# Writing about COVID-19 vaccines: Emotional profiling unravels how mainstream
and alternative press framed AstraZeneca, Pfizer and vaccination campaigns

Alfonso Semeraro  Salvatore Vilella  Giancarlo Ruffo  Massimo Stella

###### Abstract

Ever since their announcement in November 2020, COVID-19 vaccines were largely
debated by the press and social media. With most studies focusing on COVID-19
disinformation in social media, little attention has been paid to how
mainstream news outlets framed COVID-19 narratives compared to alternative
sources. To fill this gap, we use cognitive network science and natural
language processing to reconstruct time-evolving semantic and emotional frames
of 5745 news about COVID-19 vaccines. Our dataset covers 17 outlets over 8
months and includes Italian news articles that were massively re-shared on
Facebook (5 mil. total shares) and Twitter (200k total shares). We found
consistently high levels of trust/anticipation and less disgust in the way
mainstream sources framed the general idea of ”vaccine/vaccino”. These
emotions were crucially missing in the ways alternative sources framed
COVID-19 vaccines. More differences were found within specific instances of
vaccines. Alternative news included titles framing the AstraZeneca vaccine
with strong levels of sadness, absent in mainstream titles. Mainstream news
initially framed ”Pfizer” along more negative associations with side effects
(e.g. ”allergy”, ”reaction”, ”fever”) than ”AstraZeneca”. With the temporary
suspension of the latter vaccine, on March 15th 2021, we identified a
semantic/emotional shift: Even mainstream article titles framed ”AstraZeneca”
as semantically richer in negative associations with side effects, while
”Pfizer” underwent a positive shift in valence, mostly related to its higher
efficacy. Thrombosis entered the frame of vaccines together with fearful
conceptual associations, while the word death underwent an emotional shift,
steering towards fear in alternative titles and losing its hopeful connotation
in mainstream titles, with the lack of anticipation. Our findings expose
crucial aspects of the emotional narratives around COVID-19 vaccines adopted
by the press, highlighting the need to understand how alternative and
mainstream media report vaccination news.

###### keywords:

natural language processing , text analysis , complex networks , cognitive
network science , COVID-19 , COVID-19 vaccines

††journal: peer review.

[a]organization=Department of Computer Science, University of
Turin,city=Turin, country=Italy

[b]organization=CogNosco Lab, Department of Computer Science, University of
Exeter,city=Exeter, country=UK

##  1 Introduction

Vaccination campaigns are quickly turning the table against COVID-19. Massive
efforts were put in place by most countries to acquire and distribute millions
of doses all over the world (Rolland et al., 2021). Ever since their
announcement in November 2020, vaccines were largely covered, described and
debated by news and social media, creating a deluge of information consumed by
individuals (Puri et al., 2020, Stella et al., 2021b, Murphy et al., 2021).

Whereas many studies focused on the structural and dynamical features of
COVID-19 knowledge flows in social media (Castioni et al., 2021, Puri et al.,
2020, Jiang et al., 2021), less well explored is the other half of news media
consumption, represented by constellations of highly credible/mainstream and
lowly credible/alternative journal venues (Bridgman et al., 2020, Cinelli et
al., 2020). Newspapers not only convey succinct information about the
happening of events but can often bolster awareness about specific aspects of
events (Apuke and Omar, 2021, Gozzi et al., 2020), e.g., bolstering the fatal
consequences of statistically rare side effects of vaccines, or promote
specific emotional perceptions (Stella, 2021), e.g., painting the announcement
of vaccine shortages with concern or hopefulness for the future.

Ultimately both social media and newspapers represent key components of
information consumption (Castioni et al., 2021, Vilella et al., 2021): they
promote knowledge and specific perceptions about real-world events like
vaccines. Exploring the semantic and emotional profiles of knowledge
disseminated by such venues becomes therefore essential to reconstruct key
ideas read by and influencing massive audiences (Stella, 2021). Specifically
for COVID-19 vaccines, this reconstruction is urgently needed to rethink how
different news outlets, over time, structured knowledge around vaccinations
that reached massive audiences (Bridgman et al., 2020). While monitoring the
presence of emotions like anger or trust in massively re-shared knowledge is
important (Yang et al., 2021), the main challenge to understand how these
emotions can affect individuals is finding out which are the main concepts
eliciting such emotions in massively read content (Sulis et al., 2016, Stella,
2021). Identifying the specific emotional and conceptual associations promoted
by news media remains a crucial achievement for fighting misinformation and
social manipulation (Cinelli et al., 2020).

This manuscript adopts an interpretable natural language processing (NLP)
framework of narratives centered around COVID-19 vaccines, promoted by
mainstream/alternative media outlets subsequently re-shared on social media
like Twitter and Facebook. We select the Italian news system as a complex yet
relatively unexplored case study (cf. Stella et al. 2021b). The current
investigation adopts the recent frameworks of cognitive network science (Siew
et al., 2019) and forma mentis networks (Stella, 2021) to interpret language
processing and unveil the structure of knowledge embedded in news articles as
syntactic/semantic networks of conceptual associations. Adopting semantic
frame theory from psycholinguistics (Fillmore, 2006), we reconstruct the
meaning attributed to vaccines and other terms in language by looking at their
semantic and emotional associations. Finding these associations
operationalises a model for meaning reconstruction from text, a cognitive task
where networks of conceptual associations between concepts cast meaning and
emotional context to each individual concept (Fillmore, 2006, Carley, 1993).
This approach has been commonly used as content mapping through human coding
(Carley, 1993), which is clearly impractical to analyse thousands of news
papers. The methodology outlined in this manuscript describes how artificial
intelligence methods can automatise meaning reconstruction through cognitive
networks and thus parse large volumes of texts at once.

Reconstructing the semantic and emotional frames surrounding COVID-19 vaccines
is fundamental for understanding which perceptions pressured individuals
taking part in vaccination campaigns. A lack of trust towards something can
crucially inhibit adherence to norms promoted by institutions, slowing down
vaccination enrolment. Similarly, semantic associations linking “vaccine” with
“hoax” or “conspiration” could bolster conspiracy theories, altering the risk-
perception of individuals and ultimately exposing them to contagion. Despite
the analysis of these patterns often faces the complexity of trends and
behaviours present in social media data, that might be unaffected by
flickering emotions or contrasting content (Valensise et al., 2021), the
scientific community must consider richer semantic and emotional maps of
knowledge exchange to better understand how social media impact real-world
behaviours.

The manuscript is organised in the following way. We first review relevant
works that motivate our cognitive network approach to reconstructing vaccine
perceptions in alternative and mainstream news media. Our results
interestingly unveil drastic differences in the ways alternative and
mainstream outlets framed COVID-19 vaccines within their news titles. We also
detect structural shifts in the semantic frames of AstraZeneca and Pfizer, two
specific types of COVID-19 vaccines. Our cognitive networks also highlighted
the emergence of strong debates in news about the side effects of specific
types of vaccines: Not all COVID-19 vaccines received the same
treatment/semantic framing from alternative and mainstream news. We discuss
our results in light of related literature and outline our machine learning
methods at the end of the manuscript.

Figure 1: (A) Mechanisms of data creation and key differences highlighted by
them in mainstream and alternative news. Data is grouped by weeks. Figure 1
(a) highlights how our approach gives structure to knowledge in news. — (B)
Each tile $<i,j>$ is coloured after the number of urls that have been shared
$i$ times on Facebook and $j$ times on Twitter. Both axes are logarithmically
binned.— (C) Z-scores of emotions in news that contain the word “vaccin”
against a neutral sample, grouped by day and smoothed with a weekly rolling
average. Mainstream news about the word “vaccin” show consistent high levels
of Trust and Anticipation, conveying hope for the vaccination campaign, and
significantly less Disgust. This positive leaning is not visible in
alternative news. —(D) Distribution of z-scores of bodies (right) and titles
(left) into the emotional frame of the word “astrazenec”, divided by
mainstream and alternative outlets. There is a striking difference in how
titles are written by mainstream and alternative news outlets, with the latter
evoking more sadness but also significantly less disgust than the former. Such
a difference is not visible in the articles bodies. White filled petals
represent emotions that are not significantly over- or under-represented in
the corpus, if compared with a neutral baseline.

###  1.1 Related Works

Humans can communicate their ideas through language (Aitchison, 2012), which
makes it key to use linguistic data for mapping perceptions and attitudes.
Both in computer science and psycholinguistics, the problem of detecting
positive or negative perceptions from language is known as stance detection
(Küçük and Can, 2020). Although historically psycholinguistic focused on human
coding of texts (Biber and Finegan, 1989), in the last decade several
approaches from artificial intelligence in computer science have brought to
accurate predictions of stances in texts via machine learning techniques like
deep neural networks (Küçük and Can, 2020, Zarrella and Marsh, 2016) or
sentiment analysis (Rudkowsky et al., 2018). A key limitation of these
approaches is their black-box nature, where the experimenter cannot easily
interpret the inner structure of the model fitted from the trained data and
its relationship with the specific stance under scrutiny (Rudin, 2019). To
overcome this limitation, a novel stream of stance detection approaches adopts
complex networks as tools for representing and visualising key aspects of
text-based discourse (Saif et al., 2012, Radicioni et al., 2021, Stella et
al., 2018, Lai et al., 2019, 2020). Complex networks have the distinctive
advantage of highlighting topological patterns of connectivity between
interconnected entities, be it online users engaging in replies/mentions/re-
sharing (Vilella et al., 2020, Radicioni et al., 2021) or specific hashtags or
words co-occurring together within the same text (Akimushkin et al., 2017,
Amancio, 2015, Stella et al., 2018). Complex networks represent a powerful
tool for enriching machine learning approach and highlighting the inner
structure of a given stance promoted by online users, i.e., for understanding
how ideas were associated and framed by online audiences (Stella, 2021).

