11institutetext: Department of Ophthalmology and Optometry, Medical University
of Vienna, Austria
11email<EMAIL_ADDRESS>
# Time-Equivariant Contrastive Learning for Degenerative Disease Progression
in Retinal OCT

Taha Emre  Arunava Chakravarty  Dmitrii Lachinov  Antoine Rivail
Ursula Schmidt-Erfurth  Hrvoje Bogunović

###### Abstract

Contrastive pretraining provides robust representations by ensuring their
invariance to different image transformations while simultaneously preventing
representational collapse. Equivariant contrastive learning, on the other
hand, provides representations sensitive to specific image transformations
while remaining invariant to others. By introducing equivariance to time-
induced transformations, such as disease-related anatomical changes in
longitudinal imaging, the model can effectively capture such changes in the
representation space. In this work, we propose a Time-equivariant Contrastive
Learning (TC) method. First, an encoder embeds two unlabeled scans from
different time points of the same patient into the representation space. Next,
a temporal equivariance module is trained to predict the representation of a
later visit based on the representation from one of the previous visits and
the corresponding time interval with a novel regularization loss term while
preserving the invariance property to irrelevant image transformations. On a
large longitudinal dataset, our model clearly outperforms existing equivariant
contrastive methods in predicting progression from intermediate age-related
macular degeneration (AMD) to advanced wet-AMD within a specified time-window.

###### Keywords:

Equivariance Contrastive Learning AMD Prediction.

##  1 Introduction

The advent of contrastive self-supervised learning (SSL) methods showed that
it is possible to learn informative and discriminative image representations
by learning invariances to image transformations that do not alter their
semantics. The invariance is achieved by increasing the representational
similarity between the original image and its transformed versions in the
representation space, while pushing all other pairs apart as negatives using a
Siamese network [1]. However, sensitivity (equivariance) to certain
transformations may be crucial for specific downstream tasks, such as rotation
degree information in histopathology slides [21]. Introducing equivariance to
time-induced transformations, such as anatomical changes due to degenerative
disease progression in longitudinal medical images could be pivotal in
downstream tasks such as predicting the future risk of conversion of a patient
from an early to a late-stage.

Retinal degenerative diseases such as Age-related Macular Degeneration (AMD)
are frequently identified through non-invasive optical coherence tomography
(OCT) scans, made up of sequential 2D cross-sectional slices (B-scans). As a
leading cause of vision loss and blindness [3], AMD leads to irreversible
tissue damage, implying that the severity of the disease can be represented as
a monotonically non-decreasing function over time. However, the speed of
disease progression and the associated anatomical changes vary widely between
patients. Also, the longitudinal nature of the AMD datasets makes them costly
to label by clinicians. Consequently, a Time-Equivariant SSL stage could be
crucial in identifying the patients at a higher risk of progression to an
advanced disease stage from a single visit to provide timely treatment and
personalized management.

Existing Equivariant SSL methods aim to learn such sensitivity, unlike
contrastive methods that seek total invariance. Indeed, invariance to certain
transformations is not always desirable [22]. This sensitivity can be achieved
with pretext SSL, by learning to recover the parameter of the target
transformation from the representations. Image rotation prediction [11], video
clip ordering [23] and distinguishing close and far clips [13] for temporal
equivariance can be given as examples. In medical imaging, the time difference
[19] or ordering two visits (scans) of a patient [16] is commonly used as a
pretraining step for temporal tasks such as disease forecasting. In [27] they
decoupled the effect of aging from disease-related changes in brain MRI scans
by carefully curating a temporal dataset consisting of old subjects without
any underlying diseases. Then, they learned an aging trajectory in the
representation space shared by all patients. Although these methods learn
sensitivity to the transformation, they lack the desired invariance to image
perturbations.

To address these issues, Equivariant contrastive methods have been explored to
learn equivariance for specific transformations while ensuring invariance for
the others. Unlike methods with specific architecture [6] for achieving
equivariance, equivariant contrastive learning methods are mostly architecture
agnostic. Dangovski et al. [7] combined transformation parameter prediction
with a contrastive loss for the transformation invariance, requiring four
additional encoders for the rotation prediction task [11], increasing the
computational overhead substantially (ESSL). Similarly, in [14], they proposed
an additional temporal pretext task, such as frame ordering, to the
contrastive task for videos. Emre et al. [9] introduced TINC for OCT scans
where they exploited the time difference between visits by applying a
similarity insensitivity increasing with the difference. In [17], AugSelf is
proposed to enforce equivariance on the representation space by predicting
transformation parameters between input pairs, then projected the learned
representations into an invariant space for the contrastive task. However,
transformation parameter recovery does not guarantee explicit equivariance,
because it’s possible for transformation parameters to be confined to a single
dimension of the representation, leaving the rest completely invariant. In
addition, parameter prediction models do not learn any explicit function for
directly transforming the representations. To address these limitations,
Devillers et al. [8] proposed utilizing two different projections of the
representation space; one for the invariance, and the second for the
equivariance with a trainable module allowing direct manipulation of the
projections with the image transformation parameter (EquiMod). Since
invariance is a trivial solution for the equivariance where the equivariant
representation transformation collapses to identity, they introduced an
additional contrastive loss for the equivariance branch at the cost of an
increased computational overhead. These methods aim to achieve equivariance
using transformation parameter recovery, additional contrastive tasks,
projection spaces, or pre-defined image transformation (e.g. rotation matrix)
[10].

