperson. A
normal person has no medical knowledge and requires a generic summary about
the medical document and does not require extensive medical details.

Instructions: 1\. Carefully review the Source Document and the Persona.
Consider the specific needs and understanding level of each persona while
generating the summaries.

2\. No additional software download is required. Use a browser, preferably
Google Chrome, and ensure a stable internet connection.

3\. Allocate time judiciously for crafting each of the three summaries based
on the provided 2 SD instances.

4\. After completion, you will be asked to provide feedback on the generation
exercise, platform interaction, and details about your academic background,
age, country of birth, and any medical background or experience with model-
generated summaries.

Payment Requirements: Upon completing the study, click on the provided link
containing the completion code to redirect you to the Prolific platform.
Payment will be processed within one to two weeks.

Ethical Considerations:

Adhere to strict confidentiality and data protection standards to ensure the
privacy of medical information. If you have concerns or questions, feel free
to reach out, as this study aligns with ethical guidelines.

This study aims to harness diverse perspectives, including those of medical
professionals, to refine the generation of personified summaries for enhanced
utility in various contexts.

Next, the details while providing the documents - “You will be given 2
documents in the next 2 pages and you need to write the summaries with respect
to Doctor, Patient and Normal Person (as in example).

Please do not use ChatGPT/GPT-4 or any Large Language Models - all summaries
should be generated by human properly. It is a strict instruction and will be
checked manually - if found any issue: it will be rejected and re-doing will
be required.

Your summary length (word count) is approximately 15% - 20% of the document
length (word count) for three different types.”

##  Appendix G Examples

Two human annotated examples in doctor persona is shown in Fig 3 where the
GPT-4 generated summary is better and Fig 4 where LLAMA-2 generated summary is
better. Two examples of human annotation interface is shown in Fig 5 and Fig 6
respectively.

Figure 3: GPT-4 generated summary better than LLAMA2-13b model generated
summary[persona : doctor] Figure 4: LLAMA2-13b model generated summary better
than GPT-4 generated summary[persona : doctor] Figure 5: Persona identify
experiment example snapshot Figure 6: Llama2-13b finetune and GPT-4 summary
comparison experiment example snapshot


# On hyperbolic characteristic functions from an analytic and a free-
probability point of view

Zbigniew J. Jurek (University of Wrocław)

(May 25, 2020)

> Abstract. For free-probability Voiculescu transforms, analogous to
> hyperbolic characteristic functions, we show how to get their representing
> measures in an integral form. For that purpose it is enough to know those
> transforms only on the imaginary axis. This is in a contrast to a complex
> analysis where one needs to know them in some domains in the complex plane.
>
> _Mathematics Subject Classifications_(2020): Primary 60E10, 60E07, 44A20.
>
> _Key words and phrases:_ a characteristic function; hyperbolic
> characteristic functions; infinite divisibility; Lévy measure; Voiculescu
> free-infinite divisibility; Laplace transform; convolution.
>
> _Abbreviated title: On hyperbolic characteristic function._

Author’s address:

Institute of Mathematics, University of Wrocław, Pl. Grunwaldzki 2/4,

50-384 Wrocław, Poland;

Email<EMAIL_ADDRESS>; www.math.uni.wroc.pl/$\sim$zjjurek

In a classical complex analysis one of the fundamental results is the integral
representation of _analytic functions_ from the upper to the lower complex
half-plane. Those functions, say $H$, admit unique canonical form

$H(z)=a+\int_{\mathbb{R}}\frac{1+zx}{z-x}\rho(dx)=a-\int_{\mathbb{R}}\big{[}\frac{1}{x-z}-\frac{x}{1+x^{2}}\big{]}(1+x^{2})\rho(dx),\
(\star)$

where $a\in\mathbb{R}$ is a constant, $\rho$ is a finite (Borel) measure on a
real line $\mathbb{R}$; (in a literature $H$ are coined as Pick functions and
a representation $(\star)$ is called Nevalinna Theorem.) One simply notes that
a constant $a=\Re H(i)$ (a real part), and a total mass
$\rho(\mathbb{R})=-\Im(H(i)$. Finally, for a measure $\rho$ we have an
inversion formula

$\rho([c,d])=\lim_{\epsilon\to 0^{+}}\frac{1}{\pi}\int_{c}^{d}\Im
H(x+i\epsilon)dx,\ \mbox{whenever }\ \rho(\\{c,d\\})=0;\qquad\ (\star\star)$

cf.Akhiezer(1965), p.125, or Lang(1975), p.380, or Bondensson(1992), p.21.

However, what can be said about a measure $\rho$ if we only have values
$H(it)$, for $t\neq 0,$ and we don’t know if it is a restriction of an
analytic function to the imaginary axis ? Note that in $(\star\star)$ we need
to know $H$ in some strips of the complex plane to retrieve a measure $\rho$.
Never the less, in Jankowski and Jurek (2012), Theorem 1, there is an
inversion procedure that allows us to identify a measure $\rho$ or more
precisely its characteristic function $\hat{\rho}$.