Complex networks of conceptual associations can also give stances a measurable
structure, highlighting how ideas/concepts/emotions were associated and framed
by specific textual narratives (Stella et al., 2021a, de Arruda et al., 2019).
Among many options, associations between concepts can be reconstructed mainly
in three ways: through word co-occurrences (de Arruda et al., 2019, Amancio et
al., 2012), e.g., detecting adjacent words in sentences; (ii) through semantic
associations (Kenett et al., 2017, Akimushkin et al., 2017), e.g., words being
synonyms or reminding of each other; or (iii) through syntactic relationships,
e.g., words specifying the meaning of each other. Since word co-occurrences
can approximate syntactic relationships once words devoid of meaning are
filtered out (Cancho and Solé, 2001, Amancio, 2015), the above methods can
transform unstructured texts into networks of interconnected words/concepts.
These structures can then be enriched with emotional data, identifying the key
emotions (Mohammad and Turney, 2013a, Mokryn et al., 2020) or sentiment
patterns (Mohammad, 2016) evoked by concepts. The resulting network structure
is indicative of associative knowledge and emotional signatures embedded in
texts (Stella, 2020), partially reflecting the way stances were organised in
the authors’ psychology (Jackson et al., 2020). In fact, both the frameworks
of content mapping in communication science (Carley, 1993) and frame semantics
in cognitive psychology (Fillmore, 2008) indicate that the way words are
linked in semantic/syntactic networks provides key insights for reconstructing
the meaning attributed to words in texts by authors.

This relationship between network structure and meaning in narratives has been
used for a variety of automatic tasks. Amancio and colleagues used co-
occurrence networks of words to perform author identification of novels
(Amancio, 2015, Amancio et al., 2012). Stella and colleagues (Stella et al.,
2018) used hashtag co-occurrence networks to highlight the negative, hatred-
inspiring stances injected by automated accounts on social discourse about the
Catalan referendum in 2018. Colladon (Colladon, 2018) combined multiple
network metrics of word-word co-occurrences in social discourse to identify
key emotional and semantic features of news relative to brand advertisement.
Ferrara and colleagues (Ferrara and Yang, 2015) found that in social media,
textual posts richer in positive emotions could reach larger audiences whereas
a faster spreading rate was found for stances richer in negative emotional
content. Radicioni and colleagues (Radicioni et al., 2021) combined hashtag
co-occurrences and online social interactions to identify discursive
communities engaging in different stances of immigration. The authors
identified a strong polarisation in favour and against humanitarian
interventions to immigration mainly correlated with political factions, in
agreement with previous social network approaches (Vilella et al., 2020).
Teixeira and colleagues (Teixeira et al., 2021) used semantic/syntactic
networks to reconstruct stances expressed in suicide notes and found negative
perceptions of concepts like “love”, distorted by suicide ideation when
compared to control data. Mokryn and colleagues found that the emotional words
expressed in movie reviews were predictive of the emotions inspired by those
movies, further strengthening a connection between textual data and
cognitive/emotional content.

Although differing in their scope and methods, the above approaches have a key
common element: They all perform quantitative measurements of the semantic,
syntactic and emotional content of texts. This automation makes it possible to
perform stance detection through volumes of data/texts that would be
intractable with human coding (Biber and Finegan, 1989).

In this work, we build upon the above past approaches by reconciling
interpretable machine learning and networks of conceptual associations within
the framework of textual forma mentis networks (TFMN) (Stella, 2020). These
networks perform dependency parsing - powered by recurrent neural networks
(Dozat and Manning, 2016) \- to identify how individual words are
syntactically related in sentences. Syntactic connections are also enriched
with synonym relationships - indicating which words can overlap in meaning
according to WordNet 3.0 (Miller, 1998) \- and emotional data - indicating how
words were positively/negatively/emotionally perceived in validated
psycholinguistic experiments (Mohammad and Turney, 2013a, Mohammad, 2016). The
resulting multi-layer, feature-rich network structure contains insights about
how text authors organised their knowledge and perceptions in texts. Through
the lens of cognitive network science (Siew et al., 2019, Kenett et al., 2017,
Stella, 2021), discourse content, centrality and frames can all be measured
via interpretable network metrics and, importantly, visualised. These aspects
provide experimenters direct access into the structure of stances expressed in
texts (Stella, 2020) and also in the psychology of text authors (Jackson et
al., 2020, Teixeira et al., 2021). A key advantage of TFMNs is their ability
to unveil semantic frames surrounding specific concepts in discourse as
network neighbourhoods of concept associations. This is a crucial
methodological feature for investigating specific aspects of phenomena as
complex as vaccination campaigns.

##  2 Methods

### Data collection

We collected 5745 news articles about vaccines that circulated in Italy in a
time window that spans from October 2020 to May 2021, along with their number
of shares on social networks as Twitter and Facebook. We divided the dataset
in articles coming from mainstream media sources and alternative sources.
Mainstream sources are three among the most visited Italian newspapers, while
alternative sources include a list of blogs, underground information and
pseudo-newspapers already blacklisted by fact checkers of Bufale.net as
persistent spreaders of mis- and disinformation. It must be noted, however,
that none of the news in the dataset was verified by fact checkers nor by us.
Thus, the two source categories are not indicative of true versus false
contents, but rather of newspapers that have previously earned trust from the
public opinion as authoritative and credible, versus unreliable and possibly
partisan forms of news outlets. The full list of news sources is reported in
Table 1 together with their categorisation. The news selection was initially
operated through the Twitter APIs, by retrieving tweets that contained a word
related to vaccines and a url from the above list. We considered to be related
to vaccines keywords such as the Italian word “vaccino” itself, plus all the
names of vaccines available worldwide by the time we collected data, i.e.,
“pfizer”, “astrazeneca”, “vaxzevria”, “moderna”, “johnson”, “sputnik” and
“sinovac”. We then scraped the articles, downloading the date, the title and
the text content of the news. After discarding miscast articles, cancelled
articles, articles protected by a paywall, and articles that were dated before
the observation period (and merely re-tweeted in the observation period), we
collected in total 3447 news from mainstream sources and 2298 from alternative
sources. The number of shares on Twitter for a single article was inferred by
the number of tweets we downloaded. Due to the limited amount of data
retrieved and the current Twitter’s policies about APIs and rate limits, we
can safely assume that our procedure downloaded most, if not all, the tweets
that responded to the criteria introduced above. Last, we inferred for each
url the number of shares on Facebook trough the tool for monitoring social
media reactions Sharescore (sha, 2008). Fig. 1 (b) shows a hint of the
efficacy of the data retrieval procedure, displaying a good level of
correlation between number Twitter and Facebook shares.

Domain | Type
---|---
ilfattoquotidiano.it | Mainstream
repubblica.it | Mainstream
lastampa.it | Mainstream
imolaoggi.it | Alternative
voxnews.info | Alternative
renovatio21.com | Alternative
byoblu.com | Alternative
maurizioblondet.it | Alternative
scenarieconomici.it | Alternative
mag24.es | Alternative
irresponsabile.com | Alternative
disinformazione.it | Alternative
internapoli.it | Alternative
centrometeoitaliano.it | Alternative
essere-informati.it | Alternative
dionidream.com | Alternative
fonteverificata.it | Alternative
Table 1: List of the media outlets web domains analysed.

###  2.1 Emotion detection and analysis of semantic frames

The main goal of this work is the analysis of the emotional fingerprint of
news articles about vaccines and related concepts. Generally speaking, this
task is performed by checking the texts against a lexicon of word-emotion
associations (the NRC Lexicon (Mohammad and Turney, 2013b)), but we included
into our methodology two additional steps that increase the sensitivity and
the significance of the analyses.

First, we pre-processed all texts and extracted their Textual FormaMentis
Networks (TFMNs) (Stella, 2020). TFMNs, as we can see in Fig 1(a), provide a
method for determining meaningful relationships between words, allowing for a
fine-grain analysis about a single concept: by extracting the neighbourhood of
a word from the TFMN of a (collection of) article(s), we were able to identify
the words specifically associated to our target in the texts, filtering out
words that merely co-occur in the same text but that have not a direct
semantic or syntactic link with the target. Newspapers’ texts can cover a wide
spectrum of emotions, due to the length and the variety of subjects and facts
within the same article. For instance, while talking positively about the
impact of vaccines on daily casualties rates, an article could convey negative
emotions about the deaths, or concerns about the future evolution of the
outbreak. Overall, positive or negative framing of a concept may be diluted
into the numerous traces of different emotions, thus being of primarily
importance to explore semantic frameworks of words.

To do so, we extracted the emotion distribution of the words belonging to the
TFMN neighbourhood of a concept, by checking them against the Italian
translation of the NRCLex lexicon. Last, we compared the emotions in the
semantic frame to a null model, i.e., a random selection of words and their
associated emotions. We computed the z-scores of the distribution of emotions
that we found in TFMNs against the emotion distribution of 300 random samples
of the lexicon itself. This methodology yields a numeric score for each
emotion, which is an indication of how much that emotion is under-represented
or over-represented. Fig. 1(D) and Fig. 4(C) have been generated using a
slight modification of the visualisation library PyPlutchik (Semeraro et al.,
2021), which allows for a quantitative representation of the Plutchik’s wheel
of emotions. Petals were sized after the z-score of how much an emotion has
been detected in the TFMNs against a neutral baseline, and coloured only when
they were at greater than 1.96 (or lower that -1.96), making it simple to
identify emotions significantly over/under-expressed. A grey shadowed ring in
such plots represents the space within 1.96 standard deviations from the
average, where z-scores are not statistically odd. Similarly, in Fig. 1(C), we
coloured only the lines representing those emotions that were below -1.96 or
above 1.96 standard deviations from the average at least 50% of the time,
again emphasising odd patterns that were consistent over time.