We propose to learn time-sensitive/equivariant representations directly on the
representation space and learn to propagate representations in time, using
longitudinal medical datasets in a contrastive SSL setting. Given that
degenerative diseases either progress or stagnate over time, we have modeled
disease progression in the representation space as an additive transformation
parameterized by the time difference of a patient’s visits, enabling direct
temporal transformation of representations. We adapted a regularization loss
on the calculated additive term to prevent the network from ignoring the time
difference and producing identity transformation. Finally, we evaluated the
learned representations for AMD progression prediction across multiple time-
windows from a single visit scan. In summary, our contributions are (a)
constructing contrastive pairs from different visits of a patient, (b)
training a novel equivariance module to propagate representation in the future
without accessing future visit scan, (c) introducing a novel regularization
loss to prevent invariance to time.

##  2 Methodology

Our approach learns temporal equivariance in representation space, followed by
a projection to invariant space within a contrastive SSL framework (Fig. 1).
We propose a novel loss term specifically tailored for the equivariance (Eq.
5), utilizing a longitudinal retinal OCT dataset (Sec. 3.1.1) collected across
multiple patient visits. First, we define the term time equivariant
representation. Then, we detail our approach to learn the equivariance.

Figure 1: An overview of TC architecture. (A) Representations of two scans
acquired at two time points and the time difference are fed into $h_{\psi}$
for equivariance, then projected for the invariance loss. (B) Equivariance
module with an additive displacement map.

####  2.0.1 Time-Equivariant Representations

The disease progression in a patient is routinely monitored using a series of
medical scans ${\bm{x}}\in\mathbb{I}\subseteq\mathbb{R}^{H\times W}$, and
${\bm{x}}_{t}$ denotes the scan acquired at a time $t$ sampled from
$\mathbb{T}=\\{t\in\mathbb{N}_{0};t\leq b\\}$, $b$ being the last visit date.
Let $\mu:\mathbb{I}\times\mathbb{T}\rightarrow\mathbb{I}$ represent a
transformation in the image space for all possible time differences, which
captures the complex anatomical changes between ${\bm{x}}_{t}$ and
${\bm{x}}_{t+\Delta t}$ due to the disease progression (and/or normal aging),
where $t+\Delta t\in\mathbb{T}$. Although the actual transformation $\mu$ is
difficult to model in the image space, the result of the transformation is
available as ${\bm{x}}_{t+\Delta t}$ for some specific $\Delta t$, from
available visits of the patient, such that ${\bm{x}}_{t+\Delta
t}:=\mu({\bm{x}}_{t},\Delta t)$. Let
$f_{\theta}:\mathbb{I}\rightarrow\mathbb{R}^{D}$ with learnable parameters
$\theta$ be a deep learning based encoder, which can embed ${\bm{x}}_{t}$ as a
representation $r_{t}:=f_{\theta}({\bm{x}}_{t})$ and ${\bm{x}}_{t+\Delta t}$
as $r_{t+\Delta t}:=f_{\theta}({\bm{x}}_{t+\Delta t})$. As $\mu(\cdot,\Delta
t)$ transforms ${\bm{x}}_{t}$ to ${\bm{x}}_{t+\Delta t}$ in the image space, a
corresponding transformation $h(\cdot,\Delta t)$ operates in the
representation space such that $r_{t+\Delta t}=h(r_{t},\Delta t)$. If
$f_{\theta}$ is equivariant in time, the equivariant relation is defined as:

$\exists h:\forall\Delta t\in\mathbb{T},\quad
f_{\theta}(\mu({\bm{x}}_{t},\Delta t))=f_{\theta}({\bm{x}}_{t+\Delta
t})\approx h(f_{\theta}({\bm{x}}_{t}),\Delta t)$ (1)

An undesired trivial solution for Eq. 1 would be time-invariance of
$f_{\theta}$ such that $f_{\theta}({\bm{x}}_{t})=f_{\theta}({\bm{x}}_{t+\Delta
t})$ with disease progression being ignored, in turn making $h(\cdot,\Delta
t)$ an identity mapping. This implies that $f_{\theta}$ learns only patient-
specific features staying constant with respect to time while ignoring changes
due to the progression. The transformation $h(\cdot,\Delta t)$ should be able
to capture the future changes in the scans specific to each patient, and we
propose to approximate it with a deep neural network that can model complex
changes.

