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##  Appendix A Impact of redundant features on feature importance

Machine-learning methods that build models using decision-tree ensembles have
the potential to provide insights into the structure of the training data via
analyses of the feature-importances. When all features in the dataset are
fully independent of each other, the feature-importance can provide a
relatively straightforward indication of how useful each feature is for the
model to predict the target labels. However, when there is significant co-
linearity between features, the importance may be shared between co-linear
features, resulting in potentially misleading feature-importance information.
Indeed, it is likely that significant co-linearity exists among the various
broadband magnitudes and colours used in this work. Therefore, here we examine
how the feature importance calculations used by the RandomForestClassifier,
CatBoostClassifier, and XGBoostClassifier tree-based algorithms are affected
by the presence of multiple co-linear features.

We first take the Int Wide catalogue, select features derived from $u$,
$I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$,
$J_{\scriptscriptstyle{\rm E}}$, or $H_{\scriptscriptstyle{\rm E}}$
photometry, and add five identical copies of the $u-I_{\scriptscriptstyle{\rm
E}}$ feature to the dataset. We then train RandomForestClassifier,
CatBoostClassifier, and XGBoostClassifier models to select quiescent galaxies
in the $0<z<0.25$ range. We selected the $u-I_{\scriptscriptstyle{\rm E}}$
colour for duplication because RandomForestClassifier, CatBoostClassifier, and
XGBoostClassifier all find this feature to be the most important for this
particular classification problem. For these model runs, no information was
provided regarding the redshifts of the galaxies. The results are shown in
Figs. 11– 13.

The impact on the feature importances of duplicating the most important single
feature ($I_{\scriptscriptstyle{\rm E}}-u$ in this example) is somewhat
different for each of the learning algorithms, presumably due to: (i)
differences in the way the algorithms select features for constructing
individual decision trees; (ii) how they deal with co-linearity among the
input features; and (iii) the different methods used to calculate feature
importance values.

In the case of RandomForestClassifier, the impact of including duplicates of
$I_{\scriptscriptstyle{\rm E}}-u$ is for this feature and its duplicates to be
demoted to a significantly lower rank of importance (Fig. 11), with all six
features occupying a similar position within the feature importance ranking.
Clearly, it is risky to rely on the feature importances produced by
RandomForestClassifier, since they are a function of how informative a
features is and the uniqueness of the information it provides. When using
CatBoostClassifier, the inclusion of duplicates of $I_{\scriptscriptstyle{\rm
E}}-u$ also results in the demotion of $I_{\scriptscriptstyle{\rm E}}-u$ and
its duplicates to significantly lower ranks of importance (Fig. 12), with
several being placed at the very end of the importance ranking.

In contrast, the feature importance values provided by XGBoostClassifier are
much more robust against the presence of co-linearity among features. As
illustrated in Fig. 13, $I_{\scriptscriptstyle{\rm E}}-u$, and/or several
copies thereof are consistently assigned the highest values of feature
importance, or else are simply ignored (feature importance = 0). As such, the
feature importances provided by XGBoostClassifier are likely to offer a
relatively robust method to determine the most relevant observables for the
selection of particular galaxy types, even when there is significant co-
linearity between the observables.

Figure 11: Feature importance values when using RandomForestClassifier to
select quiescent galaxies in the redshift range $0<z<0.25$. The results in the
upper panel are for the case where none of the features have been duplicated.
The lower panel shows the feature importance values for the case where five
copies of $I_{\scriptscriptstyle{\rm E}}-u$ have been injected into the
dataset prior to model training. The $x$ axis labels correspond to feature
names used by the pipeline after the pre-processing steps outlined in Sect.
4.2 have been applied, and should be self-explanatory (see the caption of Fig.
7. For example, the feature named ‘VIS-u_copy1’ is a copy of the feature named
’VIS-u’, etc.

Figure 12: Similar to Fig. 11, but for models trained using
CatBoostClassifier.

Figure 13: Similar to Fig. 11, but for models trained using XGBoostClassifier.

##  Appendix B Further analysis and tests

In this appendix we provide full details of the analyses and tests that were
summarised in Sect. 7.

Figure 14: Examples of the F1-scores from individual base-learners and the
model ensembling methods. Left: Selection of quiescent galaxies at $z=2$–2.5
from the Int Wide catalogue using Euclid photometry, without foreknowledge of
galaxy redshifts. As described in the text, the meta-learner performs a non-
linear fusion of the individual classifiers, resulting in a significantly
higher F1-score than obtained by any of the individual base learners or the
two other ensemble methods (averaging and hard-voting). Centre: Impact of
ensembling a LightGBMClassifier model, the hyperparameters of which are well-
tuned for this problem (LightGBM 1), with four other LightGBM models that have
poorly tuned hyperparameters (LightGBM 2,3,4,5). In this case, averaging the
model predictions and hard-voting both produce poor results, but the meta-
learner is able to identify and weight accordingly the low quality class
predictions. Right: Application of a meta-learner to a classification model
produced by a single base-learner, in this case XGBoostClassifier. In this
circumstance, the meta-learner performs ‘error correction’, resulting in a
significant improvement in the quality of the classifier. Figure 15:
Distribution of incorrect classifications (FP and FN) with respect to
$I_{\scriptscriptstyle{\rm E}}-Y_{\scriptscriptstyle{\rm E}}$, sSFR, and
stellar mass, when selecting quiescent galaxies from the Int catalogues. Also
shown is the overall distribution of galaxies in the catalogue (or catalogue
subset) on which the selection was performed. Top row: Using the $ugriz$,
Euclid, $W1$, $W2$, and 20 cm photometry from the Int Wide catalogue,
excluding only those galaxies with a non-detection in any Euclid band. Centre
row: As above, but excluding galaxies without a non-detection in any of the
$ugriz$ or Euclid bands. Bottom row: Using the $ugriz$, Euclid, $W1$, $W2$,
and 20 cm photometry from the Int Deep catalogue, excluding only those
galaxies with a non-detection in any Euclid band. In each case, we include the
values of the F1-score, precision, recall, and $\bar{I}_{\rm FP}$ metrics
Figure 16: Similar to Fig. 15, but instead using the SED catalogues. Figure
17: Distribution of incorrect classifications for the
$I_{\scriptscriptstyle{\rm E}}-Y_{\scriptscriptstyle{\rm E}}$,
$J_{\scriptscriptstyle{\rm E}}-H_{\scriptscriptstyle{\rm E}}$ method of B20
((left column) and the $u-I_{\scriptscriptstyle{\rm E}}$,
$I_{\scriptscriptstyle{\rm E}}-J_{\scriptscriptstyle{\rm E}}$ method of B20
(centre column) To allow a direct comparison between the
$u-I_{\scriptscriptstyle{\rm E}}$, $I_{\scriptscriptstyle{\rm
E}}-J_{\scriptscriptstyle{\rm E}}$ and our machine-learning selection method,
we show results from our pipeline under conditions equivalent to those used
for the B20 $u-I_{\scriptscriptstyle{\rm E}}$, $I_{\scriptscriptstyle{\rm
E}}-J_{\scriptscriptstyle{\rm E}}$ method: Galaxy photometric redshifts are
included as a feature to be trained on; only galaxies detected in $u$,
$I_{\scriptscriptstyle{\rm E}}$, and $J_{\scriptscriptstyle{\rm E}}$ are used;
only galaxies in the redshift ranges $0<z<1$ and $0<z<1.5$ are used for the
Wide and Deep catalogues, respectively. In each panel, we include the values
of the F1-score, precision, recall, and $\bar{I}_{\rm FP}$ metrics.  Figure
18: Precision (purity), recall (completeness), and the F1-score as a function
of the probability threshold used to separate quiescent and star-forming
galaxies, for Int Wide (top), SED Wide (centre), and the average result
(bottom).

Figure 19: Impact of including source redshifts as an additional feature in
the input data. Left: Case where the dataset is binned by photometric redshift
prior to model training, with 100 per cent, 50 per cent, or none of the
redshifts included as a feature in the data. Right: Illustrating the case
where no redshift binning is performed, with classifiers being trained to
identify quiescent galaxies at specific redshift intervals. For this test, the
Int Wide mock catalogue was used. Figure 20: Testing the impact of photometry
measurement uncertainties on the results from our classification pipeline.
Top: Example of how the F1-score is reduced when Gaussian noise is added to
the optical and NIR photometry in the Int Wide catalogue. Bottom: Example of
how the F1-score is increased when galaxies fainter than an arbitrary
$I_{\scriptscriptstyle{\rm E}}$ threshold are excluded from the Int Wide
catalogue.

###  B.1 Stacking versus individual learners

We discuss the benefits of our implementation of the generalised stacking
method, in which meta-learners are trained to fuse the output from several
base-learners into a single classifier. We find that, with very few
exceptions, our stacking method consistently outperforms each of the
individual base-learners, in addition to outperforming the traditional
ensembling methods of model averaging and hard-voting. This is illustrated in
Fig. 14 (left panel), where we show results from a single run of our pipeline,
in this case applied to the selection of quiescent galaxies at $z=2$–2.5 from
the Int Wide catalogue using Euclid photometry, and without foreknowledge of
galaxy redshifts. In this example, averaging the predictions across the base-
learners results in an ‘averaging-down’, while the hard-vote ensemble method
results in an F1-score that matches that of the best individual base-learner
(in this case LightGBMClassifier). In contrast, the meta-learner yields an
F1-score that is substantially higher ($\sim 46$ per cent in this case) than
any of the individual base-learners or other ensemble methods.

In Fig. 14 (centre panel) we illustrate the robustness of the generalised
stacking method against pollution by multiple low-quality classifier models.
We ensembled a LightGBM model with hyperparameters that are well tuned for
this problem (LightGBM 1), with four other LightGBM models that have
purposefully poorly tuned hyperparameters (LightGBM 2–5). Whereas the average
and hard-voting ensembles give poor results, the meta-learner is able to
discard low-quality class predictions and also make a significant improvement
over the single high-quality model (LightGBM 1).

In addition, we illustrate the usefulness of generalised stacking when applied
to a single classifier model. In Fig. 14 (right panel), we show the result of
selecting quiescent galaxies in the $z=2.5$–3 redshift range, using $ugriz$,
Euclid, $W1$, and $W2$ photometry from the Int Wide catalogue, with
foreknowledge of redshifts. The meta-learner is able to substantially improve
on the F1-score of the XGBoostClassifier, increasing the score by 61 per cent,
effectively turning a rather poor classifier into a potentially much more
useful one.

###  B.2 The nature of the false positives

Although the $P$, $R$, and F1-score classification metrics are informative
about whether galaxies are correctly or incorrectly classified, they do not
provide information about the incorrectness of the incorrect classifications.
For instance, the F1-score is insensitive to whether false positives are
marginally non-quiescent (e.g. ${\rm sSFR}\sim 10^{-10.4}\,{\rm yr}^{-1}$) or
are in fact powerful starburst galaxies (e.g. $10^{-8}{\rm yr}^{-1}$); when
selecting samples of quiescent galaxies, the former case is clearly less
baneful than the latter. Therefore, we now examine the nature of the non-
quiescent contaminants in samples selected as quiescent by our classification
pipeline, utilising our metrics of incorrectness $I_{\rm FP}$ and
$\bar{I}_{\rm FP}$ (see Eqs. 7 and 8).

In Figs. 15 and 16 we show the distribution of incorrect classifications with
respect to $I_{\scriptscriptstyle{\rm E}}-H_{\scriptscriptstyle{\rm E}}$,
sSFR, and stellar mass, for the Int and SED catalogues. As is often the case
when performing a binary classification on a continuously distributed sample,
the incorrectly classified objects cluster around the class threshold value
($10^{-10.5}{\rm yr}^{-1}$), with a density that is highest in the bins
immediately adjacent to the class boundary. However, the precise distribution
of classification errors varies between the different catalogues (and subsets
thereof).

When using the Wide survey mock catalogues, $\ga 50$ per cent of false
positives are what we consider to be marginal classification errors, that is,
their sSFR is within 0.5 dex of the class boundary. Furthermore, there are
very few false positives with high values of sSFR: fewer than 25 per cent of
the false positives are at sSFR $\geq 10^{-9.5}{\rm yr}^{-1}$, while false
positives with sSFR $\geq 10^{-9}{\rm yr}^{-1}$ are negligible ($\la 5$ per
cent). On the other hand, when the Deep survey mock data are used, the
fraction of false positives at relatively high values of sSFR ($\geq
10^{-9}{\rm yr}^{-1}$) is non-negligible ($\sim 25$ per cent). This is true
regardless of whether the Int Deep or SED Deep catalogue is used. The
distribution of the incorrect classifications with respect to the stellar mass
shows a significant diversity and depends strongly on which mock catalogue is
used. In general, the false positives are biased towards relatively high
stellar mass (i.e. $\geq 10^{9}M_{\odot}$).