##  3 Results

Results are organised along the following timeline. We first start by
providing evidence that social engagement is not enough to identify
differences between mainstream and alternative sources of information relative
to COVID-19 vaccines. In other words, users tend to post about alternative and
mainstream news at similar rates. However, the emotional content of these two
sources of news differ drastically in the way they frame the idea of vaccines.
Subsequently, we focus our attention on perceptions about specific vaccines
like AstraZeneca. Lastly, we outline how news media reported negative concepts
related to vaccines, like “mort” (stem for the Italian for death) and
“trombos” (thrombosis).

###  3.1 Prevalence of the discourse about vaccines on social networks and
peaks of activity.

Vaccines have been extensively debated both on social and on news media.
Keeping track of the popularity of the articles reshared on social media can
disclose insights about the ways mainstream and alternative news media
described vaccination campaigns. Figure 1(b) displays a correlation heatmap of
the popularity of posts on the same set of news on Facebook and Twitter (see
Methods), namely articles mentioning “vaccin” in their titles. It can be
observed a positive correlation among the two measures of popularity
(Pearson’s coefficient 0.38, p-value $\ll 0.001$). The correlation vanishes
for extremely popular content, which is quite rare, and for URLs unpopular on
Twitter, which can have diverse outcomes on Facebook. The densest mass in the
plot is in the middle, where news are shared around 14 times on Twitter and
300 to 600 times on Facebook, confirming an order of magnitude of shift
between the volume of the two social networks. Further details on social media
users posting activity about vaccine-related news are provided in the
Supplementary Information. Roughly the same positive correlation between
Facebook and Twitter resharings persisted for both mainstream (Pearson’s
coefficient 0.44, p-value $\ll 0.001$) and alternative (Pearson’s coefficient
0.41, p-value $\ll 0.001$) news outlets. This lack of differences indicates
that both mainstream and alternative news were re-shared in similar ways
across social media platforms. This also means that differences between
mainstream and alternative news is to be searched not only within their online
spread but rather in their semantic and emotional content, as investigated in
the following subsection.

###  3.2 Differences and similarities in the narrative about vaccine in
mainstream and alternative news media articles

Unable to distinguish between alternative and mainstream news, it has to be
underlined that user activity on social media provides also little to no
information on the narrative in which vaccines are framed. Hence, we rather
focus our attention on quantifying the semantic/emotional content of news. To
this end, we extract the semantic frameworks of selected concepts following a
working pipeline explained in Figure 1(a) and in Sec. 2 and, for starting, we
explore the semantic framework of the term “vaccin”.

Figure 1(c) reports that different kinds of media framed the concept of
“vaccin” with wildly different emotions. In fact, mainstream Italian news
media expressed significantly more trust and anticipation, as well as less
disgust, than expected at random (see Methods). Such richly emotional framing
of COVID-19 vaccines is completely absent in alternative sources. Whereas
mainstream sources consistently framed “vaccin” with mostly positive/trustful
jargon for more than 50% of our sampling time window, alternative news framed
the same concept as emotionless. Our results provide strong evidence that it
is not social engagement but rather emotional profiling that strongly
characterises sources coming from alternative and mainstream outlets.

However, Fig. 1(c) considers both titles and news bodies. Would these
differences persist in case we focused on either of these elements? More
importantly, would these differences be shared by specific types of vaccines
like AstraZeneca?

In Fig. 1(d) we focus on the AstraZeneca vaccine. Petals without colours in
Fig. 1(d) represent emotions that are not significantly over- or under-
expressed in the corpus, if compared with a neutral baseline (See. Section 2).
We analyse the time-aggregated emotions that are expressed on this noun over
the whole time window, always referring to a neutral baseline. It is also
interesting to differentiate not only between mainstream and alternative
sources, but also among bodies and titles of the articles, since article
headlines are meant to convey information and to catch the reader’s eye in a
very limited number of words. Using Plutchik’s flower to represent the eight
primary emotions of the model, we find interesting differences. While both
mainstream and alternative media adopt for their narratives the same emotional
structure in the main bodies of their articles, different choices were made
for the titles. Alternative sources expressed way more sadness compared to the
baseline, while mainstream ones adopt a more positive attitude, expressing
trust and anticipation.

###  3.3 Perception of vaccine-related dangers: Before and after March 15,
2021

We can further investigate the emotional approaches to vaccines of mainstream
and alternative news media by studying the narratives before and after 15
March 2021. On this day the administration of the VaxZevria (then AstraZeneca)
vaccine was temporarily suspended due to a small number of suspect side-effect
thrombosis111https://www.aifa.gov.it/-/aifa-sospensione-precauzionale-del-
vaccino-astrazeneca, Last Accessed: 7/12/2021. An important media fuss was
raised over this event, since the AstraZeneca vaccine was, at that time, one
of the most used vaccine in Italy on several public worker
categories222According to the National Vaccination Guidelines valid at that
time, that can be found at the following URL (Italian only):
shorturl.at/cBNT0, Last Accessed: 9/12/2021. To do so, we conduct a two-fold
analysis by analysing the semantic frameworks around the words “pfizer” and
“astrazenec” through TFMNs, as well as around the words “mort” (death) and
“thrombos” (thrombosis).

Figure 2: Semantic frames of tightly linked concepts around “astrazenec” in
journal news titles from mainstream sources (right) and alternative sources
(left), before (top) and after (bottom) the temporary suspension of 15 March
2021. Figure 3: Semantic frames of tightly linked concepts around “pfizer” in
journal news titles from mainstream sources (right) and alternative sources
(left), before (top) and after (bottom) the temporary suspension of 15 March
2021.

Figures 2 and 3 display the communities of tightly connected concepts
surrounding, respectively, “astrazenec” and “pfizer” in their semantic frames
as reconstructed from the TFMN. To highlight community structure we used the
Louvain algorithm (Blondel et al., 2008) as in previous works (Stella, 2020).
Clustering concepts together in communities can better highlight more tightly
connected and thus more semantically related concepts as debated in texts
(Stella, 2020). These network visualisations report the semantic content
associated with the above two entities in the corpus of news titles. Each
network visualisation compared titles from mainstream sources (right) and
alternative sources (left), before and after (top and bottom) the temporary
suspension of VaxZevria on March 15th 2021.

Investigating the semantic content of these networks reveals interesting
insights. Before the suspension, alternative journals framed “astrazenec” with
way less negative associations than “pfizer”. These journal venues
concentrated negative associations like allergies, reactions and deaths mostly
when reporting about Pfizer’s vaccine (cf. 3 top left). These associations are
absent in the titles surrounding “astrazenec” (cf. 2 top left). This indicates
that alternative journal venues produced titles giving more semantic
prominence to allergic/negative reactions to the vaccines mostly when
mentioning Pfizer and more rarely when mentioning “astrazenec”. This pattern
is flipped when considering titles from mainstream journal venues, which
feature more negative, reaction-related jargon when mentioning “astrazenec”
(cf. 2 top right) rather than when talking about “pfizer” (cf. 3 top right).
On top on this disparity in reporting negative side effects of different
brands of vaccines, titles from mainstream journals framed Pfizer with a
positive cluster of concepts that is missing from AstraZeneca’s semantic frame
and relative to trust spawning from Pfizer’s approval from experts and
institutions.

After the temporary suspension of AstraZeneca, the semantic frames of
“astrazenec” and “pfizer” underwent some drastic changes in the TFMNs obtained
from news titles. Pfizer underwent a drastic drop of network degree (-79%) in
the titles from alternative journal venues, indicating a reduced semantic
richness of language surrounding “pfizer” in those titles and, consequently, a
reduced semantic prominence of Pfizer’s vaccine in such titles (cf. 3 bottom
left). Always within titles coming from alternative news media, the semantic
community of “astrazenec” underwent a densification of negative associations.
This included links with thrombosis, threat and dangerous that were not
present before. Mainstream journals produced semantic communities framing
“astrazenec” in similar ways before and after the temporary suspension of the
vaccine. Noticeably, mainstream venues featured associations with clusters of
concepts related to bureaucracy, underlining how the vaccine was under further
scrutinise.

Figure 4: (A) Fraction of news that mention “mort” (death) and “trombos”
(thrombosis) in their bodies and titles over the total of mainstream news
(blue) and alternative news (red), smoothed by a moving average of the last 7
days. (B) Distribution of emotions in the semantic frame of “mort” (death) in
the titles of mainstream and alternative articles.

A further analysis driven by the events of March 15 can be conducted by
tracking the usage of words associated to the negative events: death and
thrombosis, the main alleged side effect that was put under the spotlight. An
analysis of the prevalence of these two words, as shown in Fig. 4(a), suggests
us that both mainstream and alternative media covered these events in a
similar way. The words “mort” (death) and “trombos” (thrombosis) have a long
history due to its association with the COVID-19 outbreak, especially during
the peak of casualties in Fall 2020, in both mainstream and alternative news
outlets. The two plots show how many of the news produced by mainstream and
alternative outlets everyday included the words “mort” (death) and “trombos”
(thrombosis) respectively. Daily values were averaged with a moving average of
7 days, reducing the noise and increasing the readability of the figures.
Indeed, they both show a peak of the said words around or after March 15,
mentioning death in almost 35% of their articles and thrombosis in the 25%
(mainstream) and 30% (alternative) of their articles.