####  2.0.2 Time-Equivariant Contrastive Learning (TC)

Our model extends the existing contrastive methods with the introduction of a
temporal equivariance property. The input pair to the Siamese network is
created from two different visits of a patient and both are transformed with
contrastive augmentations (Fig.1.A). Recent equivariant contrastive methods
with a learnable transformation [8, 10] on features, relied on two projection
spaces, one for equivariance and the other for invariance, preventing them
from directly steering the representations. Instead, we propose to enforce
equivariance directly on the representation space, and then to learn the
invariance on a projection space [15] (Fig.1.A). This allows for the
preservation of the time-sensitive information in the representation space
while enforcing invariance in the projection space. A logical order since
invariance is a trivial solution of equivariance. It also simplifies the
network architecture and computational overhead, and facilitates transferring
the encoder weights as a backbone.

We introduce an equivariance module $h_{\psi}$ containing the learned
transformation (predictor) with learnable parameters $\psi$, which takes the
concatenation of representation $r_{t}$ and time difference $\Delta t$,
normalized between 0-1, as input, and transforms $r_{t}$ to
$r^{\prime}_{t+\Delta t}$ to satisfy the equivariance property defined in Eq.
1. The predictor enables the model to generate future visits’ representations
by forwarding available scans in time to be used in predictive tasks. The
equivariance loss [8, 10] for $h_{\psi}$ is defined as:

$\ell_{\text{equiv}}=\|f_{\theta}({\bm{x}}_{t+\Delta
t})-h_{\psi}(f_{\theta}({\bm{x}}_{t}),\Delta t)\|_{2}^{2}=\|r_{t+\Delta
t}-h_{\psi}(r_{t},\Delta t)\|_{2}^{2}$ (2)

As it is highlighted in [8, 10], the contrastive loss imposes invariance to
the image transformations including time-related changes. The learned
invariance results in a trivial solution for the equivariance loss where
$h_{\psi}$ collapses to an identity mapping, ignoring $\Delta t$. The
aforementioned methods rely on another set of computationally heavy
contrastive task on $r^{\prime}_{t+\Delta t}$ to prevent collapse. For an MLP
$h_{\psi}$, the collapse manifests as $\frac{\partial h_{\psi}}{\partial t}$
becoming 0. To avoid an additional contrastive task or a regularization term
on the gradients to prevent $h_{\psi}$ from collapsing, we reparametrized
$h_{\psi}$ as an additive displacement map (DM):

$r^{\prime}_{t+\Delta t}=h_{\psi}(r_{t},\Delta
t)=r_{t}\oplus\rho_{\psi}(r_{t},\Delta t)$ (3)

Now, the equivariant transformation $h_{\psi}$ has two parts: an MLP
$\rho_{\psi}$ for predicting DM as an additive term and direct sum of the DM
with the representation (Fig. 1.B). In Eq. 3, the collapse condition is much
easier to regularize since it is the zero norm for the predicted DM. The ideal
regularization should prevent DM norm becoming zero, while not encouraging
high norm values. The representation space should respond to the time change
smoothly without predicting large DMs easily. We adapted a regularization loss
term for preventing collapse from pair-wise loss of RankNet [4]. The original
loss function aims to increase the probability of ranking two embeddings
correctly. In our case, the ranking is always one-sided, because the norm
cannot take negative values. Thus, the final DM regularization term becomes:

$\ell_{\text{reg}}=\log\left(1+\exp(-||\rho_{\psi}(r_{t},\Delta
t)||_{2})\right)$ (4)

The $\ell_{\text{reg}}$ penalizes null displacements without encouraging very
high norms.

For the invariant contrastive loss, we used VICReg [2]. It is calculated using
projections $z_{t}$ and $z_{t+\Delta t}$, obtained by projecting
representations $r_{t}$ and $r_{t+\Delta t}$ with an MLP projector
$g_{\gamma}$. As a non-contrastive method, it does not require very large
batch sizes. It has 3 loss terms; an $\ell_{2}$ term for increasing similarity
within pair, a covariance term to prevent dimensional collapse [26, 15] and a
variance term for implicitly pushing the negatives apart which is calculated
along a batch of projections, where each pair could have different $\Delta t$
values.