In a couple of last decades representation of the form $(\star)$ appeared in
so-called _free-probability_ as a free-analog of the classical Lévy-Khintchine
formula for infinite divisible characteristic functions (Fourier transforms).

In this note we show applications of the inversion formula from Jankowski and
Jurek (2012) for $\tilde{C},\tilde{S},\tilde{T}$ free-analogs of the classical
hyperbolic functions $C,S$ and $T$. Recall that $C,S$ and $T$ are defined by
their characteristic functions as follows

$\phi_{C}(t):=\frac{1}{\cosh(t)},\ \ \phi_{S}(t):=\frac{\sinh(t)}{t},\ \
\phi_{T}(t):=\frac{\tanh(t)}{t},\ t\in\mathbb{R}.$

In free-probability variables $\tilde{C},\tilde{S},\tilde{T}$ are given by
their Voiculescu transforms $V_{\tilde{C}}(z),V_{\tilde{S}}(z)$ and
$V_{\tilde{T}}(z),z\in\mathbb{C}^{+}$, although in our approach to free-
probability theory we consequently use only purely imaginary $z=it,t\neq 0$;
cf. Jurek (2019), Corollaries 3 ,4 and 5, respectively.

Let us recall that hyperbolic characteristic functions, from an infinite
divisibility point of view, were studied in Pitman and Yor (2003) and from a
selfdecomposability point of view in Jurek (1997) (as infinite series of
independent exponentially distributed variables), and in Jurek-Yor (2004)
(from stochastic representations of their background driving processes). The
last description can be done as all hyperbolic characteristic functions are
selfdecomposable ones and therefore they admit a representation by random
integrals; cf. Jurek and Mason (1993), Chapter 3, Theorem 3.6.8 or Jurek and
Vervaat (1983), Theorem 3.2.

1\. Introduction.

Let us for an index $X$ (where $X$ can be a random variable or a measure or a
characteristic function) define a function $V_{X}$ on the imaginary axis
$i(\mathbb{R}\setminus{\\{0\\}})$ as follows

$V_{X}(it):=a_{X}+\int_{\mathbb{R}}\frac{1+itx}{it-x}m_{X}(dx),\ \ t\neq 0,$
(1)

where $a_{X}\in\mathbb{R}$ and $m_{X}$ is a non-negative, finite Borel
measure. Then note that $V_{X}(i)=a_{X}-i\,m_{X}(\mathbb{R})$ and hence we get

$a_{X}=\Re V_{X}(i)\in\mathbb{R},\ \ \ m_{X}(\mathbb{R})=-\Im
V_{X}(i)\in[0,\infty).$ (2)

Furthermore, if $\hat{m}_{X}(s):=\int_{\mathbb{R}}e^{isx}m_{X}(dx)$ denotes _a
characteristic function of a measure $m_{X}$_ then its _Laplace transform
$\mathfrak{L}[\hat{m}_{X};w]$, for $w>0$_, satisfies equality

$\mathfrak{L}[\hat{m}_{X};w]=\int_{0}^{\infty}\hat{m}_{X}(x)e^{-wx}dx=\frac{iV_{X}(-iw)-i\Re
V_{X}(i)-w\Im V_{X}(i)}{w^{2}-1};$ (3)

cf. Theorem 1 in Jankowski and Jurek (2012). Equivalently,

$\mathfrak{L}[\hat{m_{X}}(s)+ia_{X}\sinh(s)-m_{X}(\mathbb{R})\cosh(s);w]=\frac{iV_{X}(-iw)}{w^{2}-1},\
\ \ \mbox{because}\\\ \mathfrak{L}\big{[}\sinh x;w]=\frac{1}{w^{2}-1},\
\mbox{and}\ \mathfrak{L}[\cosh x;w]=\frac{w}{w^{2}-1},\ \mbox{for}\ w>1.$ (4)

In propositions below our aim is to show that functions
$w\to\frac{iV_{X}(-iw)}{w^{2}-1}$,

(on the right hand side in (4)), are indeed Laplace’s transform of some
functions or measures. This , in principle, enables us to identify a
representing measure $m_{X}$ in the canonical form (1).

2\. Results. In order to present our results we will need some special
functions.. Therefore before each proposition we recall their definitions
and/or characterizations. Many of those functions are derived from _Euler’s
$\Gamma$ gamma function_: $\Gamma(z):=\int_{0}^{\infty}x^{z-1}e^{-x}dx;\ \Re
z>0$ and _digamma function_ $\psi(z):=d/dz\log\Gamma(z)$. For more facts and
formulas we refer to the Appendix at the end of this article.

In the first proposition we need _$\beta$ beta function_ which admits a
representation $\beta(z)=\int_{0}^{\infty}(1+e^{-x})^{-1}\,e^{-zx}dx,\ \ \Re
z>0$, and originally was defined via digamma function $\psi$; cf. Appendix
(A).