##  4 Discussion

News media shape the public opinion over many topics, including general
health-related issues (Gollust et al., 2013, 2019) as well as specific aspects
of the fight against the pandemic, such as vaccine hesitancy (Catalan-
Matamoros and Elías, 2020, Carrieri et al., 2019). Needless to say, during the
last two years COVID-19 has been one of the major talking points in all kind
of news media. With this regard, many studies have been conducted on both the
nature of COVID-related news and the effects such news have on the audience.
While many have already pointed out the burst in COVID-19 online
misinformation (Yang et al., 2021, Briand et al., 2021), it has also been
shown that users’ online activity related to COVID-19 is strongly driven by
media coverage (Gozzi et al., 2020). Therefore, general mass media play a
fundamental role in steering the unfolding of the social consequences of the
pandemic: being constantly exposed to COVID-related news might impact not only
the pandemic itself, modifying the people’s behaviour, but also the mental
health of the audience (Su et al., 2021). This is a crucial aspect that should
not be neglected: fear and anxiety can affect the general audience as well as
those who are first in line in the response to the pandemic (Coelho et al.,
2020) and, particularly, using caution relatively to the COVID-19 media
coverage is advisable (Coelho et al., 2020). This is especially true when
considering that social networks, which as we saw are an important channel to
share news articles, are a powerful tool for emotional contagion, that can
also happen without direct interaction between people (Kramer et al., 2014).

Indeed we found that, within the time window we considered, vaccines have been
consistently portrayed with significantly more trust and anticipation in
mainstream news, with no significantly emotional language displayed in
alternative news. This general feeling towards the vaccines in alternative
news was not the same reserved to the AstraZeneca vaccine for which, overall,
carries significantly more sadness. This propensity for negative emotions in
alternative news is reinforced as soon as we focus on the analysis around the
date of 15 March 2020. This day has become a milestone in the Italian - and
European - vaccination program, since the administration one of the most
popular vaccines (AstraZeneca) underwent a series of disruptions due to a
small number of serious side effects possibly related to the vaccine. Here,
the usage of danger-related words such as death or thrombosis exploded in both
mainstream and alternative media outlets, as did the emotional load, but it
was in the latter where negative emotions prevailed, with a significant amount
of fear dominating over the others. The analysis of the semantic neighbourhood
of key concepts before and after the 15 March 2021 confirms that, after that
date, there was a shift in the media’s attention from the vaccine Pfizer to
AstraZeneca, that was framed under an increasingly negative light, with new
associations with negative words.

Overall, we can appreciate significant differences between mainstream and
alternative media sources. It is important to note though, that both kind of
outlets insisted in presenting some topics (especially the AstraZeneca
vaccine) in a strongly emotional way, with an almost always significantly
higher emotional load with respect to a neutral language baseline. This is
particularly true for the articles’ titles: since they have to convey a short,
effective message, they are - predictably - more loaded with emotional content
than the articles’ bodies, where the differences tend to taper. The media
coverage of vaccines, throughout the whole pandemic, was emotionally intense;
in a moment of crisis, such was the 15 March 2021, the media responded with a
further diversification of the emotions, always higher than the neutral
baseline. This fosters the findings presented at the beginning of this
Section, where the authors highlight the strong interplay between media
coverage of events and the mental health and response of the audience.
Particular attention should be put both by the audience, for a well-reasoned
consumption of media content, and by the media outlets for a careful choice of
the narrative under which to describe sensitive matters, as are the events
related to the vaccination campaign during a global pandemic. Given the proven
influence of media coverage on vaccination campaigns (Yoo et al., 2010, Ma et
al., 2006, Tchuenche et al., 2011), conveying positive emotions, such as trust
and anticipation whose combination, according to Plutchik’s model (Plutchik,
2001), can be seen as an expression of hope, could potentially have positive
effects. Indeed, at the present date Italy shows an above-the-average adoption
rate of vaccines against COVID-19, with 81% coverage of the total population,
performing better than United Kingdom (%76), United States of America (%74)
and the average of the European Union (73%)333According to
https://ourworldindata.org/covid-vaccinations, last accessed: 12/01/2021. On
the other hand, an excess in optimism can also yield negative drawbacks
(McColl et al., 2022), proving once again how delicate is the balance in
crisis communication and how important is the role of mass media in its
management.

###  4.1 Limitations and Future Work

The present work has a number of limitations that should be taken into account
while interpreting the results. The main limitation lies in the number of
third-party resources used throughout all the study. Particularly, the URLs of
the news media articles analysed have been collected by tracking down a
selection of web domains and their diffusion on Twitter. These domains were
flagged by independent annotators as either mainstream sources or
misinformation spreaders, as per Table 1; to collect the URLs, we resorted to
the Twitter APIs. This procedure brings two inherent limitations, related to
both how up-to-date the independent lists are and how representative is the
portion of tweets retrieved through the APIs. As for the former issue, we made
sure that the references used to identify mainstream and alternative news
media, commonly used by many other studies about Italian infodemics, were as
current as possible at the moment of the data collection.

Thanks to TFMNs we were able to avoid a further limitation that we could have
encountered by quantifying emotions with a mere count of emotional words,
referring to an external lexicon (such as the NRC Lexicon (Mohammad and
Turney, 2013b) that we used). Indeed, by applying this method - either to
articles or to smaller excerpts, like titles or sentences - we would have
likely lost sight of the context in which the words are used. TFMNs are
instead a way to reconstruct such context, taking into account syntactic and
semantic relationships between words. Such a results could be achieved by
means of other NLP techniques, such as attention-based deep neural networks,
more and more often used for emotions detection in texts (Acheampong et al.,
2021). Comparing the results yielded by these different methods could be very
interesting, and a possible future research direction; as of now, we preferred
to adopt a simple, light and - most importantly - entirely explainable
approach, also in light of the sensible topic under discussion during these
troubled times.

From a psychological perspective, it is important to underline that the
current method highlights only emotions as described in text and not as felt
by individual readers (Stella, 2021). This difference means that our study
cannot, on its own, determine the consequences of being exposed to specific
emotional profiles, e.g. did the negative associations attributed to ”Pfizer”
elicit vaccine hesitancy? Future research should be devoted to measure
specifically how exposure to specific semantic/emotional content influences
ways of thinking and, subsequently, behaviour in individuals.

##  5 Conclusions

News diffused via social media can portray the same event in different ways.
Focusing on over 5000 Italian articles, re-shared more than 5 million of times
on Facebook and Twitter, we show that mainstream and alternative news outlets
framed COVID-19 vaccines in different ways. Whereas mainstream media adopted
an overall positive stance towards vaccines, alternative outlets adopted less
optimistic and more neutral semantic/emotional frames. This phenomenon is even
more pronounced when it comes to specific pharmaceutical brands, particularly
Pfizer and AstraZeneca, with news titles about the latter being characterised
by strong sadness (in alternative news) as opposed to the presence of hope (in
mainstream titles). Our results also highlighted how emotions evolved within
the narrative about COVID-19 vaccines in the press: after a crisis like the
temporary suspension of the AstraZeneca vaccine in March 2021, words like
death and thrombosis came more frequently into play when debating vaccination
campaigns. Supported by artificial intelligence, network science and
psychologically validated emotional norms, the framework we introduced here
provides a powerful tool for automatically unveiling emotional/semantic
features of news content in large datasets. This tool opens the way to
measuring differences in the way news media portray news as crucial and
delicate as vaccination campaigns during a global pandemic.

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## Appendix: Supplementary Information

We first showcase some general, descriptive statistics about the prevalence
and popularity of the discourse about vaccines on social media. As explained
in Sec. 2, when collecting articles we distinguished between mainstream and
alternative media outlets, having set the objective of finding whether there
are significant differences in the narrative around vaccines between the two
categories.

Figure 5: Posts on Twitter and Facebook in the dataset. A peak is registered
in week 25 of observation, around the suspension of AstraZeneca on March 15.

Figure 5 reports the amount of shares on Twitter (top) and Facebook (bottom),
divided by mainstream and alternative media sources, and grouped by week. The
activity on the two social networks is substantially different in numbers,
with an average of 153 thousand shares by week on Facebook and only 5
thousands on Twitter, but it shows some resemblances. The number of daily
posts slowly rises in late 2020, when the first news about the trial of Pfizer
and AstraZeneca appeared; it bumps at the end of December 2020, when first
Pfizer was approved for subministration444Source:
https://www.aifa.gov.it/-/autorizzato-il-vaccino-biontech-pfizer, Last
Accessed: 2/12/2021., and later at the end of January 2021, when AstraZeneca
(now VaxZevria) was also approved555Source: https://www.aifa.gov.it/-/aifa-
autorizzato-vaccino-astrazeneca, Last Accessed: 2/12/2021. by AIFA, the
Italian Agency for Drugs. The official press releases from AIFA are
responsible for outstanding peaks of activity on Facebook (bottom), exactly
around the approval of both vaccines. The number of posts diminishes in
February, but it quickly rises to the highest level on March, around the
suspension of AstraZeneca of March 15 (red dotted vertical line) due to a
small number of suspect cases of thrombosis in
Europe666https://www.aifa.gov.it/-/aifa-sospensione-precauzionale-del-vaccino-
astrazeneca, Last Accessed: 7/12/2021. While consistently higher than before,
the volume of the discussion around both vaccines decreased steadily after
March 15. While the general trends of engagement are similar on both social
networks, there is a striking difference on what kind of sources drove the
conversation about vaccines. The shares of news published by alternative
sources can be considered negligible on Facebook, but on Twitter such news
represent a massive quota of the total shares. To give a numeric comparison,
total shares of alternative sources contents on Facebook were 8.36% of the
total, while on Twitter 43.61%. Notice that Facebook counts might be
underestimated because of sampling issues (see Methods) that originate from
alternative sources being spread in smaller social pages that can go
undetected by our external data gathering tools. On Twitter, where user
activities are transparently monitored by the Twitter API (i.e., no third-
party software) data exhibit no evident posting differences between
alternative and mainstream news sources. The same high level of agreement is
consistently present in mainstream news across Facebook and Twitter.