For comparison, in Fig. 17 we also show the distribution of errors with
respect to sSFR for the $I_{\scriptscriptstyle{\rm
E}}-Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm
E}}-H_{\scriptscriptstyle{\rm E}}$ (left column) and
$u-I_{\scriptscriptstyle{\rm E}}$, $I_{\scriptscriptstyle{\rm
E}}-J_{\scriptscriptstyle{\rm E}}$ (middle column) colour-colour methods
developed by B20. To allow a relevant comparison between the
$u-I_{\scriptscriptstyle{\rm E}}$, $I_{\scriptscriptstyle{\rm
E}}-J_{\scriptscriptstyle{\rm E}}$ colour-colour method and our machine-
learning pipeline, Fig. 17 (right column) also shows results from applying our
pipeline with conditions equivalent to those used for the
$u-I_{\scriptscriptstyle{\rm E}}$, $I_{\scriptscriptstyle{\rm
E}}-J_{\scriptscriptstyle{\rm E}}$ method: (i) Galaxy redshifts are included
as a feature to be trained on; (ii) only galaxies detected in $u$,
$I_{\scriptscriptstyle{\rm E}}$, and $J_{\scriptscriptstyle{\rm E}}$ are used;
(iii) only galaxies in the redshift ranges $0<z<1$ and $0<z<1.5$ are used for
the Wide and Deep catalogues, respectively.

Our classification pipeline results in significantly lower $\bar{I}_{\rm FP}$
than the colour-colour methods. This is due to a reduction in the fraction of
false positives that are located at high sSFR (e.g. $\ga 10^{-9}{\rm
yr}^{-1}$). In other words, our pipeline not only offers a significant
improvement over colour-colour methods, in terms of $P$, $R$, and the
F1-score, but also significantly reduces the degeneracy between quiescent
galaxies and dusty, star-forming galaxies.

There is another potential source of classification errors that should be
mentioned. Throughout this work we have tacitly assumed that the target
variable accurately represents the ground truth. This assumption is clearly
valid for the SED catalogues, since they are derived from templates
corresponding to known physical properties. However, for the Int catalogues
there is the possibility that some instances flagged as classification errors
are, in actual fact, instances where our pipeline provides the correct
classification and the target variable is incorrect.

###  B.3 Reconciling the Int and SED results

As discussed above, our machine-learning models were trained and evaluated
using one of the four mock catalogues, but there are often significant
differences between the results obtained using the Int or SED catalogues for a
given Euclid survey (e.g. Fig. 8). In particular, the precision, recall, and
F1-scores for quiescent galaxy selection tend to be higher when using an SED
mock catalogue, compared to its corresponding Int catalogue (i.e. SED Wide vs.
Int Wide; SED Deep vs. Int Deep).

This is due to the different methods used in the construction of the
catalogues (see Sect. 2 and B20). The Int catalogues have an observationally
more realistic starting point since they are constructed with real photometry,
albeit twice convolved with a filter, but the signal-to-noise ratio of the
data is significantly lower than will be the case for the actual Euclid
photometry, likely increasing the difficulty of the classification problem
compared to when the real Euclid (and ancillary) survey data are used.
Conversely, the mock photometry in the SED catalogues has noise properties
that match those expected for Euclid observations, but the SEDs themselves are
forced to conform to a restricted set of simplified templates, which could
potentially simplify the classification problem.

Therefore, results obtained using the Int catalogues are likely to be
pessimistic, and results obtained with the SED catalogues are likely to be
optimistic with regard to the performance of our pipeline when applied to real
Euclid (and ancillary) survey data. Thus, to estimate the performance of our
pipeline when selecting quiescent galaxies from Euclid (and ancillary) survey
data, we use performance metrics averaged over the Int catalogue and the
corresponding SED catalogue (i.e. Int Wide and SED Wide; Int Deep and SED
Deep). Therefore, Tables 3 and 4 include averaged metrics, where appropriate.
We consider results obtained with the Int and SED catalogues to bracket the
likely range of performance of our pipeline when it is applied to real Euclid
(plus LSST, etc.) photometry.

###  B.4 Tuning the probability threshold

In some circumstances, it is desirable to maximise either the precision
(purity) or the recall (completeness) of the selected quiescent galaxy
samples, according to the nature of one’s scientific objectives. Thus, we
investigate and illustrate the impact on the precision and recall of tuning
the value of the class probability threshold, instead of adopting the default
threshold value of 0.5.

In Fig. 18 we show how precision ($P$), recall ($R$), and the $F1$-score vary
with the class probability threshold when selecting quiescent galaxies from
the Int Wide mock catalogue (top) or the SED Wide catalogue (middle). Also
shown are the scores when averaged between the two catalogues (bottom). The
redshifts are included as a feature in the input data. There exists a trade-
off between precision and recall such that one may be increased, but at the
cost of reducing the other. For example, from the averaged scores (bottom
panel), we find that adopting a probability threshold of 0.85 yields a sample
of quiescent galaxies that is very pure ($P=0.98$) but somewhat incomplete
($R=0.56$). Conversely, using a probability threshold of 0.05 results in a
sample with moderate purity ($P=0.61$) but high completeness ($R=0.97$).

Tuning the probability threshold allows a balance to be struck between $P$ and
$R$ that is suitable for different scientific needs. For instance, using a
probability threshold of 0.3 gives a sample of quiescent galaxies that is both
reasonably pure ($P=0.8$) and reasonably complete ($R=0.9$). While the
examples given here pertain to selection in the redshift range $0\leq z\leq
3$, this exercise can, of course, also be performed for the selection of
quiescent galaxies in narrower redshift bands. As an example of this, when
selecting quiescent galaxies in the redshift range $1\leq z\leq 2$ using a
probability threshold of 0.9, we obtain $P=0.98$ and $R=0.51$, while using a
threshold of 0.1 yields $P=0.70$ and $R=0.95$.

###  B.5 Impact of including redshift as a feature

Here we investigate the impact of several different methods for the treatment
of redshift information in our pipeline. Thus far, we have included redshift
information by pre-binning the mock catalogues using the Laigle et al. (2016)
COSMOS2015 photometric redshifts (Sect. 5.1). Alternatively, we ignored
redshift information and instead performed a global selection of quiescent
galaxies, deriving photometric redshifts subsequently (Sect. 5.2). A further
possibility that we now also examine is the inclusion of redshifts as a
feature in the input data for model training (e.g. Simet et al., 2021).

In Fig. 19 we show F1-score versus redshift when selecting quiescent galaxies
from the Int Wide catalogue, considering several different configurations for
the included redshift information. As before, we exclude galaxies that have a
non-detection in one or more Euclid bands. The left panel of Fig. 19 shows
results from our pipeline when the mock catalogue data are pre-binned by
redshift as described in Sect. 5.1, and all available photometry is used (i.e.
$ugriz$, Euclid, $W1$, $W2$, 20 cm). We find that including the Laigle et al.
(2016) photometric redshifts in the input data significantly increases the
F1-scores in bins at $z\leq 0.5$ or $z\geq 2.5$, by reducing the degeneracy
between redshift and sSFR. However, the F1-score is not significantly changed
for bins in the range $0.5<z<2.5$.

We also experimented with the inclusion of redshifts for a fraction of
galaxies only, by randomly replacing redshift values with $-99.9$. The left
panel of Fig. 19 also shows the result of including only 50 per cent of the
redshifts. At the low end of the redshift range, we find that the inclusion of
50 per cent of the redshifts provides a small but significant improvement in
the F1-score. However, at the high endpoint of the redshift range, there is no
noticeable improvement compared to the F1-scores obtained when no redshifts
are included.

We repeated the same experiment, but without pre-binning the galaxies by
redshift (see Sect. 5.1.3). As before, we excluded objects with a non-
detection in any of the Euclid bands and used all available photometry from
the Int Wide catalogue. The results are shown in the right panel of Fig. 19.
Within the $0<z<2$ range, there is essentially no penalty in terms of F1-score
associated with not pre-binning by redshift, provided all redshifts are
included as a feature in the input data (nevertheless, pre-binning does allow
considerably faster training of models, since the training set is now much
smaller). However, at $z=2$ and above, significantly lower F1-scores are
obtained compared to the case where the data are pre-binned by redshift. When
only 50 per cent of redshift values are included, we find a significant
decrease in F1-scores compared to the case where 100 per cent of redshift
values are included, but the F1-score are still usually significantly above
those obtained when none of the redshift values are included. Classification
metrics for the global selection of quiescent galaxies, including 100 per
cent, 50 per cent, or none of the galaxy redshifts are included in Table 4.

Figure 21: Results from our transfer learning method. When we train models on
SED Wide data, and use them to select quiescent galaxies from Int Wide data
(dashed orange curve), the F1-scores are only slightly lower than those we
obtain from models trained on Int Wide data (solid blue curve). A generally
similar result is obtained when we train models on Int Wide data and use them
to select quiescent galaxies from SED Wide data (dotted black curve). For
comparison we also show the results from the LePhare template fitting method
applied to the SED Wide mock catalogue (see Sect. 6.4), using Euclid, $ugriz$,
$W1$, $W2$, and 20 cm photometry, and with redshifts fixed at the Laigle et
al. (2016) values.

###  B.6 The impact of noise

In this subsection we explore the impact of different types of noise that are
expected to be present within the data. The experiments presented here are
intended to be informative and illustrative, but not necessarily exhaustive.

####  B.6.1 Adding noise to the photometry

In Fig. 20 we show results from modelling the impact of the addition or the
reduction of noise in the data. For this experiment, we select quiescent
galaxies at $0\leq z\leq 3$ from the Int Wide mock catalogue, using our
pipeline in fast mode. The upper panel of Fig. 20 shows how the F1-score
decreases when each magnitude measurement has a random offset, drawn from a
Gaussian distribution of $\sigma$, applied. Interestingly, even when the data
are extremely noisy, the pipeline remains nominally functional. For instance,
even when the photometry has been degraded to a signal-to-noise ratio of $\sim
3$, it is nonetheless still able to perform a global selection of quiescent
galaxies, albeit with somewhat reduced precision, recall, and F1-scores of
$\sim 0.67$.

To simulate a reduction in noise, we perform a cut to remove galaxies fainter
than an arbitrary $I_{\scriptscriptstyle{\rm E}}$ threshold, where lower
values of this threshold result in a higher average signal-to-noise ratio for
the mock catalogue (Fig. 20, lower panel). Despite the crudeness of these
tests, it is clear that reducing (increasing) the signal-to-noise ratio of the
data results in lower (higher) quality classification models, as evaluated by
the F1-scores. While the different noise characteristics probably play a
significant role in the differences in pipeline performance between the Int
and SED catalogues, additional effects may also be important.

####  B.6.2 Label noise

Heretofore, we have tacitly assumed that the quiescent and star-forming labels
used in the training and evaluation of our classification models give an
accurate representation of the ground truth. However, there is the possibility
that some labels are incorrect, that is, quiescent galaxies labelled as star-
forming or vice versa. We examine the potential impact of incorrect labels
using two slightly different approaches.

In the first approach, we assess the impact on model quality from introducing
incorrect labels at random. To do this, we select a subset of galaxies in the
mock photometry catalogue, and for these galaxies we replace all occurrences
of the value 0 with the value 1, and all occurrences of the value 1 with 0. We
then perform our standard preprocessing and model training steps as described
in Sect. 4. For these tests, the ARIADNE pipeline is used in fast mode. The
classification metrics are evaluated for two different cases, where (i) the
test set class predictions are compared with the original unaltered test set
labels, or (ii) the test set class predictions are compared with the altered
(noisy) test set labels.

In the context of this experiment we point out that the ‘random errors’
discussed here can either be truly random errors arising from the methodology
used to generate the labels (e.g. LePhare template fitting) or else can be
systematic errors that the machine-learning algorithms are unable to model and
reproduce. An example of the latter type of error might be a systematic error
driven by photometric bands that are present in the multi-wavelength dataset
used to generate the labels, but which do not appear in the data that are seen
by our pipeline.

Under these conditions, the presence of noisy labels varies depending on the
particular classification problem that is being addressed. For instance, when
selecting quiescent galaxies at $0\leq z\leq 3$ from the Int Wide catalogue,
without foreknowledge of redshifts, and with 33 per cent of the labels having
been flipped, the metrics obtained when the test set labels contain errors are
relatively poor, at $P=0.62$, $R=0.12$, and ${\rm F1{\text{-}}score}=0.21$.
This is to be expected, because the classification results are being evaluated
against a ground truth that contains many incorrect labels, leading to
artificially poor metrics.

Conversely, the metrics we obtain are substantially better if we instead
evaluate the same classification results against the original ground truth,
yielding $P=0.86$, $R=0.68$, and ${\rm F1{\text{-}}score}=0.76$; these values
are only slightly different compared to the case where label errors are not
introduced at all ($P=0.84$, $R=74$, ${\rm F1{\text{-}}score}=0.79$). In other
words, for this case the presence of randomly incorrect labels in the training
data did not have a substantial impact on the classification of sources in the
test set.