######  Proposition 1.

For a free-infinitely divisible Voiculescu transform

$V_{\tilde{C}}(it)=i[1-t\beta(t/2)],t\neq 0,$

we have that in its representation (1) a real parameter $a_{\tilde{C}}=0$ and
a measure $m_{\tilde{C}}$ is such that a total mass
$m_{\tilde{C}}(\mathbb{R})=\pi/2-1\approx 0.57079$ and its characteristic
function $\hat{m}_{\tilde{C}}$ is equal to

$\hat{m}_{\tilde{C}}(s)=2\sinh(s)\tan^{-1}(e^{-s})+\frac{\pi}{2}e^{-s}-1\\\
=\int_{0}^{\infty}\cos(sx)\frac{|x|}{1+x^{2}}\frac{1}{\sinh(\pi|x|/2)}dx$ (5)

($\tilde{C}$ indicates a free-probability analogue of a classical hyperbolic
cosine characteristic function $\phi_{C}(t)=1/\cosh(t).$)

To formulate next proposition we need two special functions. Namely, _a
digamma function_ $\psi$ , which is defined as
$\psi(z):=\frac{d}{dz}\log\Gamma(z)$, and _an exponential integral function_
$Ei(x):=-\int_{-x}^{\infty}\frac{e^{-t}}{t}dt$, for $x<0$; for more see
Appendix (B).

######  Proposition 2.

. For a free-infinitely divisible Voiculescu transform

$V_{\tilde{S}}(it)=i[t\psi(t/2)-t\log(t/2)+1],t\neq 0,$

we have that in its representation (1) a parameter $a_{\tilde{S}}=0$ and a
measure $m_{\tilde{S}}$ is a such that $m_{\tilde{S}}(\mathbb{R})=\gamma+\log
2-1\approx 0.270362$ and its characteristic function $\hat{m}_{\tilde{S}}$ is
of a form

$\hat{m}_{\tilde{S}}(s)=-1+\cosh(s)\big{(}\log(1+e^{-s})-\log(1-e^{-s})\big{)}+\frac{e^{-s}}{2}Ei(s)+\frac{e^{s}}{2}Ei(-s)\\\
=2\int_{0}^{\infty}\cos(sx)\frac{x}{1+x^{2}}\frac{1}{e^{\pi x}-1}dx,\ \
s>0,\qquad\qquad$ (6)

and $Ei(x)$ is an exponential integral function.

($\tilde{S}$ indicates a free-probability counter part of the hyperbolic sine
characteristic function $\phi_{S}(t)=t/\sinh(t)$.)

In next statements appear special functions from two previous propositions
because of the elementary relation: $\phi_{C}(t)=\phi_{S}(t)\cdot\phi_{T}(t)$.

######  Proposition 3.

For a free-infinitely divisible Voiculescu transform

$V_{\tilde{T}}(it)=V_{\tilde{C}}(it)-V_{\tilde{S}}(it)=it\,[\,\log(t/2)-\beta(t/2)-\psi(t/2)\,],t\neq
0,$

we have that in its a representation (1) a parameter $a_{\tilde{T}}=0$ and for
a measure $m_{\tilde{T}}$ we have that
$m_{\tilde{T}}(\mathbb{R})=\pi/2-\gamma-\log 2\,\,(\approx 0.3004)$, and its
characteristic function $\hat{m}_{\tilde{T}}$ has a form

$\hat{m}_{\tilde{T}}(s)=\frac{\pi}{2}e^{-s}+2\sinh(s)\tan^{-1}(e^{-s})-\cosh
s\log(1+e^{-s})\\\
-\frac{e^{-s}}{2}\big{(}Ei(s)-\log(1-e^{-s})\big{)}-\frac{e^{s}}{2}\big{(}Ei(-s)-\log(1-e^{-s})\big{)}\\\
=\int_{0}^{\infty}\cos(sx)\frac{|x|}{1+x^{2}}\frac{e^{-\pi|x|/4}}{\cosh(\pi|x|/4)}dx,\
\ s>0.\qquad\qquad$ (7)

($T$ indicates a hyperbolic tangent characteristic function
$\phi_{T}(t)=\tanh(t)/t$.)

All three Voiculescu transforms from Propositions 1, 2 and 3 are free-
probability analogs of selfdecomposable characteristic functions. Therefore
they have so called _background driving terms_ from corresponding random
integral representations. In particular, these are background driving
characteristic functions $\psi_{\tilde{C}},\psi_{\tilde{S}}$ and
$\psi_{\tilde{T}}$, and Lévy (spectral) measures $N_{C},N_{S}$ and $N_{T}$;
cf. Jurek (2019), Section 2.1 or Jurek-Yor (2004).

As in previous propositions we have similar results for them as well, although
we computed it only for a background driving characteristic function
$\psi_{\tilde{C}}$, only; cf. Proposition 4 below.

For a following proposition, we need another two special functions. Namely,
_Riemann’s zeta function_ , $\zeta$, and _polylogarithm functions $Li_{n}(z)$
_ (in Wolframalpha.com language written as $PolyLog[n,z]$); cf. Appendix (C).