# Discrete orthogonality of hypergeometric polynomial sequences on linear and
quadratic lattices

Luis Verde-Star  Department of Mathematics, Universidad Autónoma
Metropolitana, Iztapalapa, Apartado 55-534, Mexico City 09340, Mexico
<EMAIL_ADDRESS>
###### Abstract

We present a method to obtain weight functions associated with linear and
quadratic lattices that yield discrete orthogonality with respect to a quasi-
definite moment functional of the orthogonal polynomial sequences in the Askey
scheme, with the exception of the Jacobi, Bessel, Laguerre, and Hermite
polynomials.

AMS classification: 33C45, 33D45.

Keywords: Hypergeometric orthogonal polynomials, discrete orthogonality,
generalized moments, linear and quadratic lattices, Askey scheme.

##  1 Introduction

The families of hypergeometric orthogonal polynomial sequences included in the
Askey scheme [7] are some of the most important and have been studied for a
long time. Some of those families have an orthogonality determined by a weight
function $w(x_{k})$ defined on a finite or infinite set of points $x_{k}$ that
belong to a linear or quadratic lattice. In such cases the orthogonality of a
polynomial sequence $\\{u_{n}(t):n\in\mathbb{N}\\}$ is given by

$\sum_{k\in M}u_{n}(x_{k})u_{m}(x_{k})w(x_{k})=K_{n}\delta_{n,m},\qquad n,m\in
M,$ (1)

where $M=\\{0,1,2,\ldots,N\\}$, for some positive integer $N$, or
$M=\mathbb{N}$, and $K_{n}$ is a nonzero constant for $n\in M$.

The polynomial families that satisfy certain types of differential or
difference equations and have a discrete orthogonality with positive weights
$w(x_{k})$ have been characterized and studied in great detail. The main
references for discrete orthogonal polynomials are the books [9] and [7]. See
also [1], [4], [5], and [10].

In the present paper we present a method to find a weight function for each
polynomial family in a subset $\mathcal{B}$ of the Askey scheme that
determines a discrete orthogonality with respect to a quasi-definite moment
functional. The families that are not included in $\mathcal{B}$ are the ones
of Jacobi, Bessel, Laguerre, and Hermite polynomials. The weights $w(x_{k})$
are the values at $t=1$ of a sequence of hypergeometric functions $f_{k}(t)$
that are of type ${}_{3}F_{2}$ or type ${}_{2}F_{1}$. The parameters of
$f_{k}(t)$ depend on the parameters that determine the coefficients in the
three-term recurrence relation of the corresponding polynomial family. We show
that the hypergeometric functions $f_{k}(t)$ are convergent at $t=1$ and that
they are obtained from $f_{0}(t)$ by repeated differentiation and
multiplication by certain factors. In this paper we do not deal with the
problem of characterizing the cases for which the weights are positive.

For the classical families of discrete orthogonal polynomials of the Hahn,
Meixner, Krawtchouk, and Charlier polynomials, the orthogonality that we
obtain coincides with the classical ones presented in [7], with a different
normalization in some cases.

We obtain our results using the linear algebraic approach of our previous
papers [12], [13], [14], and [15]. In [15] we presented a unified construction
of all the hypergeometric and basic hypergeometric orthogonal polynomial
sequences that uses three linearly recurrent sequences of numbers that satisfy
certain difference equation of order three, with constant coefficients. The
initial terms of such numerical sequences determine the sequences of
orthogonal polynomials and provide us with a uniform parametrization of all
the hypergeometric and basic hypergeometric sequences. In the present paper we
used several results from [15].

In Section 2 we present some results related with the construction of the
hypergeometric orthogonal polynomials and their uniform parametrization from
[15]. In Section 3 we present the main result, whose proof uses infinite
systems of linear equations and properties of hypergeometric functions. In
Section 4 we find the weight functions for some families of orthogonal
polynomials from the Askey scheme. Analogous results for the families of basic
hypergeometric polynomial families in the $q$-Askey scheme will be presented
elsewhere.

##  2 The class $\mathcal{H}_{1}$ of hypergeometric orthogonal polynomial
sequences

In this section we present some results about the class $\mathcal{H}_{1}$ of
the hypergeometric orthogonal polynomial sequences that were obtained in [15].

Consider the homogeneous difference equation

$s_{k+3}=3(s_{k+2}-s_{k+1})+s_{k},\qquad k\geqslant-1.$ (2)

The characteristic polynomial of this equation has $1$ as a root of
multiplicity 3, and therefore the general solution is a quadratic polynomial
in $k$. Let $x_{k}$, $h_{k}$, and $e_{k}$ be solutions of (2). Then

$x_{k}=b_{0}+b_{1}k+b_{2}k^{2},\qquad h_{k}=a_{0}+a_{1}k+a_{2}k^{2},\qquad
e_{k}=d_{0}+d_{1}k+d_{2}k^{2}.$ (3)

The sequence $x_{k}$ determines the Newtonian basis $\\{v_{n}(t):n\geqslant
0\\}$ of the complex vector space of polynomials in $t$, defined by

$v_{n}(t)=(t-x_{0})(t-x_{1})\cdots(t-x_{n-1}),\qquad n\geqslant 1,$ (4)

and $v_{0}(t)=1.$ We define the sequence $g_{k}$ by

$g_{k}=x_{k-1}(h_{k}-h_{0})+e_{k},\qquad k\geqslant 1,$ (5)

and $g_{0}=0$. Therefore we must have $e_{0}=d_{0}=0$. In addition, we suppose
that $h_{k}\neq h_{j}$ if $k\neq j$, and $g_{k}\neq 0$ for $k\geqslant 1$.

Let $\mathcal{D}$ be the linear operator on the space of polynomials defined
by

$\mathcal{D}v_{k}=h_{k}v_{k}+g_{k}v_{k-1},\qquad k\geqslant 0.$ (6)

Since $g_{0}=0$ we see that $\mathcal{D}t^{n}=h_{n}t^{n}+$ polynomial of lower
degree. For $n\geqslant 0$ let $u_{n}$ be a monic polynomial of degree $n$
which is an eigenfunction of $\mathcal{D}$ with eigenvalue $h_{n}$. That is

$\mathcal{D}u_{k}=h_{k}u_{k},\qquad k\geqslant 0.$ (7)

The operator $\mathcal{D}$ is a generalized difference operator, which in
concrete examples becomes a second order differential operator or a difference
operator on a linear or quadratic lattice. In [15] we showed that

$u_{n}(t)=\sum_{k=0}^{n}c_{n,k}v_{k}(t),\qquad n\geqslant 0,$ (8)

where the coefficients $c_{n,k}$ are given by

$c_{n,k}=\prod_{j=k}^{n-1}\frac{g_{j+1}}{h_{n}-h_{j}},\qquad 0\leqslant
k\leqslant n-1,$ (9)

and $c_{n,n}=1$ for $n\geqslant 0$. This expression for $u_{n}(t)$ was also
obtained by Vinet and Zhedanov in [16] using a different approach. The idea of
representing orthogonal polynomials in terms of a Newtonian basis was
introduced by Geronimus in [6].

The matrix of coefficients $C=[c_{n,k}]$ is an infinite lower triangular
matrix and the coefficients of $u_{n}$ appear in the $n$-th row of $C$. Since
$c_{n,n}=1$ for $n\geqslant 0$, $C$ is invertible. Let
$C^{-1}=[\hat{c}_{n,k}]$ and define the polynomials

$w_{n,k}(t)=\prod_{j=k}^{n-1}(t-h_{j}),\qquad 0\leqslant k\leqslant n.$ (10)

Using divided differences we obtain

$\hat{c}_{n,k}=\frac{\prod_{j=k+1}^{n}g_{j}}{w_{n+1,k}^{\prime}(h_{k})}=\prod_{j=k+1}^{n}\frac{g_{j}}{h_{k}-h_{j}},\qquad
0\leqslant k\leqslant n-1,$ (11)

and $\hat{c}_{n,n}=1$ for $n\geqslant 0.$

The entries in the $0$-th column of $C^{-1}$ are given by

$\hat{c}_{k,0}=\prod_{j=1}^{k}\frac{g_{j}}{h_{0}-h_{j}},\qquad k\geqslant 1,$
(12)

and $\hat{c}_{0,0}=1$. We denote them by $m_{k}=\hat{c}_{k,0}$ for $k\geqslant
0.$ They satisfy $m_{0}=1$ and

$\sum_{k=0}^{n}c_{n,k}m_{k}=0,\qquad n\geqslant 1.$ (13)

Note that the sequence $m_{n}$ satisfies a recurrence relation of order one.