In another example, selecting quiescent galaxies at $1\leq z\leq 1.25$ with no
foreknowledge of redshifts, and with 10 per cent of labels in error, results
in an F1-score of 0.13 when evaluated using the test set ground truth with
label errors injected. Remarkably, when evaluating the same classification
results using the ground truth test set without label errors, the F1-score is
0.69, effectively unchanged from the case where no label errors are injected
at all (${\rm F1{\text{-}}score}=0.69$). In this case, the presence of random
errors in the training set labels had no detrimental effect on the final
classifications of sources in the test set.

One of the interesting implications of these results is that our pipeline is
generally able to ignore random label errors. Furthermore, when random label
errors are present, our pipeline may even produce predictions that are of
higher quality than the labels in the input data, in terms of how close the
predictions are to the real ground truth. Further research (beyond the scope
of this paper) is needed to test whether machine-learning methods such as
those discussed herein may be able to improve on traditional SED fitting
methods, rather than simply emulating them.

The second approach is similar to the first, but aims to simulate systematic
errors in the class labels. For this, we used the KMeans clustering algorithm
from Scikit-Learn to separate galaxies from the Int Wide catalogue into an
arbitrary number of clusters, with an arbitrary number of the clusters being
selected to have an arbitrary fraction of their class labels inverted. For
illustrative purposes, we separated the data into 100 clusters, and inverted
99 per cent of the labels in three randomly selected clusters; the inverted
labels represent 11 per cent of all labels. We then selected quiescent
galaxies at $0\leq z\leq 3$, again using features derived from the $ugriz$,
Euclid, $W$1, $W$2, and 20 cm bands, and without foreknowledge of galaxy
redshifts. Evaluating the metrics when using the modified labels for the test
set, we obtain $P=0.89$, $R=85$, and an F1-score of 0.87. In contrast,
evaluating the metrics using the original test set labels, we obtain the
substantially lower values $P=0.36$, $R=0.69$, and an F1-score of 0.47. In
this case, our pipeline is able to emulate the systematic label errors present
in the training set, resulting in test set class predictions that contain
similar systematic errors.

####  B.6.3 Redshift noise

As we have demonstrated heretofore, the inclusion of redshift information in
the training data often results in classification models that are better able
to correctly identify quiescent galaxies (e.g. Appendix B.5). A key point to
be considered is whether our results are significantly affected by the
accuracy of the photometric redshifts used, especially since the 30-band
COSMOS2015 redshifts (Laigle et al., 2016) we have used could potentially be
more accurate than the redshifts that will be estimated from Euclid photometry
and the anticipated ancillary data (see e.g. Euclid Collaboration: Desprez et
al., 2020). Therefore, we explore the impact of adding Gaussian noise to the
COSMOS2015 redshifts prior to model training. While a full treatment of this
issue is beyond the scope of the present work, we consider several different
cases in order to obtain indicative results.

Random samples were drawn from a Gaussian distribution with standard deviation
$\sigma_{z}$; these values were multiplied by $1+z$ and then added to the
photometric redshift values after the Target variable was set, simulating the
addition of Gaussian noise. The global selection of quiescent galaxies in the
range $0<z<3$ was then repeated with the (now noisier) redshifts included as a
feature, along with the $ugriz$, Euclid, and Wise photometry, and colours
derived therefrom. This test was performed for the Int Wide and SED Wide
catalogues, for the values $\sigma_{z}=0.025,0.05,0.075$.

The results from this test, averaged over an equal number of pipeline runs on
the Int Wide and SED Wide catalogues, are shown in Table 4. While $P$ is
essentially unchanged, $R$, and the F1-score are slightly reduced by $\sim
0.01$–0.02.

Next, we consider the impact of redshift noise on the selection of quiescent
galaxies in the narrower redshift bins used in Sect. 5.1. As one might expect,
it is more difficult to select quiescent galaxies inside these narrower
redshift bins when the photometric redshifts are noisier. This is because the
models, in addition to separating quiescent and star-forming galaxies, must
now also separate quiescent galaxies by their redshift. For example, in the
case where $\sigma_{z}=0.05$, the F1-score is typically reduced by $\sim 0.07$
compared to the case where no noise is added. Nevertheless, the F1-scores are
still significantly higher than when the redshifts are not included at all,
with the improvement ranging from $\sim 0.02$ at $1<z<1.5$, to $\sim 0.2$ at
$z<0.5$ and $z>2$. Interestingly, even in the case where $\sigma_{z}=0.1$,
representing rather noisy redshifts (NMAD $\sim 0.1$), the F1-scores at
$z<0.5$ and $z>2$ are still $\sim 0.1$ higher than when redshifts are not
included. In other words, even when photometric redshifts are somewhat noisy,
their inclusion in the data can nevertheless result in significantly stronger
classification models, compared to when photometric redshifts are not used.

###  B.7 Transfer learning experiments

####  B.7.1 Training on templates and predicting on real SEDs

We also experimented with the possibility of using classification models
trained on spectral templates to select quiescent galaxies from catalogues of
observed photometry. To explore this, our pipeline trained its classification
models on the SED Wide mock catalogue, and selected quiescent galaxies from
the Int Wide catalogue using the resulting classifier. Essentially, we gave
our pipeline the task of identifying and weighting the defining
characteristics of quiescent and star-forming galaxies from a set of
simplifying abstractions (galaxy SED templates), rather than from observed
SEDs. The train-test split was performed as described in Sect. 4.1, ensuring
that each galaxy is present in either the training set or the test set, but
not both. For this experiment, the data are pre-binned by redshift as
described in Sect. 5.1, but we did not include the redshift values as a
feature in the input data for the classification pipeline.

The results are shown in Fig. 21, where it can be seen that classifiers
trained on the SED Wide catalogue are indeed able to select quiescent galaxies
from the Int Wide catalogue, albeit with marginally lower F1-scores compared
to models trained on the Int Wide catalogue itself. This opens up the
interesting possibility of using machine-learning models trained on synthetic
galaxy SEDs as a potential alternative to traditional colour-colour or
template fitting methods (e.g. Girelli, Bolzonella, & Cimatti, 2019; Cecchi et
al., 2019) of selecting quiescent galaxies that have redshifts (or other
properties) for which there are no (or few) known examples.

Figure 21 also reveals that the transfer learning method can also function in
reverse. When the training and test datasets are swapped with each other, such
that classification models are trained on the Int Wide catalogue and are then
used to select quiescent galaxies from the SED Wide catalogue, very similar
results are obtained. The exception is in the lowest redshift bin
($0<z<0.25$), where the F1-score is now reduced by $\sim 0.12$ compared to the
previous case. Classification metrics for the global selection of quiescent
galaxies using transfer learning are also included in Table 4.

We also show in Fig. 21 (red dashed line) the F1-scores obtained when using
LePhare to fit templates to SED Wide mock data using the same photometry, and
with redshifts fixed to the values from Laigle et al. (2016, see also Sect.
6.4). This template fitting method clearly provides results of similar quality
to our pipeline in the $1.0\la z\la 2.0$ redshift range, but at $z\la 1.0$ or
at $z\ga 2$ the method significantly under-performs our transfer learning
method. The under-performance of our LePhare template fitting method is likely
caused, at least partly, by the absence of priors concerning (i) the known (or
suspected) distribution of the galaxy classes within the redshift and colour-
spaces, and (ii) the relative importance (or weighting) that should be given
to each data point in the broadband SED, aside from their signal-to-noise
ratio.

####  B.7.2 Training on Deep and predicting on Wide survey data

The Euclid Deep Survey will provide photometry (and spectra) in several fields
for which there are pre-existing multi-wavelength observations, allowing the
construction of source catalogues with high-quality labels. In turn, this is
expected to facilitate the training of classifiers, which can then be used to
predict labels for the enormous number of sources that will be detected in the
Euclid Wide Survey. An important question, however, is whether models trained
using the deep field photometry are suitable for use in selecting quiescent
galaxies in the wide field survey. While the Deep and Wide Surveys are
expected to be somewhat similar (but clearly not identical) within the
magnitude-space covered by the Wide Survey, at least 60 per cent of the
galaxies in the Deep Survey are below (or close to) the
$I_{\scriptscriptstyle{\rm E}}$ $3\sigma$ detection threshold of the Wide
Survey. It is not clear a priori how the presence of these faint galaxies will
affect the quality of the classification models.

To test this, we train our pipeline using the Int Deep catalogue, and use the
resulting classification model to predict classes for the galaxies in the Int
Wide catalogue. No galaxy was permitted to be present in both the training and
the test set. In addition, no foreknowledge of redshifts was assumed, all
available photometry bands were used ($ugriz$, Euclid, $W$1, $W$2, and 20 cm),
and only galaxies detected in all four Euclid bands were included. Under these
conditions, the resulting F1-score for selection of quiescent galaxies from
the Int Wide catalogue is 0.74, moderately lower than the F1-score of 0.80
obtained using models trained using Int Wide (see Table 4). Conducting this
test instead using the SED catalogues resulted in a similarly reduced F1-score
of 0.76, compared to 0.82 when using only SED Wide catalogue.

Thus, although it is possible to separate quiescent and star-forming galaxies
in the Wide Survey mocks, the F1-score suffers a significant penalty and is
reduced by at least $\sim 0.06$. Clearly, the presence of a large number of
additional, faint galaxies in the training set exacerbates at least some of
the degeneracies described in Sect. 1, and induces the learning algorithms to
place undue weight on galaxies near the faint end of the magnitude
distribution.

In this study we are, of course, only able to make use of mock Euclid
photometry catalogues, but we must also consider how the training set for this
task will be constructed from real Euclid data. Based on the analyses
presented heretofore, we propose constructing the training set(s) from Deep
Survey photometry from fields that have high-quality multi-wavelength
observations (and spectroscopy), using the deepest available data in order to
generate high-quality ground truth labels, but then training classification
models on Wide Survey-depth observations.

###  B.8 Alternative targets

In this work, we have used target labels that were generated using an sSFR
threshold to differentiate between quiescent and star forming galaxies.
However, it is also interesting to consider other methods for generating the
labels. Thus, here we explore the impact of using labels derived from the
${\rm NUV}-r$ versus $r-J_{\scriptscriptstyle{\rm E}}$ (Ilbert et al., 2010,
2013) or ${\rm NUV}-r$ versus $r-K$ (Arnouts et al., 2013) colour-colour
methods, instead of the sSFR-based labels used throughout this work. The mock
catalogue used for these tests is Int Wide, and features derived from the
$ugriz$, $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$,
$J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$, $W$1, $W$2
and 20 cm bands were used, with detections required in all of the Euclid
bands. Our pipeline was used in fast mode. The results are summarised in Table
5.

In the global selection of quiescent galaxies ($0\leq z\leq 3$), the impact of
using labels the alternative labels is limited. In the case of the labels
derived from the ${\rm NUV}-r$ versus $r-J_{\scriptscriptstyle{\rm E}}$
method, the F1-score is improved by $\sim$0.03 compared to the original (sSFR)
labels. On the other hand, the F1-score when using the labels derived from the
${\rm NUV}-r$ versus $r-K$ method is $\sim$0.02 lower compared to when the
original labels are used.

When selected quiescent galaxies in narrower redshift ranges, having first
removed sources whose photometric redshifts place them outside the range of
interest, we usually obtain slightly higher F1-scores when using labels from
the ${\rm NUV}-r$ versus $r-J_{\scriptscriptstyle{\rm E}}$ method, compared to
when the original labels are used. in the $z=2.5-3$ bin, there is a
particularly large improvement in the F1-score (from $\sim$0.4 to $\sim$0.6).
Conversely, in the $z=0-0.25$ bin we obtain a significantly lower F1-score
when using when using labels from the ${\rm NUV}-r$ versus
$r-J_{\scriptscriptstyle{\rm E}}$ method (0.53 compared to 0.69). When labels
derived from the ${\rm NUV}-r$ versus $r-K$ method are used, the resulting
F1-scores are consistently lower compared to the case where the original
labels (sSFR) are used.