######  Proposition 4.

For a free-infinitely divisible Voiculescu transform

$V_{\psi_{\tilde{C}}}(it)=i\,[\frac{t^{2}}{2}\zeta(2,t/2)-\frac{t^{2}}{4}\zeta(2,t/4)+1]$

we have that in its representation (1) a parameter $a_{\psi_{\tilde{C}}}=0$, a
total mass $m_{\psi_{\tilde{C}}}(\mathbb{R})=2C-1\,(\approx 0.83193)$ and for
a measure $m_{\psi_{\tilde{C}}}$ we have its characteristic function

$\hat{m}_{\psi_{\tilde{C}}}(t)=-1-t\tanh(t)-\cosh(t)\big{(}\,i(Li_{2}(ie^{t})-Li_{2}(-ie^{t}))+2t\arctan(e^{t})\,\big{)}\\\
=\frac{\pi}{2}\int_{0}^{\infty}\cos(tx)\,\frac{x^{2}}{1+x^{2}}\,\frac{\cosh(\pi
x/2)}{\sinh^{2}(\pi x/2)}dx.\qquad\qquad$

In particular, we have:

$i(Li_{2}(ie^{t})-Li_{2}(-ie^{t}))=2\sum_{k=1}^{\infty}(-1)^{k}\frac{e^{(2k-1)t}}{(2k-1)^{2}};\
\ i(Li_{2}(i)-Li_{2}(-i))=-2C.$

As a by-product of our Proposions 1 and 4, we have

######  Corollary 1.

For a hyperbolic cosine function $\phi_{C}(t)=1/\cosh t$ we have

$\hat{m}_{\psi_{\tilde{C}}}(t)+\hat{m}_{\tilde{C}}(t)=2\int_{0}^{\infty}\cos(tx)\frac{x^{3}}{1+x^{2}}\big{(}k_{C}(x)\big{)}^{\prime}dx$

where a function $k_{C}(x):=(2x\sinh(\pi x/2))^{-1}$ is a denstity of Lévy
measure of a hyperbolic cosine function. $\phi_{C}$.

3\. PROOFS

All boldface numbers appearing below refer to the corresponding fromulae in
Gradstheyn-Ryzhik (1994).

_Proof of Proposition 1._

First, note that $V_{\tilde{C}}(i)=i(1-\beta(1/2))=-i(\pi/2-1)$ and therefore
$m_{\tilde{C}}(\mathbb{R})=\pi/2-1$. Second, since by (8),

$\beta(s)=\int_{0}^{\infty}\frac{1}{1+e^{-x}}e^{-sx}dx=\mathfrak{L}[(1+e^{-x})^{-1};s],\
\ \Re s>0,\ \textbf{8.371}(2)$

consequently

$\frac{iV_{\tilde{C}}(-iw)}{w^{2}-1}=\frac{1-w\beta(w/2)}{w^{2}-1}=\mathfrak{L}[\sinh
x;w]-\mathfrak{L}[\cosh x;w]\,\mathfrak{L}[2/(1+e^{-2x});w]\\\
=\mathfrak{L}[\sinh
x;w]-\mathfrak{L}[\,\underline{(\cosh(s)\ast(2/(1+e^{-2s})))}(x);w],\ \
\qquad$ (8)

where $\ast$ denotes a convolution of functions on positive half-line.

Third, one checks by a differentiation (or by WolframAlpha or Mathematica)
that for $x>0$ we have

$\big{(}\underline{\cosh(s)\ast(2/(1+e^{-2s}))}\big{)}(x):=\int_{0}^{x}\cosh(x-s)\,\frac{2}{1+e^{-2(x-s)}}ds\\\
=2\sinh(x)\arctan(e^{s-x})-e^{-s}+constant|_{s=0}^{s=x}\\\
=2\sinh(x)\arctan(e^{x-x})-e^{-x}-\big{(}2\sinh(x)\arctan(e^{-x})-1\big{)}\\\
=2\sinh(x)(\pi/4-\arctan(e^{-x}))-e^{-x}+1,\qquad\qquad$

and inserting it into (9) we get

$\frac{iV_{\tilde{C}}(-iw)}{w^{2}-1}=\mathfrak{L}[\sinh x-2\sinh
x(\pi/4-\arctan(e^{-x}))+e^{-x}-1;w]$ (9)

Finally, since $m_{C}(\mathbb{R})=\pi/2-1$ and taking into account (4) we have

$\hat{m}_{C}(x)=(\pi/2-1)\cosh x+\sinh x-2\sinh
x(\pi/4-\arctan(e^{-x})+e^{-x}-1\\\ =2\sinh
x\arctan(e^{-x})+\pi/2e^{-x}-1,\qquad$

which gives a first equality in (5).

On the other hand, from Jurek (2019), Example 1 we know that
$V_{\tilde{C}}(it)$ is a free-probability analog of a classical hyperbolic
characteristic function $1/\cosh(t)$ whose (finite) Khintchine measure $m_{C}$
has a density $\frac{1}{2}\frac{|x|}{1+x^{2}}\frac{1}{\sinh(\pi|x|/2)}$, for
$x\in\mathbb{R}$. Thus

$\hat{m}_{C}(t)=\int_{\mathbb{R}}e^{itx}\frac{1}{2}\frac{|x|}{1+x^{2}}\frac{1}{\sinh(\pi|x|/2)}dx=\int_{0}^{\infty}\cos(tx)\frac{|x|}{1+x^{2}}\frac{1}{\sinh(\pi|x|/2)}dx,$

which completes proof of Proposition 1.