In [15] we also proved that the polynomial sequence $u_{n}(t)$ satisfies a
three-term recurrence relation of the form

$u_{n+1}(t)=(t-\beta_{n})u_{n}(t)-\alpha_{n}u_{n-1}(t),\qquad n\geqslant 1,$
(14)

where the coefficients are given by

$\beta_{n}=x_{n}+\frac{g_{n+1}}{h_{n}-h_{n+1}}-\frac{g_{n}}{h_{n-1}-h_{n}},$
(15)

and

$\alpha_{n}=\frac{g_{n}}{h_{n-1}-h_{n}}\left(\frac{g_{n-1}}{h_{n-2}-h_{n}}-\frac{g_{n}}{h_{n-1}-h_{n}}+\frac{g_{n+1}}{h_{n-1}-h_{n+1}}+x_{n}-x_{n-1}\right).$
(16)

By Favard’s theorem [3, Thm. 4.4], [8], if all the $\alpha_{n}$ are positive
and the $\beta_{n}$ are real then $\\{u_{n}\\}$ is orthogonal with respect to
a positive-definite moment functional, and if all the $\alpha_{n}$ are nonzero
then $\\{u_{n}\\}$ is orthogonal with respect to a quasi-definite moment
functional.

Let $\mathcal{B}$ be the class of all the families of hypergeometric
orthogonal polynomial sequences whose recurrence coefficients are determined
by a sequence of nodes $x_{k}=b_{0}+b_{1}k+b_{2}k^{2}$, where at least one of
$b_{1}$ or $b_{2}$ is nonzero. The families in the Askey scheme that are not
included in $\mathcal{B}$ are the families of Jacobi, Bessel, Laguerre, and
Hermite. For these families $x_{k}=x_{0}$ for $k\geqslant 0$.

##  3 Generalized moments and discrete orthogonality

Let us suppose that the parameters that determine the sequences $x_{k},h_{k},$
and $e_{k}$ are such that $h_{k}\neq h_{j}$ if $k\neq j$, and the coefficients
$\alpha_{n}$, given by (16), are nonzero for $n\geqslant 1$. Then, by Favard’s
theorem there exists a unique quasi-definite moment functional $\tau$ on the
space of polynomials such that the polynomial sequence $\\{u_{n}(t):n\geqslant
0\\}$ is orthogonal with respect to $\tau$, that is,

$\tau(u_{n}(t)u_{m}(t))=K_{n}\delta_{n,m},\qquad K_{n}\neq 0,\ \
n,m\in\mathbb{N}.$ (17)

Since $u_{0}(t)=1$ we have

$\tau(u_{n}(t))=\sum_{k=0}^{n}c_{n,k}\tau(v_{k}(t))=0,\qquad n\geqslant 1,$
(18)

and by the uniqueness of the inverse of $C$ we must have
$\tau(v_{k}(t))=\hat{c}_{k,0}=m_{k}$ for $k\geqslant 0$. This means that the
numbers $m_{k}$ are the generalized moments of $\tau$ with respect to the
Newtonian basis $\\{v_{k}(t):k\geqslant 0\\}$. Note that $\tau(1)=m_{0}=1.$

######  Theorem 3.1.

Suppose that at least one of $b_{1}$ and $b_{2}$ is nonzero, at least one of
$a_{1}$ and $a_{2}$ is nonzero, and $\alpha_{n}\neq 0$ for $n\geqslant 1$.
Then there exists a unique weight function $w$, defined on the nodes $x_{k}$,
such that for every polynomial $p(t)$ we have

$\tau(p(t))=\sum_{k=0}^{\infty}p(x_{k})w(x_{k}).$ (19)

Proof: From (3) we see that the hypothesis implies that $\\{x_{k}\\}$ and
$\\{h_{k}\\}$ are sequences of pairwise distinct numbers, and that the
polynomial sequence $\\{u_{n}(t):n\geqslant 0\\}$ is orthogonal with respect
to the quasi-definite moment functional $\tau$.

We will write $r_{j}=w(x_{j})$ for $j\geqslant 0$. Since
$\tau(v_{k}(t))=m_{k}$ for $k\geqslant 0$, the numbers $r_{j}$ that we want to
find must satisfy

$m_{k}=\tau(v_{k}(t))=\sum_{j=0}^{\infty}v_{k}(x_{j})\,r_{j},\qquad k\geqslant
0.$ (20)

This is an infinite system of linear equations. Let us denote by $P$ the
matrix of coefficients. Then $P=[v_{k}(x_{j})]$, where $j$ is the index for
rows, $k$ is the index for columns and $j$ and $k$ are non-negative integers.
From the definition of the polynomials $v_{k}(t)$ in (4) we can see that $P$
is an infinite lower triangular matrix and its entries in the main diagonal
are

$v_{k}(x_{k})=(x_{k}-x_{0})(x_{k}-x_{1})\cdots(x_{k}-x_{k-1}),$

and they are nonzero. Therefore $P$ is invertible.

Using basic properties of divided differences with respect to the nodes
$x_{j}$ it is easy to show that $P^{-1}$ is the lower triangular matrix whose
$(j,k)$ entry equals $1/v_{j+1}^{\prime}(x_{k})$. Therefore, from the system
(20) we obtain

$r_{k}=\sum_{j=k}^{\infty}\frac{m_{j}}{v_{j+1}^{\prime}(x_{k})},\qquad
k\geqslant 0.$ (21)

We will show that if the parameters $a_{1},a_{2},b_{1},b_{2},d_{1},d_{2}$
satisfy certain conditions, then all the series in (21) are convergent (or
terminating) hypergeometric series.

Let us define the power series

$f_{k}(t)=\sum_{j=k}^{\infty}\frac{m_{j}}{v_{j+1}^{\prime}(x_{k})}t^{j},\qquad
k\geqslant 0.$ (22)

The denominators in the previous equation are

$v_{j+1}^{\prime}(x_{k})=\prod_{i=0,i\neq k}^{j}(x_{k}-x_{i}),\qquad
k\geqslant j\geqslant 0.$ (23)

By substitution of (23) and (12) in (22), writing all the sequences using the
formulas (3) and (5), we find an expression for $f_{k}(t)$ in terms of the
parameters $a_{1},a_{2},b_{0},b_{1},b_{2},d_{1},d_{2}$. For example, the first
3 terms of $f_{0}(t)$ are

$\begin{split}f_{0}(t)=1+&\frac{a_{1}b_{0}+a_{2}b_{0}+d_{1}+d_{2}}{(a_{1}+a_{2})(b_{1}+b_{2})}t+\\\
&\frac{(a_{1}b_{0}+a_{2}b_{0}+d_{1}+d_{2})((b_{0}+b_{1}+b_{2})(a_{1}+2a_{2})+d_{1}+2d_{2})}{(a_{1}+a_{2})(a_{1}+2a_{2})(b_{1}+b_{2})(b_{1}+2b_{2})}\frac{t^{2}}{2!}+\cdots\end{split}$
(24)

Note that $a_{0}$ does not appear here because the sequence of eigenvalues
$h_{k}$ enters only trough the differences $h_{k}-h_{j}$. Therefore the value
of $a_{0}$ is not relevant and we can suppose that $a_{0}=0$.

The series in (24) looks like a hypergeometric function. We can confirm that
it is indeed a hypergeometric function through the changes of parameters that
we define next. We will consider four cases, corresponding to whether $a_{2}$
and $b_{2}$ are zero or nonzero.

Case 1. If $a_{2}\neq 0$ and $b_{2}\neq 0$ we introduce the parameters
$p,r,y_{1},y_{2}$ as follows

$\begin{split}a_{1}=&(r-1)a_{2},\\\ b_{1}=&(p-1)b_{2},\\\
d_{1}=&a_{2}(b_{2}((y_{1}-1)(y_{2}-1)(p+r-y_{1}-y_{2}-2)+(r-1)(p-2))-(r-1)b_{0}),\\\
d_{2}=&a_{2}(b_{2}((y_{1}+y_{2})(p+r-y_{1}-y_{2}-1)+y_{1}y_{2}-r(p-1))-b_{0}).\end{split}$
(25)

The new parameters $y_{1}$ and $y_{2}$ can be expressed in terms of the
original parameters by solving for $y_{1}$ and $y_{2}$ the last two equations
in (25).

Substitution of $a_{1},b_{1},d_{1},d_{2}$ in $f_{0}(t)$ using (25) gives us

$f_{0}(t)=F(y_{1},y_{2},p+r-y_{1}-y_{2}-1;r,p;t),$ (26)

where $F$ is the hypergeometric function of type ${}_{3}F_{2}$ defined by

$F(y_{1},y_{2},y_{3};r,p;t)=\sum_{k=0}^{\infty}\frac{(y_{1})_{k}(y_{2})_{k}(y_{3})_{k}}{(r)_{k}(p)_{k}}\frac{t^{k}}{k!},$
(27)

where $(a)_{k}$ denotes the shifted factorial, defined by $(a)_{0}=1$ and

$(a)_{k}=a(a+1)(a+2)\cdots(a+k-1),\qquad a\in\mathbb{C},\ k\geqslant 1.$

For any values of the parameters $p,r,y_{1}$ and $y_{2}$ the parametric excess
(the sum of the lower parameters minus the sum of the upper parameters) of the
hypergeometric function in (26) is equal to one. Therefore it is convergent at
$t=1$, since a hypergeometric function converges at $t=1$ if the real part of
the parametric excess is positive. See [2, Thm. 2.1.2, p.62] or [11, p.45 ].