Table 5: Selection of quiescent galaxies in the redshift range $0\geq z\geq
3$, using the Int Wide catalogue with its labels replaced by labels derived
from the ${\rm NUV}-r$ vs. $r-J_{\scriptscriptstyle{\rm E}}$ or ${\rm NUV}-r$
vs. $r-K$ colour-colour methods.

|  | $\overbrace{\rule{100.00015pt}{0.0pt}}^{\text{\small${\rm NUV}-r$ vs. $r-J_{\scriptscriptstyle{\rm E}}$ labels}}$ | $\overbrace{\rule{100.00015pt}{0.0pt}}^{\text{\small${\rm NUV}-r$ vs. $r-K$ labels}}$ | $\overbrace{\rule{100.00015pt}{0.0pt}}^{\text{\small sSFR labels}}$
---|---|---|---|---
Redshift range | Redshift binning | $P$ | $R$ | F1-score | $P$ | $R$ | F1-score | $P$ | $R$ | F1-score
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11)
$0-3$ | No | 0.86 | 0.79 | 0.82 | 0.79 | 0.74 | 0.77 | 0.84 | 0.74 | 0.79
$0-0.25$ | Yes | 0.51 | 0.56 | 0.53 | 0.41 | 0.46 | 0.43 | 0.72 | 0.67 | 0.69
$0.25-0.5$ | Yes | 0.82 | 0.81 | 0.82 | 0.78 | 0.74 | 0.76 | 0.81 | 0.80 | 0.80
$0.5-0.75$ | Yes | 0.86 | 0.85 | 0.86 | 0.84 | 0.81 | 0.83 | 0.86 | 0.85 | 0.85
$0.75-1$ | Yes | 0.90 | 0.90 | 0.90 | 0.80 | 0.85 | 0.82 | 0.85 | 0.87 | 0.86
$1-1.25$ | Yes | 0.90 | 0.90 | 0.90 | 0.79 | 0.86 | 0.82 | 0.88 | 0.88 | 0.88
$1.25-1.5$ | Yes | 0.81 | 0.85 | 0.83 | 0.69 | 0.80 | 0.74 | 0.82 | 0.84 | 0.83
$1.5-2$ | Yes | 0.77 | 0.79 | 0.78 | 0.61 | 0.72 | 0.66 | 0.80 | 0.82 | 0.81
$2-2.5$ | Yes | 0.70 | 0.69 | 0.69 | – | – | – | 0.62 | 0.69 | 0.65
$2.5-3$ | Yes | 0.62 | 0.65 | 0.62 | – | – | – | 0.40 | 0.57 | 0.37

151515For this test the $ugriz$, $I_{\scriptscriptstyle{\rm E}}$,
$Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$,
$H_{\scriptscriptstyle{\rm E}}$, $W$1, $W$2 and 20 cm bands were used, with
detections required in all of the Euclid bands. Here, the ARIADNE pipeline was
used in its fast mode. No results are shown for the ${\rm NUV}-r$ vs. $r-K$
case in the redshift ranges $2-2.5$ or $2.5-3$, due to the very low number of
sources labelled as quiescent by this method.
The columns are as follows:
(1) Redshift range in which the test was conducted;
(2) information on whether photometric redshifts were used to restrict the
dataset to source in the specified redshift range (‘Yes’ or ‘No’);
(3) precision $P$ for quiescent galaxy selection when using labels derived
from the ${\rm NUV}-r$ vs. $r-J_{\scriptscriptstyle{\rm E}}$ method;
(4) recall $R$ for quiescent galaxy selection when using labels derived from
the ${\rm NUV}-r$ vs. $r-J_{\scriptscriptstyle{\rm E}}$ method;
(5) the F1-score for quiescent galaxy selection when using labels derived from
the ${\rm NUV}-r$ vs. $r-J_{\scriptscriptstyle{\rm E}}$ method;
(6) precision $P$ for quiescent galaxy selection when using labels derived
from the ${\rm NUV}-r$ vs. $r-K$ method;
(7) recall $R$ for quiescent galaxy selection when using labels derived from
the ${\rm NUV}-r$ vs. $r-K$ method;
(8) the F1-score for quiescent galaxy selection when using labels derived from
the ${\rm NUV}-r$ vs. $r-K$ method;
(9) precision $P$ for quiescent galaxy selection when using labels derived
from the sSFR;
(10) recall $R$ for quiescent galaxy selection when using labels derived from
the sSFR;
(11) the F1-score for quiescent galaxy selection when using labels derived
from the sSFR.

###  B.9 Which observables are useful to select quiescent galaxies?

Figure 22: Visual representation of the feature importances derived from
XGBoostClassifier models trained to select quiescent galaxies using only
Euclid photometry and colours (top left panel), or Euclid photometry with the
addition of one ground-based optical band, and the relevant broadband colours.
The optical bands are $u$ (top right), $g$ (middle left), $r$ (middle right),
$i$ (bottom left), and $z$ (bottom right). The XGBoostClassifier models were
trained without foreknowledge of the redshifts. Figure 23: Effect on the
F1-score due to the including one additional band with the Euclid photometry
set. The ARIADNE pipeline was run in fast mode using the LightGBMClassifier
base learner, or XGBoostClassifier at $2.5<z<3$, without foreknowledge of
galaxy redshifts.

Figure 24: Similar to Fig. 22, but showing the feature importances when the
$W$1 (top), $W$2 (middle) or 20 cm (bottom) bands are included with the Euclid
photometry.

Heretofore, we have approached the question of the usefulness of different
features (colours, magnitudes, etc.) in terms of whether they are informative
and whether their inclusion significantly improves the quality of our
classification models (see Sect. 4.2.4). However, it is also desirable to
reach a deeper understanding of the usefulness of each feature, and how their
usefulness depends on the circumstances under which quiescent galaxies are to
be selected. For instance, the colours that are most useful for selecting
quiescent galaxies in one redshift range may be less useful in another.

Moreover, while it may be self-evident that the inclusion of photometry in
various optical bands allows for better characterisation of galaxy SEDs, and
thus more accurate classification of galaxies, the degree of improvement may
not always justify the cost of acquiring additional observations. Figure 8
illustrates such an example, where the inclusion of $ugriz$ photometry
provides little or no significant improvement in the selection of quiescent
galaxies at $z\ga 1.25$, compared to when only the Euclid photometry is used
(see Sect. 5.1.3 for further details).

Our objective for this section, therefore, is to provide analyses that can
inform the planning of future surveys regarding which filters or frequencies
would be important to include, and to guide the construction of new selection
methods, be they colour-colour, template- fitting, or machine-learning-based.
For this we employ feature importance analysis (see also Sect. 4.2.4). The
machine-learning models we use for these analyses are trained using the
XGBoostClassifier learning algorithm and the Int Wide mock catalogue, because
its feature importance values are more robust, as discussed in Appendix A. We
also perform split-run tests using our pipeline in fast mode to examine the
impact on the F1-score from the addition of ancillary bands to the Euclid
$I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$,
$J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$photometry.

It is worth noting that these analysis methods provide information about
machine-learning models and how they make use of the input data, rather than
the data itself. As such, the information provided is limited by what the
learning algorithms were able to learn from the data, and it is quite possible
that additional relationships exist between features and the target that the
learning algorithm was unable to find. Thus, while features found to be
important are highly likely to be useful for the selection of quiescent
galaxies, features found to be unimportant (or less important) may
nevertheless hold hidden information that other selection methods (e.g.
template fitting) may find useful when selecting quiescent galaxies.

####  B.9.1 Which single optical band is most useful?

In Fig. 22 we show the effect of adding one optical band to the Euclid
photometry set, when selecting quiescent galaxies in specific redshift ranges
from the Int Wide catalogue using XGBoostClassifier, with no foreknowledge of
redshifts. In addition, Fig. 23 shows the improvement in F1-score due to the
inclusion of each optical band.

First, we examine feature importance as a function of redshift when only the
Euclid data are available (Fig. 22, top left). In this circumstance, there is
no single, decisive broadband colour for the selection of quiescent galaxies
at $z\la 0.5$. The lack of sensitivity to the 4000 Å break, or to emission
bluewards thereof, results in models with relatively low F1-scores (see Fig.
8). In this regime, our XGBoostClassifier models assign a fairly similar
importance value to each feature.

Above $z=0.4$, the 4000 Å break becomes redshifted into the
$I_{\scriptscriptstyle{\rm E}}$ band, becoming potentially detectable via one
or more of the Euclid broadband colours. In the range $0.5<z<1.5$, the colour
$I_{\scriptscriptstyle{\rm E}}-Y_{\scriptscriptstyle{\rm E}}$ is the most
sensitive to the presence of the 4000 Å break, and our XGBoostClassifier
models consider this feature to be the single most important for these
redshifts. At $z>1.5$, where the 4000 Å break is now redshifted into the NIR
bands, other colours become more important: $I_{\scriptscriptstyle{\rm
E}}-J_{\scriptscriptstyle{\rm E}}$ at $1.5<z<2$, $Y_{\scriptscriptstyle{\rm
E}}-H_{\scriptscriptstyle{\rm E}}$ at $2<z<2.5$, and
$J_{\scriptscriptstyle{\rm E}}-H_{\scriptscriptstyle{\rm E}}$ at $2.5<z<3$.

The addition of $u$-band photometry allows the 4000 Å break to be detected at
$z\la 0.5$, resulting in significantly increased F1-scores in this redshift
range (see Fig. 23). In this case, $I_{\scriptscriptstyle{\rm E}}-u$ and/or
$Y_{\scriptscriptstyle{\rm E}}-u$ become the most important in the redshift
range $0<z<0.75$ by a large margin. Features using the $u$-band photometry
continue to have significant (or non-zero) importance up to $z\sim 2$, before
dropping to $\sim 0$ at $z>2.5$.

In the case where models are trained using features derived from the $g$-band
and Euclid photometry, colours involving $g$ are found to be the most
important within the range $0.25<z<1$. While the inclusion of $g$ also helps
significantly at $z<0.25$, the improvement is considerably smaller compared to
when the $u$ is used, because the 4000 Å break is bluewards of the $u$ band’s
wavelength range. While the importance values of features using the $g$ band
decrease substantially at $z>1.25$, these features remain useful even in the
$2.5<z<3$ bin.

When the $r$, $i$, or $z$ band is used together with the Euclid bands, the
situation is similar to that described above for $g$, with the main
differences being the substantially lower feature importances at $0.25<z<0.5$
for colours that use one of the three bands, and the relatively high
importance of $I_{\scriptscriptstyle{\rm E}}-z$ at $1<z<1.25$.

In summary, there is no single ‘ideal’ optical band to include alongside the
Euclid bands when selecting quiescent galaxies, and a trade-off should be made
depending on whether low-redshift ($z\la 0.25$) or high-redshift ($z\ga 2$)
galaxies are required. Of course, one may circumvent this choice if $ugriz$
photometry is available.

Figure 25: Similar to Fig. 23, but showing the effect on the F1-score due to
the including $W$1, $W$2, or the 20 cm radio band with the Euclid photometry
when training LightGBMClassifier ($z<2.5$) or XGBoostClassifier ($2.5<z<3$)
models.

####  B.9.2 Importance of long wavelength bands

We repeat the process described in Appendix B.9.1, this time considering the
addition of the $W$1, $W$2, or 20 cm radio band and related broadband colours.
The data for these three bands is very sparse, with detection fractions of
0.053, 0.024, and 0.0076 for $W$1, $W$2, and 20 cm, respectively.

The feature importance as a function of redshift is shown in Fig. 24, and the
improvement from including the $W$1, $W$2, or 20 cm bands with the Euclid
photometry is shown in Fig. 25. Despite being very sparse, each of the three
bands provide a significant improvement in F1-score within the redshift ranges
$0<z<1$ and $1.5<z<3$. We find little or no improvement evident in the
redshift range $1<z<1.5$.

The usefulness of the 20 cm band for discriminating quiescent galaxies from
star-forming galaxies is intriguing and somewhat surprising. Clearly, there is
a correlation (or correlations) between the galaxy class and the presence of
radio continuum emission. We speculate that this may be due to a subset of
star-forming galaxies being detected in radio continuum, and/or the presence
of radio-loud massive elliptical galaxies whose X-ray emission is below the
COSMOS2015 detection threshold.

###  B.10 Selection of quiescent galaxies from SPRITZ: Euclid Deep Survey

To complement the results presented in Sect. 5.1.1, we also tested the
selection of quiescent galaxies using the simulated Euclid Deep Survey from
SPRITZ (Bisigello et al., 2021). Compared to the Euclid Deep Survey
catalogues, the SPRITZ catalogue has the advantage of being complete down to
the expected depth for the actual survey Deep Survey and the expected
ancillary ground-based data. For the Euclid and $ugriz$ bands, we adopt
identical photometric uncertainties and detection limits to those used for the
SED Deep catalogue given in Sect. 2. In the case of Spitzer Space Telescope
IRAC bands, we adopt the following four cases based on Euclid Collaboration:
Moneti et al. (2022): Case 1: no IRAC photometry; Case 2: IRAC photometry
$3\,\sigma$ depths of 24.55, 24.39, 22.61, and 22.17 mag in channel 1, 2, 3
and 4, respectively; Case 3: IRAC photometry $3\,\sigma$ depths of 25.55,
25.39, 23.61, and 23.17 mag in channel 1, 2, 3 and 4, respectively; and Case
4: IRAC photometry $3\,\sigma$ depths of 26.55, 26.39, 23.61, and 23.17 mag in
channel 1, 2, 3 and 4, respectively.