######  Remark 1.

_From a formula in Proposition 1, we have an identity_

$\int_{0}^{\infty}\cos(sx)\,\frac{|x|}{1+x^{2}}\frac{1}{\sinh(\pi|x|/2)}dx=-1+\frac{\pi}{2}e^{-s}+2\sinh(s)\tan^{-1}(e^{-s}),\
s>0;$

_which is confirmed by 4.113(8). _

_Proof of Proposition 2._

Since $\psi(1/2)=-\gamma-2\log 2$, (8.366(2)) then $V_{S}(i)=-i(\gamma+\log
2-1)$. Hence a parameter $a_{\tilde{S}}=0$ and a finite measure
$m_{\tilde{S}}$ in (1) has a finite mass
$m_{\tilde{S}}(\mathbb{R})=\gamma+\log 2-1$. Finally, using an integral
formula for $\psi$ function,

$\psi(z)=\log
z+\int_{0}^{\infty}\big{(}\frac{1}{s}-\frac{1}{1-e^{-s}}\big{)}e^{-zs}ds,\ \Re
z>0;\ \ \textbf{8.361}(8),$

we have

$\frac{iV_{S}(-iw)}{w^{2}-1}=\frac{1+w(\psi(w/2)-\log(w/2))}{w^{2}-1}\\\
=\mathfrak{L}[\sinh x;w]+\mathfrak{L}[\cosh
x;w]\mathfrak{L}[\frac{1}{x}-\frac{2}{1-e^{-2x}};w]\\\ =\mathfrak{L}[\sinh
x+\underline{\big{(}\cosh
s\ast(\frac{1}{s}-\frac{2}{1-e^{-2s}})\big{)}}(x);w],$ (10)

where $"\ast"$ denotes a convolution of functions.

By a direct differentiation using properties quoted before a proof of this
proposition (or using www.Wolframalpha.com) one checks that:

$\int\cosh(x-s)\Big{[}\frac{1}{s}-\frac{2}{1-e^{-2s}}\Big{]}ds\\\
=\frac{e^{-x}}{2}Ei(s)+\frac{e^{x}}{2}Ei(-s)-e^{s-x}+\cosh(x)\,\log\frac{1+e^{-s}}{1-e^{-s}}+constant.$

Thus for $x>0$ we have

$\underline{\big{(}\cosh
s\ast(\frac{1}{s}-\frac{2}{1-e^{-2s}})\big{)}}(x)=\int_{0}^{x}\cosh(x-s)\Big{[}\frac{1}{s}-\frac{2}{1-e^{-2s}}\Big{]}ds\\\
=\big{[}\frac{e^{-x}}{2}Ei(s)+\frac{e^{x}}{2}Ei(-s)-e^{s-x}+\cosh(x)\,\log\frac{1+e^{-s}}{1-e^{-s}}\Big{]}\Big{|}^{s=x}_{s=0^{+}}\\\
=\frac{e^{-x}}{2}Ei(x)+\frac{e^{x}}{2}Ei(-x)-e^{x-x}+\cosh(x)\,\log\frac{1+e^{-x}}{1-e^{-x}}\\\
-\lim_{s\to
0^{+}}\big{[}\frac{e^{-x}}{2}(Ei(s)-\log(1-e^{-s}))+\frac{e^{x}}{2}(Ei(-s)-\log(1-e^{-s}))\\\
-e^{s-x}+\cosh
x\log(1+e^{-s})\big{]}=\frac{e^{-x}}{2}Ei(x)+\frac{e^{x}}{2}Ei(-x)-1\\\
+\cosh(x)\,\log\frac{1+e^{-x}}{1-e^{-x}}-\gamma\cosh x+e^{-x}-\cosh x\log 2\\\
=\frac{e^{-x}}{2}Ei(x)+\frac{e^{x}}{2}Ei(-x)-1+e^{-x}+\cosh(x)\big{(}\log\frac{1+e^{-x}}{1-e^{-x}}-\gamma-\log
2\big{)}.$

Inserting above line into (11) and using (4) with
$m_{S}(\mathbb{R})=\gamma+\log 2-1$ we get

$\hat{m}_{S}(x)=(\gamma+\log 2-1)\cosh(x)+\sinh(x)\\\
+\frac{e^{-x}}{2}Ei(x)+\frac{e^{x}}{2}Ei(-x)-1+e^{-x}+\cosh(x)\big{(}\log\frac{1+e^{-x}}{1-e^{-x}}-(\gamma+\log
2)\big{)}\\\
=-\cosh(x)+\sinh(x)+\frac{e^{-x}}{2}Ei(x)+\frac{e^{x}}{2}Ei(-x)-1+e^{-x}+\cosh(x)\log\frac{1+e^{-x}}{1-e^{-x}}\\\
=\frac{e^{-x}}{2}Ei(x)+\frac{e^{x}}{2}Ei(-x)-1+\cosh(x)(\log(1+e^{-x})-\log(1-e^{-x})),$
(11)

which proves first equality in Proposition 2.