The functions $f_{k}(t)$, for $k\geqslant 1$, can also be expressed in terms
of hypergeometric functions using the parameters $p,r,y_{1}$ and $y_{2}$. In
those functions the factors of the form $p+i$ in the denominators appear in a
less regular way than in $f_{0}(t)$. This is so because they come from the
derivatives $v_{j+1}^{\prime}(x_{k})$ and this produces jumps and shifts in
the index $i$ of the factors $p+i$.

We can verify by direct computations that

$\begin{split}f_{k}(t)=(-1)^{k}&\left(\frac{p+2k-1}{p+k-1}\right)\frac{t^{k}}{k!}\
\times\\\ &\qquad D_{t}^{k}F(y_{1},y_{2},r+p-y_{1}-y_{2}-1;r,p+k;t),\quad
k\geqslant 0,\end{split}$ (28)

where $D_{t}$ denotes differentiation with respect to $t$.

For $k\geqslant 0$ the parametric excess of
$F(y_{1},y_{2},r+p-y_{1}-y_{2}-1;r,p+k;t)$ is $k+1$, and thus the function
converges at $t=1$. Therefore $f_{k}(t)$ converges at $t=1$ and the weights
$r_{k}=f_{k}(1)$ are well defined for $k\geqslant 0.$

The functions $f_{k}(t)$ can also be expressed as follows

$\begin{split}f_{k}(t)=(-1)^{k}&\frac{(y_{1})_{k}(y_{2})_{k}(p+r-y_{1}-y_{2}-1)_{k}}{(r)_{k}(p-1+k)_{k}}\frac{t^{k}}{k!}\
\times\\\ &\qquad
F(y_{1}+k,y_{2}+k,r+p-y_{1}-y_{2}-1+k;r+k,p+2k;t).\end{split}$ (29)

The sequence of weights $r_{k}=f_{k}(1)$ that we have found satisfies (20) for
$k\geqslant 0$, and since the set $\\{v_{k}(t):k\geqslant 0\\}$ is a basis for
the space of all polynomials, we conclude that
$\tau(p(t))=\sum_{k=0}^{\infty}p(x_{k})r_{k}$ for every polynomial $p(t)$.

Case 2. We consider now the case with $b_{2}=0$ and $a_{2}\neq 0$. We use here
the substitutions

$\begin{split}b_{2}=&0,\\\ a_{1}=&(r-1)a_{2},\\\
d_{1}=&a_{2}(b_{1}(r-1+(y_{1}-1)(y_{2}-1))-b_{0}(r-1)),\\\
d_{2}=&a_{2}(b_{1}(y_{1}+y_{2}-r)-b_{0}),\end{split}$ (30)

and obtain

$f_{0}(t)=F_{2}(y_{1},y_{2};r;t),$ (31)

where $F_{2}$ is the hypergeometric function of type ${}_{2}F_{1}$ defined by

$F_{2}(y_{1},y_{2};r;t)=\sum_{k=0}^{\infty}\frac{(y_{1})_{k}(y_{2})_{k}}{(r)_{k}}\frac{t^{k}}{k!}.$
(32)

From (30) we can obtain $y_{1}$ and $y_{2}$ by solving a system of equations.
They satisfy

$y_{1}+y_{2}=\frac{(b_{1}r+b_{0})a_{2}+d_{2}}{b_{1}a_{2}},$

and therefore the function (31) converges at $t=1$ if the real part of
$-(a_{2}b_{0}+d_{2})/(a_{2}b_{1})$ is positive.

The substitution of (30) in $f_{k}(t)$ gives us

$f_{k}(t)=(-1)^{k}\frac{t^{k}}{k!}\,D_{t}^{k}F_{2}(y_{1},y_{2};r;t),\qquad
k\geqslant 0.$ (33)

These functions are also convergent at $t=1$ if the real part of
$-(a_{2}b_{0}+d_{2})/(a_{2}b_{1})$ is positive. We can also write the
functions $f_{k}(t)$ as

$f_{k}(t)=(-1)^{k}\frac{(y_{1})_{k}(y_{2})_{k}}{(r)_{k}}\frac{t^{k}}{k!}F_{2}(y_{1}+k,y_{2}+k;r+k;t),\qquad
k\geqslant 1.$ (34)

Case 3. Suppose now that $a_{2}=0$ and $b_{2}\neq 0$. In this case we see from
(24) that $f_{0}(t)$ becomes a simpler hypergeometric function. The parameter
substitutions

$\begin{split}a_{2}=&0,\\\ b_{1}=&(p-1)b_{2},\\\
d_{1}=&a_{1}(b_{2}((y_{1}-1)(y_{2}-1)+p-2)-b_{0}),\\\
d_{2}=&-a_{1}b_{2}(p-y_{1}-y_{2}-1),\end{split}$ (35)

in $f_{0}(t)$ give us

$f_{0}(t)=F_{2}(y_{1},y_{2};p;t).$ (36)

The parameters $y_{1}$ and $y_{2}$ can be obtained from (35) by solving a
system of equations. They satisfy

$y_{1}+y_{2}=p-1+\frac{d_{2}}{a_{1}b_{2}}.$ (37)

Therefore the parametric excess of the function $F_{2}(y_{1},y_{2};p;t)$ in
(36) is $p-y_{1}-y_{2}=1-\frac{d_{2}}{a_{1}b_{2}},$ and thus $f_{0}(t)$
converges at $t=1$ if

$\hbox{Re}\left(1-\frac{d_{2}}{a_{1}b_{2}}\right)>0.$ (38)

In this case, applying the substitutions of parameters (35) to the series
$f_{k}(t)$, we obtain hypergeometric functions similar to
$F_{2}(y_{1},y_{2};p;t)$. By a straightforward computation we can verify that

$f_{k}(t)=(-1)^{k}\left(\frac{p+2k-1}{p+k-1}\right)\frac{t^{k}}{k!}\,D_{t}^{k}F_{2}(y_{1},y_{2};p+k;t),\qquad
k\geqslant 0.$ (39)

Note that all the functions $f_{k}(t)$ are convergent at $t=1$ if (38) holds.
Another expression for $f_{k}(t)$ in this case is the following

$f_{k}(t)=(-1)^{k}\frac{(y_{1})_{k}(y_{2})_{k}}{(p-1+k)_{k}}\frac{t^{k}}{k!}F_{2}(y_{1}+k,y_{2}+k;p+2k;t).$
(40)

Case 4. In this case we have $a_{2}=0$ and $b_{2}=0$ and $f_{0}(t)$ has the
form

$f_{0}(t)=1+\frac{a_{1}b_{0}+d_{1}+d_{2}}{a_{1}b_{1}}t+\frac{(a_{1}b_{0}+d_{1}+d_{2})(a_{1}b_{0}+a_{1}b_{1}+d_{1}+2d_{2})}{a_{1}^{2}b_{1}^{2}}\frac{t^{2}}{2!}+\cdots$
(41)

The substitutions

$\begin{split}a_{2}=&0,\qquad b_{2}=0,\\\
d_{1}=&a_{1}(b_{1}(1+(y-1)z)-b_{0}),\\\ d_{2}=&(z-1)a_{1}b_{1},\end{split}$
(42)

give us

$f_{0}(t)=F_{2}(y,1;1;zt).$ (43)

By the ratio test we see that $F_{2}(y,1;1;zt)$ converges at $t=1$ if $|z|<1$.
With the change of parameters (42) the series $f_{k}(t)$ become the
hypergeometric functions

$f_{k}(t)=(-1)^{k}\frac{t^{k}}{k!}\,D_{t}^{k}(F_{2}(y,1;1;zt)),\qquad
k\geqslant 1,$ (44)

where $D_{t}$ denotes differentiation with respect to $t$. It is clear that
$f_{k}(t)$ converges at $t=1$ if $|z|<1$.

From (42) we can solve for $y$ and $z$ in terms of the other parameters and we
obtain

$y=\frac{a_{1}b_{0}+d_{1}+d_{2}}{a_{1}b_{1}+d_{2}},\qquad\ \
z=1+\frac{d_{2}}{a_{1}b_{1}}.$ (45)

Therefore the functions $f_{k}(t)$ are convergent at $t=1$ if
$|1+\frac{d_{2}}{a_{1}b_{1}}|<1$.

We have proved that in each of the four cases the weights $r_{k}=f_{k}(1)$ are
well defined if the parameters satisfy certain conditions.  $\Box$

It is easy to verify that in the four cases considered in the previous theorem
we have

$\sum_{k=0}^{n}\hbox{coefficient of }t^{n}\ \hbox{in}\ f_{k}(t)=0,\qquad
n\geqslant 1,$ (46)

and therefore

$\sum_{k=0}^{\infty}f_{k}(t)=1,$ (47)

for every $t$ in the region of convergence of $f_{0}(t)$, and in particular
$\sum_{k=0}^{\infty}r_{k}=1$. This property of the weights also follows from
$\tau(v_{0})=\tau(1)=m_{0}=1.$

Let us note that in the four cases the hypergeometric function $f_{0}(t)$
determines to a large extent the sequence of functions $f_{k}(t)$ and their
region of convergence.

######  Corollary 3.1.

With the hypothesis of the previous theorem, if the parameters satisfy the
conditions of the corresponding case then the polynomials $u_{k}(t)$ satisfy
the discrete orthogonality

$\tau(u_{n}(t)u_{m}(t))=\sum_{k=0}^{\infty}u_{n}(x_{k})u_{m}(x_{k})r_{k}=K_{n}\delta_{n,m},$
(48)

where the constants $K_{n}=\alpha_{1}\alpha_{2}\cdots\alpha_{n}$ are nonzero
for $n\geqslant 0.$

##  4  Some examples of the weights $r_{k}$

In this section we find the weights $r_{k}$ that correspond to some families
of orthogonal polynomial sequences from the Askey scheme [7].