The ARIADNE pipeline was used in its default configuration, where five base-
learners are employed. Galaxies not detected in one or more of the Euclid
bands were removed from the dataset, as were galaxies containing an active
galactic nucleus. The results are shown in Table 6 and Fig. 26.

Table 6: Selection of quiescent galaxies in the redshift range $0\geq z\geq
3$, from the SPRITZ Euclid Deep Survey simulation.

|  | $\overbrace{\rule{100.00015pt}{0.0pt}}^{\text{\small SPRITZ Case 1}}$ | $\overbrace{\rule{100.00015pt}{0.0pt}}^{\text{\small SPRITZ Case 2}}$ | $\overbrace{\rule{100.00015pt}{0.0pt}}^{\text{\small SPRITZ Case 3}}$ | $\overbrace{\rule{100.00015pt}{0.0pt}}^{\text{\small SPRITZ Case 4}}$
---|---|---|---|---|---
Redshift range | Redshift binning | $P$ | $R$ | F1-score | $P$ | $R$ | F1-score | $P$ | $R$ | F1-score | $P$ | $R$ | F1-score
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) | (12) | (13) | (14)
$0-3$ | No | 0.92 | 0.86 | 0.89 | 0.92 | 0.88 | 0.90 | 0.93 | 0.90 | 0.91 | 0.93 | 0.91 | 0.92
$0-0.25$ | Yes | 0.84 | 0.85 | 0.85 | 0.85 | 0.85 | 0.85 | 0.85 | 0.85 | 0.85 | 0.85 | 0.86 | 0.86
$0.25-0.5$ | Yes | 0.92 | 0.90 | 0.91 | 0.92 | 0.91 | 0.91 | 0.92 | 0.91 | 0.91 | 0.92 | 0.91 | 0.92
$0.5-0.75$ | Yes | 0.96 | 0.92 | 0.94 | 0.96 | 0.92 | 0.94 | 0.96 | 0.93 | 0.94 | 0.96 | 0.93 | 0.95
$0.75-1$ | Yes | 0.93 | 0.91 | 0.92 | 0.94 | 0.91 | 0.93 | 0.94 | 0.92 | 0.93 | 0.95 | 0.92 | 0.94
$1-1.25$ | Yes | 0.95 | 0.92 | 0.93 | 0.96 | 0.92 | 0.94 | 0.97 | 0.93 | 0.95 | 0.97 | 0.94 | 0.96
$1.25-1.5$ | Yes | 0.93 | 0.95 | 0.94 | 0.96 | 0.95 | 0.95 | 0.97 | 0.96 | 0.97 | 0.98 | 0.97 | 0.97
$1.5-2$ | Yes | 0.90 | 0.93 | 0.91 | 0.94 | 0.94 | 0.94 | 0.95 | 0.95 | 0.95 | 0.96 | 0.96 | 0.96
$2-2.5$ | Yes | 0.91 | 0.84 | 0.87 | 0.93 | 0.89 | 0.91 | 0.95 | 0.91 | 0.93 | 0.95 | 0.93 | 0.94
$2.5-3$ | Yes | 0.82 | 0.83 | 0.83 | 0.86 | 0.84 | 0.85 | 0.89 | 0.89 | 0.89 | 0.91 | 0.89 | 0.90

161616The columns are as follows:
(1) redshift range in which the test was conducted;
(2) information on whether photometric redshifts were used to restrict the
dataset to source in the specified redshift range (‘Yes’ or ‘No’);
(3) precision $P$ for quiescent galaxy selection for SPRITZ Case 1;
(4) recall $R$ for quiescent galaxy selection for SPRITZ Case 1;
(5) the F1-score for quiescent galaxy selection for SPRITZ Case 1;
(6) precision $P$ for quiescent galaxy selection for SPRITZ Case 2;;
(7) recall $R$ for quiescent galaxy selection for SPRITZ Case 2;
(8) the F1-score for quiescent galaxy selection for SPRITZ Case 2;;
(9) precision $P$ for quiescent galaxy selection for SPRITZ Case 3;;
(10) recall $R$ for quiescent galaxy selection for SPRITZ Case 3;;
(11) the F1-score for quiescent galaxy selection for SPRITZ Case 3;;
(12) precision $P$ for quiescent galaxy selection for SPRITZ Case 4;;
(13) recall $R$ for quiescent galaxy selection for SPRITZ Case 4;;
(14) the F1-score for quiescent galaxy selection for SPRITZ Case 4.

Figure 26: Results from applying the ARIADNE pipeline to the selection of
quiescent galaxies from the SPRITZ Euclid Deep Survey simulated catalogue.
Four cases described in Appendix B.10 are shown.

##  Appendix C Number of detections in each catalogue and band

Table 7: Number of $3\,\sigma$ detections in each of the bands, for each mock
catalogue.

Catalogue | $I_{\scriptscriptstyle{\rm E}}$ | $Y_{\scriptscriptstyle{\rm E}}$ | $J_{\scriptscriptstyle{\rm E}}$ | $H_{\scriptscriptstyle{\rm E}}$ | $u$ | $g$ | $r$ | $i$ | $z$ |  $W$1 |  $W$2 | 20 cm | Quiescent
---|---|---|---|---|---|---|---|---|---|---|---|---|---
Int Wide | 315755 | 212019 | 231039 | 250077 | 140782 | 226514 | 198564 | 194912 | 204649 | 10476 | 4704 | 1536 | 21998 (7.0 %)
Int Deep | 517890 | 486394 | 491588 | 500299 | 499565 | 504416 | 490457 | 493018 | 499140 | 10476 | 4704 | 1698 | 30990 (6.0 %)
SED Wide | 3249101 | 2056800 | 2270138 | 2455887 | 1498518 | 2330338 | 2091921 | 2040916 | 2031115 | 131703 | 65904 | – | 213837 (6.6 %)
SED Deep | 5121526 | 4763050 | 4890667 | 4963038 | 3971472 | 4796448 | 4828112 | 4802416 | 4766807 | 134656 | 69493 | – | 303761 (5.9 %)

171717Also shown are the numbers of quiescent galaxies. The SED catalogues do
not contain 20 cm radio band.


# Some numbers and polynomials related to degenerate harmonic and degenerate
hyperharmonic numbers

Dae San Kim  Department of Mathematics, Sogang University, Seoul 121-742,
Republic of Korea<EMAIL_ADDRESS>and  Taekyun Kim  Department of
Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
<EMAIL_ADDRESS>
###### Abstract.

Recently, the degenerate harmonic and the degenerate hyperharmonic numbers are
introduced respectively as degenerate versions of the harmonic and the
hyperharmonic numbers. The aim of this paper is to introduce the degenerate
harmonic-Fubini polynomials and numbers related to the degenerate harmonic
numbers and to study their properties, explicit expressions and some
identities. In addition, as generalizations of those polynomials and numbers,
we also introduce the degenerate hyperharmonic-Fubini polynomials and numbers
related to the degenerate hyperharmonic numbers and derive similar results to
the degenerate harmonic-Fubini polynomials and numbers.

###### Key words and phrases:

degenerate harmonic-Fubini polynomials; degenerate hyperharmonic-Fubini
polynomials

###### 2010 Mathematics Subject Classification:

11B73; 11B83

* is corresponding author

##  1\. Introduction

It is remarkable that various degenerate versions of many special polynomials
and numbers have been studied in recent years with regained interest on them.
This exploration for degenerate versions was initiated by Carlitz’s work on
the degenerate Bernoulli and the degenerate Euler numbers (see [5]). These
investigations have been carried out by using such diverse tools as
combinatorial methods, generating functions, umbral calculus, $p$-adic
analysis, differential equations, probability theory, operator theory,
analytic number theory, and so on.

The aim of this paper is to introduce the degenerate harmonic-Fubini
polynomials (see (20)) and numbers related to the degenerate harmonic numbers
(see (15)) and to study some properties, explicit expressions and identities
for them. As generalizations of those polynomials and numbers, the degenerate
hyperharmonic-Fubini polynomials (see (27)) and numbers related to the
degenerate hyperharmonic numbers (see (16), (18)) are also investigated and
similar results to the degenerate harmonic-Fubini polynomials and numbers are
obtained.

The outline of this paper is as follows. In Section 1, we recall the
degenerate logarithms together with their properties and the degenerate
exponentials. We remind the reader of the degenerate Stirling numbers of the
first kind and those of the second kind. We recall the degenerate Fubini
polynomials and the generalized degenerate Fubini polynomials. We remind the
reader of the degenerate harmonic numbers and the degenerate hyperharmonic
numbers. Then we state a useful lemma giving functional equations for two
power series. Section 2 is the main result of this paper. We introduce the
degenerate harmonic-Fubini polynomials and numbers related to the degenerate
harmonic numbers. In Theorem 1, we express the degenerate harmonic-Fubini
polynomial a finite sum involving the degenerate Stirling numbers of the
second kind and the degenerate harmonic numbers. In Theorem 2, we find an
expression of the degenerate harmonic number as a finite sum involving the
degenerate harmonic-Fubini numbers and the degenerate Stirling numbers of the
first kind. Some generalized degenerate Fubini polynomial is represented in
terms of the degenerate Stirling numbers of the second kind in Theorem 3. The
degenerate harmonic-Fubini polynomial is expressed as an infinite sum
involving the degenerare harmonic numbers in Theorem 4. In Theorem 5, the
degenerate hyperharmonic-Fubini polynomial is expressed in terms of the
degenerate hyperharmonic numbers and the degenerate Stirling numbers of the
second kind, and also in terms of the degenerate harmonic numbers and the
degenerate Stirling numbers of the second kind. In Theorem 6, we express the
degenerate hyperharmonic-Fubini polynomial as an infinite sum involving the
degenerate hyperharmonic numbers. Explicit expressions for the degenerate
harmonic numbers are obtained in Theorem 7. In Theorem 8, a functional
equation is obtained for any power series $f(t)$ by applying Lemma 1 with
$g(t)=-\frac{1}{1-t}\log_{\lambda}(1-t)$. By applying this functional equation
to $f(x)=x^{k},\,(k\geq 1)$, we get an identity involving the degenerate
harmonic numbers, the degenerate harmonic-Fubini polynomials and some
generalized degenerate Fubini polynomials in Theorem 9. From Theorem 9, an
identity of similar nature is derived in Theorem 10. An expression involving
the degenerate harmonic-Fubini polynomials and some degenerate Fubini
polynomials is shown to be equal to a differential operator applied to $g(x)$,
for the aforementioned $g(t)$. Finally, explicit expressions for the
degenerate hyperharmonic numbers are founded in Theorem 12. For the rest of
this section, we recall the facts that are needed throughout this paper.

For any nonzero $\lambda\in\mathbb{R}$, the degenerate logarithms are defined
by

(1)
$\log_{\lambda}(1+t)=\sum_{k=1}^{\infty}\frac{(1)_{k,1/\lambda}\lambda^{k-1}}{k!}t^{k}=\frac{1}{\lambda}\big{(}(1+t)^{\lambda}-1\big{)},\quad(\mathrm{see}\
[4]),$

where

(2)
$(x)_{0,\lambda}=1,\quad(x)_{n,\lambda}=x(x-\lambda)(x-2\lambda)\cdots(x-(n-1)\lambda),\quad(n\geq
1).$

From (1), we note that

(3)
$\log_{\lambda}(AB)=A^{\lambda}\log_{\lambda}B+\log_{\lambda}A=B^{\lambda}\log_{\lambda}A+\log_{\lambda}B,$

and

$\log_{\lambda}\bigg{(}\frac{B}{A}\bigg{)}=\frac{1}{A^{\lambda}}\Big{(}\log_{\lambda}B-\log_{\lambda}A\Big{)},\quad(\mathrm{see}\
[9-13,15]).$

For any nonzero $\lambda\in\mathbb{R}$, the degenerate exponentials
$e_{\lambda}^{x}(t)$ are defined by

(4) $e_{\lambda}^{x}(t)=(1+\lambda
t)^{\frac{x}{\lambda}}=\sum_{n=0}^{\infty}(x)_{n,\lambda}\frac{t^{n}}{n!},\quad
e_{\lambda}(t)=e_{\lambda}^{1}(t),\quad(\mathrm{see}\ [10,14]).$

Note that $\displaystyle\lim_{\lambda\rightarrow
0}\log_{\lambda}(1+t)=\log(1+t),\ \lim_{\lambda\rightarrow
0}e_{\lambda}^{x}(t)=e^{xt}\displaystyle$, and
$e_{\lambda}(\log_{\lambda}(t))=\log_{\lambda}(e_{\lambda}(t))=t$. Thus the
inverse of the degenerate logarithm $\log_{\lambda}(t)$ is the degenerate
exponential $e_{\lambda}(t)$.