Since from Jurek (2019), Corollary 4, we know that $V_{\phi_{S}}$ is a free-
analog of a classical hyperbolic sine characteristic function
$\phi_{S}(t)=t/\sinh(t)$ whose (finite) Khintchine measure is equal to

$m_{S}(dx)=\frac{1}{2}\frac{|x|}{1+x^{2}}\frac{e^{-\pi|x|/2}}{\sinh(\pi|x|/2)}dx=\frac{|x|}{1+x^{2}}\,\frac{1}{e^{\pi|x|}-1}dx,\
\mbox{on}\ \mathbb{R},$

we get second equality in Proposition 2.

######  Remark 2.

_From Proposition 2 we get an integral identity_

$2\int_{0}^{\infty}\cos(sx)\frac{x}{1+x^{2}}\frac{1}{e^{\pi x}-1}dx\\\
=-1+\cosh(s)\big{(}\log(1+e^{s})-\log(1-e^{-s})\big{)}+\frac{e^{-s}}{2}Ei(s)+\frac{e^{s}}{2}Ei(-s)\
\ s>0,$

_that might be of some interest and it seems to be new?_

Since for hyperbolic characteristic functions $\phi_{C},\phi_{S}$ and
$\phi_{T}$ we have that $\phi_{C}(t)=\phi_{S}(t)\cdot\phi_{T}(t)$ therefore
for their Khintchine measures $m_{C},m_{S},m_{T}$ we have
$m_{C}(dx)=m_{S}(dx)+m_{T}(dx)$.

_Proof of Proposition 3._

Taking into account a discussion above and the fact that free-infinitely
divisible transforms $V_{\tilde{C}},V_{\tilde{S}},V_{\tilde{T}}$, in Jurek
(2019) were defined via one - to -one correspondence with classical infinite
divisibility we get that
$V_{\tilde{C}}(it)=V_{\tilde{S}}(it)+V_{\tilde{T}}(it),t\neq 0$. Consequently,
proof of Proposition 3 follows from proofs of Propositions 1 and 2.

_Proof of Proposition 4._

Since $V_{\tilde{\psi_{C}}}(i)=-i(2C-1)\approx-i0,83193$, ( $C$ is Catalan
constant $\approx$ 0.9159), then in (1), $a_{\psi_{C}}=0$, and for a measure
$m_{\psi_{C}}$ we have $m_{\psi_{C}}(\mathbb{R})=2C-1$. Using (4) and the
integral representation for $\zeta(2,s)$ function in Appendix (C), we have

$\mathfrak{L}[\hat{m_{\psi_{C}}}(x)-(2C-1)\cosh(x);t]=\frac{t^{2}/2(\zeta(2,t/2)-1/2\zeta(2,t/4))+1}{t^{2}-1}\\\
=\frac{1}{t^{2}-1}+\frac{1}{2}\frac{t^{2}}{t^{2}-1}\Big{[}8\int_{0}^{\infty}\frac{w}{1-e^{-2w}}e^{-tw}dw-16\int_{0}^{\infty}\frac{w}{1-e^{-4x}}e^{-tw}dw\Big{]}\\\
=\mathfrak{L}[\sinh
x;t]+2(1+\frac{1}{t^{2}-1})\int_{0}^{\infty}w\frac{e^{-2w}-1}{1-e^{-4w}}e^{-tw}dw=\\\
\mathfrak{L}[\sinh
x;t]+(1+\frac{1}{t^{2}-1})\int_{0}^{\infty}w\frac{1-e^{2w}}{\sinh(2w)}\,e^{-tw}dw\\\
=\mathfrak{L}[\sinh x+x\frac{1-e^{2x}}{\sinh(2x)};t]+\mathfrak{L}[\sinh
x;t]\,\mathfrak{L}[x\frac{1-e^{2x}}{\sinh(2x)}]\\\ =\mathfrak{L}[\sinh
x+x\frac{1-e^{2x}}{\sinh(2x)}+h_{\hat{\psi_{C}}}(x);t].\qquad\qquad$

where

$h_{\hat{\psi_{C}}}(x):=(\sinh u\ast
u\frac{1-e^{2u}}{\sinh(2u)})(x)=\int_{0}^{x}\sinh(x-u)u\frac{1-e^{2u}}{\sinh(2u)}du.$
(12)

Consequently from the above calculation we get

$\hat{m_{\psi_{C}}}(x)=(2C-1)\cosh x+\sinh
x+\frac{x(1-e^{2x})}{\sinh(2x)}+h_{\hat{\psi_{C}}}(x),\ \mbox{for $x\geq 0$.}$
(13)

Using website www.Wolframalpha.com or just by elementary computations, knowing
that $d/dxLi_{2}(\pm ix)=-x^{-1}\log(1\pm ix)$, (cf. Appendix (C)), one checks
that

$\int\sinh(x-s)s\frac{1-e^{2s}}{\sinh(2s)}ds=e^{-x}(s-1)e^{s}\\\
-i\cosh(x)\big{(}-Li_{2}(-ie^{s})+Li_{2}(ie^{s})+s\log(1-ie^{s})-s\log(1+ie^{s})\big{)}+const.$