For each of the four cases we can find the coefficients of the three-term
recurrence relation of the corresponding polynomial family by applying to the
general formulas for $\beta_{n}$ and $\alpha_{n}$, given in (15) and (16), the
parameter substitutions that we used in the proof of Theorem 3.1.

Case 1. In the first case the parameter substitution (25) applied to
$\alpha_{n}$ yields

$\begin{split}\alpha_{n}=&\frac{b_{2}^{2}\,n(n+y_{1}-1)(n+y_{2}-1)(n-p+y_{1}+y_{2})}{(2n+r-3)(2n+r-2)^{2}(2n+r-1)}\
\times\\\ &(n-2+r)(n+r-y_{1}-1)(n+r-y_{2}-1)(n+p+r-y_{1}-y_{2}-2).\end{split}$
(49)

The change of parameters

$r=a+b+c+d,\quad p=2c+1,\quad y_{1}=c+d,\quad y_{2}=b+c,$ (50)

transforms (49) into the sequence $\alpha_{n}$ that corresponds to the Wilson
polynomials written in terms of the parameters $a,b,c,d$ used in [7, eq.
9.1.5]. The analogous result for the sequence $\beta_{n}$ of the Wilson
polynomials holds. Therefore the weights $r_{k}$ for the Wilson polynomials
are the values at $t=1$ of the functions $f_{k}(t)$ of (28) with the
parameters changed by the substitutions (50). We have

$\begin{split}f_{k}(t)=&(-1)^{k}\left(\frac{2c+2k}{2c+k}\right)\frac{t^{k}}{k!}\
\times\\\ &\qquad D_{t}^{k}F(c+d,b+c,a+c;a+b+c+d,2c+1+k;t).\end{split}$ (51)

For any values of $a,b,c,d$ the parametric excess of the hypergeometric
function in the previous equation is equal to $1+k$ and therefore $f_{k}(t)$
converges at $t=1$ for $k\geqslant 0$.

Note that (50) allows us to express the parameters $a,b,c,d$ in terms of
$p,r,y_{1},y_{2}$. From (29) we obtain another expression for the functions
$f_{k}(t)$ of the Wilson polynomials.

$\begin{split}f_{k}(t)=&(-1)^{k}\frac{(c+d)_{k}(b+c)_{k}(a+c)_{k}}{(a+b+c+d)_{k}(2c+k)_{k}}\frac{t^{k}}{k!}\
\times\\\ &\qquad F(c+d+k,b+c+k,a+c+k;a+b+c+d+k,2c+1+2k;t).\end{split}$ (52)

If $y_{1}$ or $y_{2}$ is a negative integer then the hypergeometric functions
$f_{k}(t)$ defined in (28) are terminating. In that case the corresponding set
of orthogonal polynomials is finite. This is the case for the Racah polynomial
family.

Case 2. We present next examples that correspond to the second case in the
proof of Theorem 3.1, that is, with $b_{2}=0$ and $a_{2}$ nonzero.

The substitution (30) applied to the general formula for $\alpha_{n}$ gives us

$\alpha_{n}=-\frac{b_{1}^{2}n(n+r-2)(n+y_{1}-1)(n+y_{2}-1)(n+r-y_{1}-1)(n+r-y_{2}-1)}{(2n+r-2)^{2}(2n+r-3((2n+r-1)}.$
(53)

The substitutions

$\begin{split}b_{1}=&i,\qquad r=a+b+c+d,\\\ y_{1}=&a+c,\qquad
y_{2}=a+d,\end{split}$ (54)

transform (53) into

$\begin{split}\alpha_{n}=&\frac{n(n+a+b+c+d-2)(n+a+c-1)}{(2n+a+b+c+d-2)^{2}(2n+a+b+c+d-1)}\times\\\
&\qquad\qquad\frac{(n+a+d-1)(n+b+c-1)(n+b+d-1)}{(2n+a+b+c+d-3)}.\end{split}$
(55)

This $\alpha_{n}$ corresponds to the continuous Hahn polynomials [7, Eq.
9.4.4].

Applying the change of parameters (54) to the functions $f_{k}(t)$ in (33) we
obtain

$f_{k}(t)=(-1)^{k}\frac{t^{k}}{k!}\,D_{t}^{k}F_{2}(a+c,a+d;a+b+c+d;t),\qquad
k\geqslant 0.$ (56)

These hypergeometric functions converge at $t=1$ if the real part of $b-a$ is
positive.

The Hahn polynomials [7, Eq.9.5.4], with parameters $\alpha,\beta,N$, are
obtained from the general formulas for $\alpha_{n}$ and $\beta_{n}$ with the
substitutions

$\begin{split}b_{2}=&0,\qquad b_{1}=1,\qquad b_{0}=0,\\\
a_{1}=&(\alpha+\beta+1)a_{2},\\\ d_{1}=&-a_{2}(N\alpha-\beta-1),\\\
d_{2}=&-a_{2}(N+\beta+1).\end{split}$ (57)

With these values of the parameters we obtain
$f_{0}(t)=F_{2}(-N,\alpha+1;\alpha+\beta+2;t)$, and this terminating
hypergeometric function determines the discrete orthogonality [7, Eq. 9.5.2]
of the Hahn polynomials.

Case 3. We consider next the case with $a_{2}=0$ and $b_{2}\neq 0$. The
continuous dual Hahn polynomials, with parameters $a,b,c$ [7, Eq.9.3,5], are
obtained from the general formulas taking

$\begin{split}a_{2}=&0,\qquad a_{1}=1,\\\ b_{2}=&1,\qquad b_{1}=2c,\qquad
b_{0}=c^{2},\\\ d_{2}=&a+b,\qquad d_{1}=ab+ac+bc-a-b.\end{split}$ (58)

Combining these equations with (35) we obtain

$p=2c+1,\qquad y_{1}=a+c,\qquad y_{2}=b+c.$ (59)

By substitution in (39) we obtain

$f_{k}(t)=(-1)^{k}\left(\frac{2c+2k}{2c+k}\right)\frac{t^{k}}{k!}\,D_{t}^{k}F_{2}(a+c,b+c;2c+1+k;t),\qquad
k\geqslant 0.$ (60)

These functions are convergent at $t=1$ if the real part of $1-a-b$ is
positive. The numbers $r_{k}=f_{k}(1)$ are the weights for the continuous dual
Hahn polynomials.

Case 4. We consider here examples with $a_{2}=0$ and $b_{2}=0$. Some of the
classical discrete orthogonal polynomials fall in this case.

If $a_{2}=0$ and $b_{2}=0$ the functions $f_{k}(t)$ are given in (44), which
is

$f_{k}(t)=(-1)^{k}\frac{t^{k}}{k!}\,D_{t}^{k}\,F_{2}(y,1;1;zt),\qquad
k\geqslant 0,$ (61)

where $y$ and $z$ are given in (45), and all the $f_{k}(t)$ are convergent at
$t=1$ if $|z|<1$. It is clear from (61) that $f_{0}(t)$ determines all the
functions $f_{k}(t)$, and the weights $r_{k}=f_{k}(1)$.

The Krawtchouk polynomials [7, Eq. 9.11.4], with parameters $p$ and $N$, are
obtained with

$\begin{split}a_{2}=&0,\qquad b_{2}=0,\qquad b_{1}=1,\\\
d_{2}=&(p-1)a_{1},\qquad d_{1}=-a_{1}(Np+p+b_{0}-1),\\\ \end{split}$ (62)

and these equations give $f_{0}(t)=F_{2}(-N,1;1;pt)$, which yields the
discrete orthogonality in [7, Eq. 9.11.2].

The Meixner polynomials, with parameters $c$ and $\beta$ [7, Eq. 9.10.4], are
obtained with

$\begin{split}a_{2}=&0,\qquad b_{2}=0,\qquad b_{1}=1,\\\
d_{2}=&\frac{a_{1}}{c-1},\qquad d_{1}=\frac{a_{1}(\beta-
b_{0})c+b_{0}-1)}{c-1},\\\ \end{split}$ (63)

and these parameters give us $f_{0}(t)=F_{2}(\beta,1;1;\frac{ct}{c-1})$, which
produces the discrete orthogonality described in [7, Eq. 9.10.2].

The Charlier polynomials, with parameter $a$ [7, Eq. 9.14.4], are obtained
with

$\begin{split}a_{2}=&0,\qquad b_{2}=0,\qquad b_{1}=1,\\\ d_{2}=&-a_{1},\qquad
d_{1}=(1-a-b_{0})a_{1},\\\ \end{split}$ (64)

which give $f_{0}(t)=\exp(-at).$ This function produces the discrete
orthogonality [7, Eq. 9.14.2].

##  5 Conclusions.

We have shown that all the hypergeometric orthogonal polynomial sequences in
the Askey scheme, other than the very classical families (Jacobi, Bessel,
Laguerre, and Hermite), satisfy a discrete orthogonality on a sequence of
distinct nodes $x_{k}$ in the complex plane. The nodes are of the form
$x_{k}=b_{0}+b_{1}k+b_{2}k^{2}$, where at least one of $b_{1}$ and $b_{2}$ is
nonzero. The weights $r_{k}$ associated with the nodes $x_{k}$ are the values
at $t=1$ of a sequence of hypergeometric functions. We have not dealt with the
problem of characterizing the cases for which the weights $r_{k}$ are
positive.

The approach used in this paper can be used to obtain discrete orthogonality
for some of the families of basic hypergeometric orthogonal polynomials in the
$q$-Askey scheme.

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