The degenerate Stirling numbers of the first kind are defined by

(5) $(x)_{n}=\sum_{k=0}^{n}S_{1,\lambda}(n,k)(x)_{k,\lambda},\quad(n\geq
0),\quad(\mathrm{see}\ [9]),$

where $(x)_{0}=1,\ (x)_{n}=x(x-1)\cdots(x-n+1),\ (n\geq 1)$.

In addition, the degenerate unsigned Stirling numbers of the first kind are
given by

(6) ${n\brack k}=(-1)^{n-k}S_{1,\lambda}(n,k),\quad(n,k\geq
0),\quad(\mathrm{see}\ [11]).$

As the inversion formula of (5), the degenerate Stirling numbers of the second
kind are given by

(7) $(x)_{n,\lambda}=\sum_{k=0}^{n}{n\brace k}_{\lambda}(x)_{k},\quad(n\geq
0),\quad(\mathrm{see}\ [9]).$

In [15], the degenerate Fubini polynomials are given by

(8)
$\frac{1}{1-x(e_{\lambda}(t)-1)}=\sum_{n=0}^{\infty}F_{n,\lambda}(x)\frac{t^{n}}{n!}.$

In particular, for $x=1$, $F_{n,\lambda}=F_{n,\lambda}(1)$ are called the
degenerate Fubini numbers.

From (8), we have

(9) $F_{n,\lambda}(x)=\sum_{k=0}^{n}{n\brace k}_{\lambda}k!x^{k},\quad(n\geq
0),\quad(\mathrm{see}\ [15]).$

Note that $F_{n}(x)=\lim_{\lambda\rightarrow 0}F_{n,\lambda}(x)$ are the
ordinary Fubini polynomials given by

(10)
$\frac{1}{1-x(e^{t}-1)}=\sum_{n=0}^{\infty}F_{n}(x)\frac{t^{n}}{n!},\quad(\mathrm{see}\
[1-8,16-20]).$

For any $\alpha\in\mathbb{R}$, the generalized degenerate Fubini polynomials
(called the degenerate Fubini polynomials of order $\alpha$) are given by

(11)
$\bigg{(}\frac{1}{1-x(e_{\lambda}(t)-1)}\bigg{)}^{\alpha}=\sum_{k=0}^{\infty}F_{k,\lambda}^{(\alpha)}(x)\frac{t^{k}}{k!},\quad(\mathrm{see}\
[15]),$

Thus, by (11), we get

(12)
$F_{k,\lambda}^{(\alpha)}(x)=\sum_{k=0}^{n}\langle\alpha\rangle_{k}x^{k}{n\brace
k}_{\lambda},\quad(\mathrm{see}\ [15]),$

where $\langle\alpha\rangle_{0},\
\langle\alpha\rangle_{k}=\alpha(\alpha+1)\cdots(\alpha+k-1),\ (k\geq 1)$.

We recall that the Stirling numbers of the first kind $S_{1}(n,k)$ and those
of the second kind ${n\brace k}$ are defined by

$(x)_{n}=\sum_{k=0}^{n}S_{1}(n,k)x^{k},\quad x^{n}=\sum_{k=0}^{n}{n\brace
k}(x)_{k},\quad(n\geq 0),\quad(\mathrm{see}\ [3,4,6,8,12,17]).$

Note that $\displaystyle\lim_{\lambda\rightarrow
0}S_{1,\lambda}(n,k)=S_{1}(n,k),\ \lim_{\lambda\rightarrow 0}{n\brace
k}_{\lambda}={n\brace k}$.

It is well known that the harmonic numbers are defined by

(13) $H_{0}=0,\quad H_{n}=1+\frac{1}{2}+\cdots+\frac{1}{n},\quad(n\geq
1),\quad(\mathrm{see}\ [6,7,17]).$

From (13), we note that the generating function of the harmonic numbers is
given by

(14) $-\frac{\log(1-t)}{1-t}=\sum_{k=1}^{\infty}H_{k}t^{k},\quad(\mathrm{see}\
[6,7,17]).$

Recently, the degenerate harmonic numbers are defined by

(15)
$-\frac{\log_{\lambda}(1-t)}{1-t}=\sum_{n=1}^{\infty}H_{n,\lambda}t^{n},\quad(\mathrm{see}\
[12,13]).$

For $n\geq 0$, $r\geq 1$, the degenerate hyperharmonic numbers are defined by

(16) $\ H_{0,\lambda}^{(r)}=0\ (r\geq 1),\quad
H_{n,\lambda}^{(1)}=H_{n,\lambda},\quad
H_{n,\lambda}^{(r)}=\sum_{k=1}^{n}H_{k,\lambda}^{(r-1)},\ (r\geq 2,\ n\geq
1),\ (\mathrm{see}\ [11,13]).$

From (16), we note that

(17)
$H_{n,\lambda}^{(r+1)}=\frac{\binom{n+r}{r}}{\binom{r-\lambda}{r}}(H_{n+r,\lambda}-H_{r,\lambda}),\quad(n,r\in\mathbb{N}),\quad(\mathrm{see}\
[11,13]).$

The generating function of the degenerate hyperharmonic numbers is given by

(18)
$-\frac{\log_{\lambda}(1-t)}{(1-t)^{r}}=\sum_{n=1}^{\infty}H_{n,\lambda}^{(r)}t^{n},\quad(r\in\mathbb{N}),\quad(\mathrm{see}\
[11,13]).$

For $f(x)=\sum_{n=0}^{\infty}a_{n}x^{n}\in\mathbb{C}[\\![x]\\!]$, we define

$f_{\lambda}(x)=\sum_{n=0}^{\infty}a_{n}(x)_{n,\lambda}\in\mathbb{C}[\\![x]\\!],$

where $\lambda$ is any fixed real number. Now, we introduce the next lemma
which contains functional equations useful for deriving identities in this
paper.

######  Lemma 1 ([11], Theorem 2).

Let
$f(x)=\sum_{n=0}^{\infty}a_{n}x^{n},\,\,g(x)=\sum_{k=0}^{\infty}b_{k}x^{k}\in\mathbb{C}[\\![x]\\!]$.
Then we have

$\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}\sum_{k=0}^{n}{n+r\brace
k+r}_{r,\lambda}x^{k}g^{(k)}(x)=\sum_{n=0}^{\infty}\frac{g^{(n)}(0)}{n!}f_{\lambda}(n+r)x^{n}.$

In particular, we also have

(19) $\sum_{m=r}^{\infty}\frac{f^{(m)}(0)}{m!}\bigg{(}\sum_{k=r}^{m}{m\brace
k}_{r,\lambda}x^{k}g^{(k)}(x)\bigg{)}=\sum_{n=r}^{\infty}\frac{g^{(n)}(0)}{n!}\binom{n}{r}r!\bigg{(}\sum_{m=r}^{\infty}\frac{f^{(m)}(0)}{m!}(n)_{m-r,\lambda}\bigg{)}x^{n},$

where ${n+r\brace k+r}_{r,\lambda}$ are the degenerate $r$-Stirling numbers
defined by

$(x+r)_{n,\lambda}=\sum_{k=0}^{n}{n+r\brace
k+r}_{r,\lambda}(x)_{k},\quad(n\geq 0).$

##  2\. Some numbers and polynomials related to degenerate harmonic and
degenerate hyperharmonic numbers

In view of (8) and (15), we consider the degenerate harmonic-Fubini
polynomials given by

(20)
$-\frac{\log_{\lambda}\big{(}1-x(e_{\lambda}(t)-1)\big{)}}{1-x(e_{\lambda}(t)-1)}=\sum_{n=1}^{\infty}HF_{n,\lambda}(x)\frac{t^{n}}{n!}.$

When $x=1$, $HF_{n,\lambda}=HF_{n,\lambda}(1)$ are called the degenerate
harmonic-Fubini numbers.

From (14), we note that

(21)
$\displaystyle-\frac{\log_{\lambda}\big{(}1-x(e_{\lambda}(t)-1)\big{)}}{1-x(e_{\lambda}(t)-1)}$
$\displaystyle=\sum_{k=1}^{\infty}H_{k,\lambda}x^{k}(e_{\lambda}(t)-1)^{k}$
$\displaystyle=\sum_{k=1}^{\infty}H_{k,\lambda}x^{k}k!\frac{1}{k!}\big{(}e_{\lambda}(t)-1\big{)}^{k}$
$\displaystyle=\sum_{k=1}^{\infty}H_{k,\lambda}k!x^{k}\sum_{n=k}^{\infty}{n\brace
k}_{\lambda}\frac{t^{n}}{n!}$
$\displaystyle=\sum_{n=1}^{\infty}\sum_{k=1}^{n}{n\brace
k}_{\lambda}H_{k,\lambda}k!x^{k}\frac{t^{n}}{n!}.$

Therefore, by (20) and (21), we obtain the following theorem.

######  Theorem 1.

For $n\in\mathbb{N}$, we have

$HF_{n,\lambda}(x)=\sum_{k=1}^{n}{n\brace k}_{\lambda}H_{k,\lambda}k!x^{k}.$

In particular, for $x=1$, we get

$HF_{n,\lambda}=\sum_{k=1}^{n}{n\brace k}_{\lambda}H_{k,\lambda}k!.$

Replacing $t$ by $\log_{\lambda}(1-t)$ in (20) and (21), we get

(22) $\displaystyle\sum_{n=1}^{\infty}x^{n}H_{n,\lambda}(-1)^{n}t^{n}$
$\displaystyle=\sum_{k=1}^{\infty}HF_{k,\lambda}(x)\frac{1}{k!}\Big{(}\log_{\lambda}(1-t)\Big{)}^{k}$
$\displaystyle=\sum_{k=1}^{\infty}HF_{k,\lambda}(x)\sum_{n=k}^{\infty}S_{1,\lambda}(n,k)(-1)^{n}\frac{t^{n}}{n!}$
$\displaystyle=\sum_{n=1}^{\infty}(-1)^{n}\sum_{k=1}^{n}HF_{k,\lambda}(x)S_{1,\lambda}(n,k)\frac{t^{n}}{n!}.$

Therefore, by comparing the coefficients on both sides of (22), we obtain the
following theorem.

######  Theorem 2.

For $n\in\mathbb{N}$, we have

$H_{n,\lambda}x^{n}=\sum_{k=1}^{n}S_{1,\lambda}(n,k)HF_{k,\lambda}(x).$

In particular, for $x=1$, we have

$H_{n,\lambda}=\sum_{k=1}^{n}S_{1,\lambda}(n,k)HF_{k,\lambda}.$

From (11), we note that

(23)
$\displaystyle\sum_{n=0}^{\infty}F_{n,\lambda}^{(1-\lambda)}(x)\frac{t^{n}}{n!}$
$\displaystyle=\bigg{(}\frac{1}{1-x(e_{\lambda}(t)-1)}\bigg{)}^{1-\lambda}=\sum_{k=0}^{\infty}\binom{k-\lambda}{k}x^{k}(e_{\lambda}(t)-1)^{k}$
$\displaystyle=\sum_{n=0}^{\infty}\langle
1-\lambda\rangle_{k}x^{k}\frac{1}{k!}(e_{\lambda}(t)-1)^{k}$
$\displaystyle=\sum_{k=0}^{\infty}\langle
1-\lambda\rangle_{k}x^{k}\sum_{n=k}^{\infty}{n\brace
k}_{\lambda}\frac{t^{n}}{n!}$
$\displaystyle=\sum_{n=0}^{\infty}\sum_{k=0}^{n}\langle
1-\lambda\rangle_{k}x^{k}{n\brace\lambda}_{\lambda}\frac{t^{n}}{n!}.$

Therefore, by comparing the coefficients on both sides of (23), we obtain the
following theorem.