Since $\lim_{x\to 0^{+}}(Ei(\pm x)-\log x)=\gamma$ (Euler constant), from
(13), for $x>0$,

$h_{\hat{\psi_{C}}}(x)=e^{-x}(s-1)e^{s}\qquad\qquad\qquad\qquad\qquad\\\
-i\cosh(x)\big{(}-Li_{2}(-ie^{s})+Li_{2}(ie^{s})+s\log(1-ie^{s})-s\log(1+ie^{s})\big{)}|^{s=x}_{s=0}$
$=e^{-x}(x-1)e^{x}-i\cosh(x)\big{(}-Li_{2}(-ie^{x})+Li_{2}(ie^{x})+x\log(1-ie^{x})\\\
-x\log(1+ie^{x})\big{)}-[e^{-x}(-1)-i\cosh(x)\big{(}-Li_{2}(-i)+Li_{2}(i)]\\\
=(x-1)-i\cosh(x)\big{(}-Li_{2}(-ie^{x})+Li_{2}(ie^{x})+x\log\frac{1-ie^{x}}{1+ie^{x}}\big{)}+e^{-x}-2C\cosh(x)\\\
=-1+x+e^{-x}-2C\cosh
x-i\cosh(x)\big{(}-Li_{2}(-ie^{x})+Li_{2}(ie^{x})-2ix\arctan(e^{x})\big{)},$

where in the last line we use
$\log(1-ie^{x})-\log(1+ie^{x})=-2i\arctan(e^{x})$; cf. Appendix C. And finally
from (14) we arrive at

$\hat{m_{\psi_{C}}}(x)=(2C-1)\cosh x+\sinh
x+\frac{x(1-e^{2x})}{\sinh(2x)}-1+x+e^{-x}-2C\cosh x\\\
-i\cosh(x)\big{(}-Li_{2}(-ie^{x})+Li_{2}(ie^{x})-2ix\arctan(e^{x})\big{)}\\\
=-1+x+\frac{x(1-e^{2x})}{\sinh(2x)}-i\cosh(x)\big{(}-Li_{2}(-ie^{x})+Li_{2}(ie^{x})-2ix\arctan(e^{x})\,\big{)}\qquad\qquad$

and since $(1+(1-e^{2x})/\sinh(2x))=-\tanh(x)$ therfore this completes a first
part of Proposition 4.

For the second one, let us recall that from Jurek and Yor (2004), Corollary 1
and a formula (7) on p. 186, that Khintchine (finite) measure corresponding to
BDCF $\psi_{C}$ is given by

$m_{\psi{{}_{C}}}(dx)=\frac{\pi}{4}\,\frac{x^{2}}{1+x^{2}}\,\frac{\cosh(\pi
x/2)}{\sinh^{2}(\pi x/2)}dx,\ \mbox{on}\ \mathbb{R}.$

Hence we get that

$\hat{m}_{\psi_{\tilde{C}}}(t)=\frac{\pi}{2}\,\int_{0}^{\infty}\cos(tx)\frac{x^{2}}{1+x^{2}}\,\frac{\cosh(\pi
x/2)}{\sinh^{2}(\pi x/2)}dx=-1+t+\frac{t(1-e^{2t})}{\sinh(2t)}\\\
-i\cosh(t)\big{(}-Li_{2}(-ie^{t})+Li_{2}(ie^{t})+t\log\frac{1-ie^{t}}{1+ie^{t}}\,\big{)},\
\mbox{for}\ t\geq 0.\qquad\qquad$

which completes a proof of Proposition 4.

######  Remark 3.

_As a consequence of Proposition 4 we have a formula_

$\frac{\pi}{2}\,\int_{0}^{\infty}\cos(tx)\frac{x^{2}}{1+x^{2}}\,\frac{\cosh(\pi
x/2)}{\sinh^{2}(\pi x/2)}dx=-1+t+\frac{t(1-e^{2t})}{\sinh(2t)}\\\
-i\cosh(t)\big{(}-Li_{2}(-ie^{t})+Li_{2}(ie^{t})-2it\arctan(e^{t})\,\big{)},\
\mbox{for}\ t\in\mathbb{R},\qquad\qquad$

_that might be of some interest as well._

Proof of Corollary 1.

In general, if $M(dx)=h(x)dx,h(x)>0$, is a spectral measure of a
selfdecomposable characteristic function, say $\phi$, then
$N(dx):=-(xh(x)^{\prime}dx$ is a spectral measure of a characteristic function
$\psi(t):=\exp[t\big{(}\log\phi(t)\big{)}^{\prime}]$; so called background
driving characteristic function; cf. Jurek (1997), Corollary 1.1., p.97 or
Jurek and Yor (2004), p. 183, formulae (d) and (e).

Consequently, on the level of Khintichine (finite) measures have;

$n(dx):=\frac{x^{2}}{1+x^{2}}N(dx)\\\
=-\frac{x^{2}}{1+x^{2}}(h(x)+xh^{\prime}(x))dx=-m(dx)-\frac{x^{2}}{1+x^{2}}xh^{\prime}(x)dx,$

which gives a proof of Corollary 1, when applied to $h(x):=k_{C}(x)$.