######  Theorem 3.

Let $n$ be a nonnegative integer. Then we have

(24) $F_{n}^{(1-\lambda)}(x)=\sum_{k=0}^{n}\langle
1-\lambda\rangle_{k}{n\brace k}_{\lambda}x^{k}.$

Now, by using (3) we observe that

(25)
$\displaystyle-\Big{(}\frac{1}{1-y}\Big{)}\frac{\log_{\lambda}\Big{(}1-\frac{y}{1-y}(e_{\lambda}(t)-1)\Big{)}}{1-\frac{y}{1-y}(e_{\lambda}(t)-1)}=-\frac{\log_{\lambda}\Big{(}\frac{1-ye_{\lambda}(t)}{1-y}\Big{)}}{1-ye_{\lambda}(t)}$
$\displaystyle=\frac{-\Big{(}\frac{1}{1-y}\Big{)}^{\lambda}\Big{(}\log_{\lambda}\big{(}1-ye_{\lambda}(t)\big{)}-\log_{\lambda}(1-y)\Big{)}}{1-ye_{\lambda}(t)}$
$\displaystyle=-\Big{(}\frac{1}{1-y}\Big{)}^{\lambda}\frac{\log_{\lambda}(1-ye_{\lambda}(t))}{1-ye_{\lambda}(t)}+\frac{\log_{\lambda}(1-y)}{(1-y)^{\lambda}}\frac{1}{1-ye_{\lambda}(t)}$
$\displaystyle=\frac{1}{(1-y)^{\lambda}}\sum_{k=0}^{\infty}H_{k,\lambda}y^{k}e_{\lambda}^{k}(t)+\frac{\log_{\lambda}(1-y)}{(1-y)^{\lambda}}\sum_{k=0}^{\infty}y^{k}e_{\lambda}^{k}(t)$
$\displaystyle=\frac{1}{(1-y)^{\lambda}}\sum_{n=0}^{\infty}\sum_{k=0}^{\infty}\Big{(}H_{k,\lambda}+\log_{\lambda}(1-y)\Big{)}y^{k}(k)_{n,\lambda}\frac{t^{n}}{n!}.$

On the other hand, by (20), we get

(26)
$-\Big{(}\frac{1}{1-y}\Big{)}\frac{\log_{\lambda}\Big{(}1-\frac{y}{1-y}(e_{\lambda}(t)-1)\Big{)}}{1-\frac{y}{1-y}(e_{\lambda}(t)-1)}=\frac{1}{1-y}\sum_{n=0}^{\infty}HF_{n,\lambda}\Big{(}\frac{y}{1-y}\Big{)}\frac{t^{n}}{n!}.$

Therefore, by (25) and (26), we obtain the following theorem.

######  Theorem 4.

For $n\geq 0$, we have

$\frac{1}{1-y}HF_{n,\lambda}\Big{(}\frac{y}{1-y}\Big{)}=\frac{1}{(1-y)^{\lambda}}\sum_{k=0}^{\infty}y^{k}(k)_{n,\lambda}\big{(}H_{k,\lambda}+\log_{\lambda}(1-y)\big{)}.$

Equivalently, we also have

$HF_{n,\lambda}(y)=(1+y)^{\lambda-1}\sum_{k=0}^{\infty}\Big{(}\frac{y}{1+y}\Big{)}^{k}(k)_{n,\lambda}\Big{(}H_{k,\lambda}+\log_{\lambda}\Big{(}\frac{1}{1+y}\Big{)}\Big{)}.$

Note from Theorem 4 and (15) that

$\frac{1}{(1-y)^{\lambda}}\bigg{\\{}\bigg{(}y\frac{d}{dy}\bigg{)}_{n,\lambda}\Big{(}-\frac{\log_{\lambda}(1-y)}{1-y}\Big{)}+\log_{\lambda}(1-y)\bigg{(}y\frac{d}{dy}\bigg{)}_{n,\lambda}\frac{1}{1-y}\bigg{\\}}=\frac{1}{1-y}HF_{n,\lambda}\Big{(}\frac{y}{1-y}\Big{)}.$

For $r\in\mathbb{N}$, we define the degenerate hyperharmonic-Fubini
polynomials given by

(27)
$-\frac{\log_{\lambda}\big{(}1-y(e_{\lambda}(t)-1\big{)}}{\big{(}1-y(e_{\lambda}(t)-1)\big{)}^{r}}=\sum_{n=1}^{\infty}HF_{n,\lambda}^{(r)}(y)\frac{t^{n}}{n!}.$

When $y=1$, $HF_{n,\lambda}^{(r)}=HF_{n,\lambda}^{(r)}(1)$ are called the
degenerate hyperharmonic-Fubini numbers.

From (27) and (17), we have

(28) $\displaystyle\sum_{n=1}^{\infty}HF_{n,\lambda}^{(r)}(y)\frac{t^{n}}{n!}$
$\displaystyle=-\frac{\log_{\lambda}(1-y\big{(}e_{\lambda}(t)-1\big{)}}{\big{(}1-y(e_{\lambda}(t)-1)\big{)}^{r}}=\sum_{k=1}^{\infty}H_{k,\lambda}^{(r)}\big{(}e_{\lambda}(t)-1\big{)}^{k}y^{k}$
$\displaystyle=\sum_{k=1}^{\infty}H_{k,\lambda}^{(r)}y^{k}k!\frac{1}{k!}\big{(}e_{\lambda}(t)-1\big{)}^{k}=\sum_{k=1}^{\infty}H_{k,\lambda}^{(r)}y^{k}k!\sum_{n=k}^{\infty}{n\brace
k}_{\lambda}\frac{t^{n}}{n!}$
$\displaystyle=\sum_{n=1}^{\infty}\sum_{k=1}^{n}H_{k,\lambda}^{(r)}y^{k}k!{n\brace
k}_{\lambda}\frac{t^{n}}{n!}$
$\displaystyle=\sum_{n=1}^{\infty}\Bigg{(}\sum_{k=1}^{n}\frac{\binom{k+r-1}{r-1}}{\binom{r-1-\lambda}{r-1}}(H_{k+r-1,\lambda}-H_{r-1,\lambda})y^{k}k!{n\brace
k}_{\lambda}\bigg{)}\frac{t^{n}}{n!}.$

Therefore, by comparing the coefficients on both sides of (28), we obtain the
following theorem.

######  Theorem 5.

Let $n,r$ be positive integers. Then we have

$HF_{n,\lambda}^{(r)}(y)=\sum_{k=1}^{n}H_{k,\lambda}^{(r)}y^{k}k!{n\brace
k}_{\lambda}=\sum_{k=1}^{n}\frac{\binom{k+r-1}{r-1}}{\binom{r-1-\lambda}{r-1}}\big{(}H_{k+r-1,\lambda}-H_{r-1,\lambda}\big{)}k!{n\brace
k}_{\lambda}y^{k}.$

In particular, for $y=1$, we get

$HF_{n,\lambda}^{(r)}=\sum_{k=1}^{n}H_{k,\lambda}^{(r)}k!{n\brace
k}_{\lambda}=\sum_{k=1}^{n}\frac{\binom{k+r-1}{r-1}}{\binom{r-1-\lambda}{r-1}}\big{(}H_{k+r-1,\lambda}-H_{r-1,\lambda}\big{)}k!{n\brace
k}_{\lambda}.$

By using (3) and (18), we observe that

(29)
$\displaystyle-\frac{1}{(1-y)^{r}}\frac{\log_{\lambda}\big{(}1-\frac{y}{1-y}(e_{\lambda}(t)-1)\big{)}}{\big{(}1-\frac{y}{1-y}(e_{\lambda}(t)-1)\big{)}^{r}}=-\frac{\log_{\lambda}\big{(}\frac{1-ye_{\lambda}(t)}{1-y}\big{)}}{\big{(}1-ye_{\lambda}(t)\big{)}^{r}}$
$\displaystyle=-\Big{(}\frac{1}{1-ye_{\lambda}(t)}\Big{)}^{r}\frac{1}{(1-y)^{\lambda}}\Big{(}\log_{\lambda}\big{(}1-ye_{\lambda}(t)\big{)}-\log_{\lambda}(1-y)\Big{)}$
$\displaystyle=\frac{1}{(1-y)^{\lambda}}\bigg{(}-\Big{(}\frac{1}{1-ye_{\lambda}(t)}\Big{)}^{r}\log_{\lambda}(1-ye_{\lambda}(t))+\frac{\log_{\lambda}(1-y)}{(1-ye_{\lambda}(t))^{r}}\bigg{)}$
$\displaystyle=\frac{1}{(1-y)^{\lambda}}\bigg{(}\sum_{k=1}^{\infty}H_{k,\lambda}^{(r)}y^{k}e_{\lambda}^{k}(t)+\log_{\lambda}(1-y)\sum_{k=0}^{\infty}\binom{r+k-1}{k}y^{k}e_{\lambda}^{k}(t)\bigg{)}$
$\displaystyle=\sum_{n=0}^{\infty}\frac{1}{(1-y)^{\lambda}}\sum_{k=0}^{\infty}y^{k}(k)_{n,\lambda}\bigg{(}H_{k,\lambda}^{(r)}+\binom{r+k-1}{k}\log_{\lambda}(1-y)\bigg{)}\frac{t^{n}}{n!}.$

Therefore, by (27) and (29), we obtain the following theorem.

######  Theorem 6.

For $n\geq 0$ and $r\in\mathbb{N}$, we have

$\frac{1}{(1-y)^{r}}HF_{n,\lambda}^{(r)}\Big{(}\frac{y}{1-y}\Big{)}=\frac{1}{(1-y)^{\lambda}}\sum_{k=0}^{\infty}(k)_{n,\lambda}\bigg{(}H_{k,\lambda}^{(r)}+\binom{r+k-1}{k}\log_{\lambda}(1-y)\bigg{)}y^{k}.$

Equivalently, we also have

$HF_{n,\lambda}^{(r)}(y)=(1+y)^{\lambda-r}\sum_{k=0}^{\infty}(k)_{n,\lambda}\bigg{(}H_{k,\lambda}^{(r)}+\binom{r+k-1}{k}\log_{\lambda}\Big{(}\frac{1}{1+y}\Big{)}\bigg{)}\Big{(}\frac{y}{1+y}\Big{)}^{k}.$

Let

$g(t)=-\frac{\log_{\lambda}(1-t)}{1-t}=-\frac{1}{\lambda}\big{(}(1-t)^{\lambda-1}-(1-t)^{-1}\big{)},\quad(\mathrm{see}\
\eqref{1}).$

Then, from (1) we have

(30) $\displaystyle g^{(k)}(t)$
$\displaystyle=\bigg{(}\frac{d}{dt}\bigg{)}^{k}g(t)=-\frac{1}{\lambda}\frac{\langle
1-\lambda\rangle_{k}}{(1-t)^{k+1}}(1-t)^{\lambda}+\frac{1}{\lambda}\frac{k!}{(1-t)^{k+1}}$
$\displaystyle=-\frac{\langle
1-\lambda\rangle_{k}}{(1-t)^{k+1}}\log_{\lambda}(1-t)+\frac{1}{(1-t)^{k+1}}\bigg{(}\frac{k!-\langle
1-\lambda\rangle_{k}}{\lambda}\bigg{)}.$

From (30), we have

(31) $\displaystyle
g^{(k)}(0)=\bigg{(}\frac{d}{dt}\bigg{)}^{k}g(t)\bigg{|}_{t=0}=\frac{k!-\langle
1-\lambda\rangle_{k}}{\lambda},\quad(k\in\mathbb{N}).$

By (15), we get

(32)
$g^{(k)}(0)=\bigg{(}\frac{d}{dt}\bigg{)}^{k}\bigg{(}-\frac{\log_{\lambda}(1-t)}{1-t}\bigg{)}\bigg{|}_{t=0}=\bigg{(}\frac{d}{dt}\bigg{)}^{k}\sum_{k=1}^{\infty}H_{k,\lambda}t^{k}\bigg{|}_{k=0}=k!H_{k,\lambda}.$

Therefore, by (31) and (32), we obtain the following theorem.

######  Theorem 7.

For $k\in\mathbb{N}$, we have

$H_{k,\lambda}=\frac{1}{k!}\bigg{(}\frac{k!-\langle
1-\lambda\rangle_{k}}{\lambda}\bigg{)}=\frac{1}{\lambda}\bigg{(}1-\binom{k-\lambda}{k}\bigg{)}.$

Let $r$ be a nonnegative integer, $f(x)\in\mathbb{C}[\\![x]\\!]$, and let
$g(t)=-\frac{1}{1-t}\log_{\lambda}(1-t)$.
By (19), (30) and (32), we get

(33)
$\displaystyle\sum_{n=r}^{\infty}\binom{n}{r}H_{n,\lambda}r!\bigg{(}\sum_{m=r}^{\infty}\frac{f^{(m)}(0)}{m!}(n)_{m-r,\lambda}\bigg{)}x^{n}$
$\displaystyle=\frac{1}{1-x}\sum_{m=r}^{\infty}\frac{f^{(m)}(0)}{m!}\sum_{k=r}^{m}{m\brace
k}_{r,\lambda}\bigg{(}\frac{k!-\langle
1-\lambda\rangle_{k}}{\lambda}\bigg{)}\Big{(}\frac{x}{1-x}\Big{)}^{k}$
$\displaystyle\quad-\frac{1}{1-x}\log_{\lambda}(1-x)\sum_{m=r}^{\infty}\frac{f^{(m)}(0)}{m!}\sum_{k=r}^{m}{m\brace
k}_{r,\lambda}\langle 1-\lambda\rangle_{k}\Big{(}\frac{x}{1-x}\Big{)}^{k}.$

Let $r=0$ in (33). Then, by Theorem1, Theorem 7 and (24), we have

(34)
$\displaystyle\sum_{n=0}^{\infty}H_{n,\lambda}\bigg{(}\sum_{m=0}^{\infty}\frac{f^{(m)}(0)}{m!}(n)_{m,\lambda}\bigg{)}x^{n}$
$\displaystyle=\frac{1}{1-x}\sum_{m=0}^{\infty}\frac{f^{(m)}(0)}{m!}\sum_{k=0}^{m}{m\brace
k}_{\lambda}\bigg{(}\frac{k!-\langle
1-\lambda\rangle_{k}}{\lambda}\bigg{)}\Big{(}\frac{x}{1-x}\Big{)}^{k}$
$\displaystyle\quad-\frac{1}{1-x}\log_{\lambda}(1-x)\sum_{m=0}^{\infty}\frac{f^{(m)}(0)}{m!}\sum_{k=0}^{m}{m\brace
k}_{\lambda}\langle 1-\lambda\rangle_{k}\Big{(}\frac{x}{1-x}\Big{)}^{k}$
$\displaystyle=\frac{1}{1-x}\sum_{m=0}^{\infty}\frac{f^{(m)}(0)}{m!}\sum_{k=0}^{m}{m\brace
k}_{\lambda}k!H_{k,\lambda}\Big{(}\frac{x}{1-x}\Big{)}^{k}-\frac{\log_{\lambda}(1-x)}{1-x}\sum_{n=0}^{\infty}\frac{f^{(m)}(0)}{m!}F_{m,\lambda}^{(1-\lambda)}\Big{(}\frac{x}{1-x}\Big{)}$
$\displaystyle=\frac{1}{1-x}\sum_{m=0}^{\infty}\frac{f^{(m)}(0)}{m!}HF_{m,\lambda}\Big{(}\frac{x}{1-x}\Big{)}-\frac{\log_{\lambda}(1-x)}{1-x}\sum_{m=0}^{\infty}\frac{f^{(m)}(0)}{m!}F_{m,\lambda}^{(1-\lambda)}\Big{(}\frac{x}{1-x}\Big{)}.$

Therefore, by (34), we obtain the following theorem.

######  Theorem 8.

Let $f(x)\in\mathbb{C}[\\![x]\\!]$. Then we have

(35)
$\displaystyle\sum_{n=0}^{\infty}H_{n,\lambda}\bigg{(}\sum_{m=0}^{\infty}\frac{f^{(m)}(0)}{m!}(n)_{m,\lambda}\bigg{)}x^{n}$
$\displaystyle=\frac{1}{1-x}\sum_{m=0}^{\infty}\frac{f^{(m)}(0)}{m!}HF_{m,\lambda}\Big{(}\frac{x}{1-x}\Big{)}-\frac{\log_{\lambda}(1-x)}{1-x}\sum_{m=0}^{\infty}\frac{f^{(m)}(0)}{m!}F_{m,\lambda}^{(1-\lambda)}\Big{(}\frac{x}{1-x}\Big{)}.$

Let $f(x)=x^{k},\ (k\geq 1)$, in (35). Then we have

(36)
$\sum_{n=0}^{\infty}H_{n,\lambda}(n)_{k,\lambda}x^{n}=\frac{1}{1-x}HF_{k,\lambda}\Big{(}\frac{x}{1-x}\Big{)}-\frac{\log_{\lambda}(1-x)}{1-x}F_{k,\lambda}^{(1-\lambda)}\Big{(}\frac{x}{1-x}\Big{)}.$

Therefore, by (35), we obtain the following theorem.

######  Theorem 9.

For $k\geq 1$, we have

$\sum_{n=1}^{\infty}H_{n,\lambda}(n)_{k,\lambda}x^{n}=\frac{1}{1-x}HF_{k,\lambda}\Big{(}\frac{x}{1-x}\Big{)}-\frac{\log_{\lambda}(1-x)}{1-x}F_{k,\lambda}^{(1-\lambda)}\Big{(}\frac{x}{1-x}\Big{)}.$

From (36), we note that

(37)
$\displaystyle\frac{1}{(1-x)^{2}}\bigg{(}HF_{k,\lambda}\Big{(}\frac{x}{1-x}\Big{)}-\log_{\lambda}(1-x)F_{k,\lambda}^{(1-\lambda)}\Big{(}\frac{x}{1-x}\Big{)}\bigg{)}=\frac{1}{1-x}\sum_{l=1}^{\infty}H_{l,\lambda}(l)_{k,\lambda}x^{l}$
$\displaystyle=\sum_{j=0}^{\infty}x^{j}\sum_{l=1}^{\infty}H_{l,\lambda}(l)_{k,\lambda}x^{l}=\sum_{n=1}^{\infty}\bigg{(}\sum_{l=1}^{n}H_{l,\lambda}(l)_{k,\lambda}\bigg{)}x^{n}.$

Therefore, by (37), we obtain the following theorem.

######  Theorem 10.

For $k\in\mathbb{N}$, we have

$\displaystyle\sum_{n=1}^{\infty}\bigg{(}(1)_{k,\lambda}H_{1,\lambda}+(2)_{k,\lambda}H_{2,\lambda}+\cdots+(n)_{k,\lambda}H_{n,\lambda}\bigg{)}x^{n}$
$\displaystyle\quad=\frac{1}{(1-x)^{2}}\bigg{(}HF_{k,\lambda}\Big{(}\frac{x}{1-x}\Big{)}-\log_{\lambda}(1-x)F_{k,\lambda}^{(1-\lambda)}\Big{(}\frac{x}{1-x}\Big{)}\bigg{)}.$

By (36), we get

(38)
$\displaystyle\sum_{n=1}^{\infty}\big{(}(n)_{1,\lambda}+(n)_{2,\lambda}+\cdots+(n)_{k,\lambda}\big{)}H_{n,\lambda}x^{n}$
$\displaystyle=\frac{1}{1-x}\sum_{l=1}^{k}\bigg{(}HF_{l,\lambda}\Big{(}\frac{x}{1-x}\Big{)}-\log_{\lambda}(1-x)F_{l,\lambda}^{(1-\lambda)}\Big{(}\frac{x}{1-x}\Big{)}\bigg{)}.$

From (15) and (36), we note that

(39)
$\displaystyle\Big{(}x\frac{d}{dx}\Big{)}_{k,\lambda}\bigg{(}-\frac{\log_{\lambda}(1-x)}{1-x}\bigg{)}$
$\displaystyle=\Big{(}x\frac{d}{dx}\Big{)}_{k,\lambda}\bigg{(}\sum_{n=1}^{\infty}H_{n,\lambda}x^{n}\bigg{)}=\sum_{n=1}^{\infty}(n)_{k,\lambda}H_{n,\lambda}x^{n}$
$\displaystyle=\frac{1}{1-x}\bigg{(}HF_{k,\lambda}\Big{(}\frac{x}{1-x}\Big{)}-\log_{\lambda}(1-x)F_{k,\lambda}^{(1-\lambda)}\Big{(}\frac{x}{1-x}\Big{)}\bigg{)}.$

Therefore, by (39), we obtain the following differential equation.

######  Theorem 11.

Let $k$ be a positive integer. Then we have

$\Big{(}x\frac{d}{dx}\Big{)}_{k,\lambda}\bigg{(}-\frac{\log_{\lambda}(1-x)}{1-x}\bigg{)}=\frac{1}{1-x}\bigg{(}HF_{k,\lambda}\Big{(}\frac{x}{1-x}\Big{)}-\log_{\lambda}(1-x)F_{k,\lambda}^{(1-\lambda)}\Big{(}\frac{x}{1-x}\Big{)}\bigg{)}.$

For an integer $r$ with $r>1$, we let

(40)
$g(t)=-\frac{1}{(1-t)^{r}}\log_{\lambda}(1-t)=-\frac{1}{\lambda}\big{(}(1-t)^{\lambda-r}-(1-t)^{-r}\big{)},\quad(\mathrm{see}\,\,\eqref{1}).$

Then, for $k\in\mathbb{N}$, we have

(41) $\displaystyle g^{(k)}(t)$
$\displaystyle=\bigg{(}\frac{d}{dt}\bigg{)}^{k}g(t)=-\frac{\langle
r-\lambda\rangle_{k}}{\lambda}(1-t)^{\lambda-r-k}+\frac{1}{\lambda}\langle
r\rangle_{k}(1-t)^{-r-k}$ $\displaystyle=-\frac{\langle
r-\lambda\rangle_{k}}{(1-t)^{r+k}}\log_{\lambda}(1-t)+\frac{1}{(1-t)^{r+k}}\bigg{(}\frac{\langle
r\rangle_{k}-\langle r-\lambda\rangle_{k}}{\lambda}\bigg{)}.$

From (18), we note that

(42) $g^{(k)}(0)=k!H_{k,\lambda}^{(r)},\quad(k\geq 1).$

Therefore, by (41) and (42), we obtain the following theorem.

######  Theorem 12.

For $k\geq 1$, we have

$H_{k,\lambda}^{(r)}=\frac{1}{k!}\bigg{(}\frac{\langle r\rangle_{k}-\langle
r-\lambda\rangle_{k}}{\lambda}\bigg{)}=\frac{1}{\lambda}\bigg{(}\binom{r+k-1}{k}-\binom{r+k-\lambda-1}{k}\bigg{)}.$

Let $f(x)=\sum_{n=0}^{\infty}a_{n}x^{n}\in\mathbb{C}[\\![x]\\!]$, and let
$g(t)=-\frac{\log_{\lambda}(1-t)}{(1-t)^{r}}$. Then, by Theorem 12, (19), (40)
and (41), we get

(43)
$\displaystyle\sum_{n=r}^{\infty}H_{n,\lambda}^{(r)}\binom{n}{r}r!\bigg{(}\sum_{m=r}^{\infty}\frac{f^{(m)}(0)}{m!}(n)_{m-r,\lambda}\bigg{)}x^{n}$
$\displaystyle=\sum_{m=r}^{\infty}\frac{f^{(m)}(0)}{m!}\sum_{k=r}^{m}{m\brace
k}_{r,\lambda}x^{k}\bigg{(}-\frac{\langle
r-k\rangle_{k}}{(1-x)^{r+k}}\log_{\lambda}(1-x)+\frac{1}{(1-x)^{r+k}}\bigg{(}\frac{\langle
r\rangle_{k}-\langle r-\lambda\rangle_{k}}{\lambda}\bigg{)}\bigg{)}$
$\displaystyle=-\frac{\log_{\lambda}(1-x)}{(1-x)^{r}}\sum_{m=r}^{\infty}\frac{f^{(m)}(0)}{m!}\sum_{k=r}^{m}{m\brace
k}_{r,\lambda}\langle r-\lambda\rangle_{k}\Big{(}\frac{x}{1-x}\Big{)}^{k}$
$\displaystyle\qquad+\frac{1}{(1-x)^{r}}\sum_{m=r}^{\infty}\frac{f^{(m)}(0)}{m!}\sum_{k=r}^{m}{m\brace
k}_{r,\lambda}\Big{(}\frac{x}{1-x}\Big{)}^{k}k!H_{k,\lambda}^{(r)}$

Let $f(x)=x^{k},\ (k\geq 1)$ in (43). Then we have

(44)
$\displaystyle\sum_{n=r}^{\infty}H_{n,\lambda}^{(r)}\binom{n}{r}r!(n)_{k-r,\lambda}x^{n}$
$\displaystyle=-\frac{\log_{\lambda}(1-x)}{(1-x)^{r}}\sum_{l=r}^{k}{k\brace
l}_{r,\lambda}\langle
r-\lambda\rangle_{l}\Big{(}\frac{x}{1-x}\Big{)}^{l}+\frac{1}{(1-x)^{r}}\sum_{l=r}^{k}{k\brace
l}_{r,\lambda}\Big{(}\frac{x}{1-x}\Big{)}^{l}l!H_{l,\lambda}^{(r)}.$

##  3\. Conclusion

The degenerate harmonic-Fubini polynomials are given by
$HF_{n,\lambda}(x)=\sum_{k=1}^{n}{n\brace k}_{\lambda}H_{k,\lambda}k!x^{k}$,
with $H_{k,\lambda}$ the degenerate harmonic numbers, while the degenerate
Fubini polynomials are given by $F_{n,\lambda}(x)=\sum_{k=0}^{n}{n\brace
k}_{\lambda}k!x^{k}$. The degenerate harmonic-Fubini polynomials are so named
for this reason. The degenerate hyperharmonic-Fubini polynomials are also so
named, as it is given by $HF_{n,\lambda}^{(r)}(x)=\sum_{k=1}^{n}{n\brace
k}_{\lambda}H_{k,\lambda}^{(r)}k!x^{k}$, with $H_{k,\lambda}^{(r)}$ the
degenerate hyperharmonic numbers.

We introduced the degenerate harmonic-Fubini polynomials and numbers and
studied their properties, explicit expressions and some identities by using
generating functions. In addition, as generalizations of those polynomials and
numbers, we also introduced the degenerate hyperharmonic-Fubini polynomials
and derived similar results to the degenerate harmonic-Fubini polynomials and
numbers.

It is one of our future projects to continue to study various degenerate
versions of some special polynomials and numbers and to find their
applications to physics, science and engineering as well as to mathematics.

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