######  Remark 4.

_Since a hyperbolic sine and a hyperbolic tangent are selfdecopmosable as well
we may have statements about $\tilde{S}$ and $\tilde{T}$ analogous to the one
in Corollary 1, a for hyperbolic cosine function._

5\. Appendix.

For a convenience of reading we collect here some useful facts. Boldface
numbers refer to formulae from Gradshteyn and Ryzhik (1994).

(A) $\beta(x):=\frac{1}{2}[\ \psi(\frac{x+1}{2})-\psi(\frac{x}{2})\ ],\
\beta(x)=\sum_{k=0}^{\infty}\frac{(-1)^{k}}{x+k},\ -x\notin\mathbb{N},\
\textbf{8.732(1)}.$

(B) For the exponential integral function $Ei$ we have :

$\ Ei(x)=-\int_{-x}^{\infty}\frac{e^{-t}}{t}\,dt,\ \mbox{for}\ \ x<0;\ \ \ \ \
\textbf{8.211}(1);\\\ \ Ei(x)=-\lim_{\epsilon\to
0}\Big{[}\int_{-x}^{-\epsilon}\frac{e^{-t}}{t}dt+\int_{\epsilon}^{\infty}\frac{e^{-t}}{t}dt,\
\mbox{for}\ \ x>0;\ \ \ \textbf{8.211}(2)\qquad\\\ \ \ Ei(x)=\gamma+\log
x+\sum_{k=1}^{\infty}\frac{x^{k}}{k\cdot k!},\ \mbox{for }x>0;\ \
\textbf{8.214}(2);\ \ (\gamma\ \mbox{is Euler constant})$

From above we get: $\ \frac{d}{dx}Ei(\pm x)=\frac{e^{\pm x}}{x}$ and
$\lim_{x\to 0^{+}}\,Ei(\pm x)-\log x)=\gamma.$

(C) For Riemann $\zeta$ functions we have:

$\zeta(s,a):=\sum_{k=0}^{\infty}\frac{1}{(k+a)^{s}},\ \Re s>1,\
-a\not\in\mathbb{N};\ \ \textbf{9.521}(1)\,$

In particular, for $s=2$, we have $\zeta$ function as a Laplace transform:

$\zeta(2,a)=\int_{0}^{\infty}\frac{x}{1-e^{-x}}e^{-ax}dx,\ \mbox{for}\ \Re
a>0;\ $ $\mbox{because}\ \ \sum_{k=0}^{\infty}xe^{-kx}=\frac{x}{1-e^{-x}};\ \
\mbox{and}\ \mathfrak{L}[xe^{-kx},w]=\frac{1}{(w+k)^{2}}.$

Polylogarithmic functions $Li_{n}(z)$, ($n$ is a fixed parameter) are defined
via series

$Li_{n}(z):=\sum_{k=1}^{\infty}\frac{z^{k}}{k^{n}}\equiv z\Phi(z,n,1)\
z\in\mathbb{C};\ \ \mbox{(Lerch function)},\textbf{9.550}$

In particular, $\frac{d}{dz}Li_{2}(z)=-z^{-1}\log(1-z);\ \
Li_{2}(i)-Li_{2}(-i)=2iC,$ where $C$ stands for a Catalan constant.

(D) For the equality, $i\log\frac{1-ie^{x}}{1+ie^{x}}=2\arctan(e^{x}),\
x\in\mathbb{R},$ firstly, note that for $x=0$, indeed we have
$i\log\frac{1-i}{1+i}=\pi/2$, and secondly, note equality of derivatives
$\frac{d}{dx}(i\log\frac{1-ie^{x}}{1+ie^{x}})=\frac{d}{dx}(2\arctan(e^{x}))$.

REFERENCES.

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[7 ] Z. J. Jurek (2019), On a relation between classical and free infinitely

divisible transforms, _Probab.Math. Statist._ , to appear;

[Also: math.arXiv:1707.02540 [math.PR],9 July 2017.]

[8] Z.J. Jurek and W. Vervaat (1983), An integral representation for

selfdecomposable Banch space valued random variables., _Z. Wahrschein_

_lichkeitstheorie verv. Gebiete_ , vol. 62, pp. 247-262.

[9] Z. J. Jurek and M. Yor (2004), Selfdecomposable laws associated with

hyperbolic functions,_Probab. Math. Statist._ , vol.24, Fasc.1, pp. 181-190.

[10] S. Lang(1975), _$SL_{2}(\mathbb{R})$_ , Addis0n-Wesley, Reading
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[11] J. Pitman and M. Yor (2003), Infinitely divisible laws associated with

hyperbolic functions,_Canad. J. Math._ 55 (2), pp. 292-330.

Author’s address:

Institute of Mathematics, University of Wrocław, Pl. Grunwaldzki 2/4,

50-384 Wrocław, Poland;

Email<EMAIL_ADDRESS>; www.math.uni.wroc.pl/$\sim$zjjurek