Thus, the final training loss with a contrastive loss $\ell_{\mathrm{contr}}$
is:

$\mathcal{L}_{\mathrm{TC}}=\ell_{\mathrm{contr}}+\beta\cdot(\ell_{\text{equiv}}+\upsilon\cdot\ell_{\text{reg}})$
(5)

##  3 Experiments and Results

We tested TC against three main equivariant contrastive methods: ESSL [7],
AugSelf [17] and EquiMod [8]. ImageNet pretrained ResNet50, VICReg [2] and
TINC [9] were used as baselines. During pretraining each batch contains only a
single image pair from a patient to avoid pushing apart the intra-patient
scans with the contrastive loss. Clinically, only future representations
matter for conversion risk assessment. Therefore, the equivariant models were
pre-trained only with image pairs with a positive $\Delta t$.

###  3.1 Experimental Setup

####  3.1.1 Dataset

Figure 2: Left: OCT scans with conversion to wet-AMD over time. Right:
Contrastive augmentations examples.

The SSL stage and the downstream evaluation were conducted on the HARBOR
dataset (Fig. 2 left), comprising OCT scans from 1,096 patients’ fellow-
eyes111The other eye that is not part of the clinical trial monitored for wet-
AMD onset, with monthly follow-ups over 24 months. Intermediate stage visits
(463 eyes - 10,108 scans) are used for predicting the onset of late, wet-AMD
stage within 6 and 12 months [24, 25, 20] (117 converter eyes) as a
forecasting task. The dataset was split with the eye level stratification into
4-fold cross-validation ($80\%$) for hyper-parameter optimization and a hold-
out test set ($20\%$) for reporting the performance. The remaining 540 eyes
(12,506 scans) are only used in contrastive pretraining step. We used the
fovea centered 2D B-scan, which is known to be most representative of the AMD
state [18], resized to $224\times 224$.

We followed the work of [9, 12] for contrastive image transformations specific
to B-scans (Fig. 2 right). During pretraining, for each input pair, $\Delta t$
is randomly sampled from an interval of 1 to 12 months and normalized between
$0-1$. The rationale behind the interval is that the set of all pair
permutations for all possible time differences is very large (276 per patient)
for a plausible training time. By limiting $\Delta t$, training $h_{\psi}$
becomes much more convenient, because it only needs to produce DMs for a
constrained but practically meaningful interval.

####  3.1.2 Implementation details

A ResNet50 with a representation dimension of 2048 is used as the encoder
$f_{\theta}$. Following the common practice of projecting representations to
higher dimensions [2, 26], the projector $g_{\gamma}$ is a three-layer MLP
with hidden dimensions of 4096. DM predictor $\rho_{\psi}$ of TC is
implemented as a two-layer MLP with an input size of ($2048+1$) for
$[r_{t},\Delta t]$, and it outputs a DM with the same dimension as the
representations. In EquiMod, ESSL and AugSelf, we followed the original works
for the predictor architecture.

All models were pretrained for 300 epochs with the AdamW optimizer with a
learning rate of 5E-4 and a batch size of 128. A cosine scheduler with a warm-
up was used for decaying learning rates. The same weight decay of 1E-6 was
applied to all pretraining setups. $\beta$ and $\upsilon$ of
$\mathcal{L}_{\mathrm{TC}}$ were set to 1 and 0.5, respectively.

###  3.2 Results

Table 1: Linear evaluation results for wet-AMD conversion prediction within
two time-windows: 6 and 12 months

Model | 6-months | 12-months
---|---|---
AUROC $\uparrow$ | PRAUC $\uparrow$ | BAcc $\uparrow$ | AUROC $\uparrow$ | PRAUC $\uparrow$ | BAcc $\uparrow$
ImageNet | $0.714\pm 0.018$ | $0.106\pm 0.007$ | $0.648\pm 0.016$ | $0.692\pm 0.014$ | $0.167\pm 0.009$ | $0.616\pm 0.045$
VICReg[2] | $0.700\pm 0.029$ | $0.137\pm 0.031$ | $0.650\pm 0.021$ | $0.671\pm 0.033$ | $0.158\pm 0.019$ | $0.622\pm 0.023$
TINC[9] | $0.729\pm 0.017$ | $0.151\pm 0.019$ | $0.652\pm 0.007$ | $0.688\pm 0.060$ | $0.210\pm 0.026$ | $0.629\pm 0.055$
ESSL[7] | $0.718\pm 0.033$ | $0.157\pm 0.034$ | $0.649\pm 0.028$ | $0.699\pm 0.041$ | $0.187\pm 0.039$ | $0.637\pm 0.032$
AugSelf[17] | $0.752\pm 0.031$ | $\mathbf{0.190\pm 0.027}$ | $0.670\pm 0.029$ | $0.735\pm 0.049$ | $0.245\pm 0.040$ | $0.651\pm 0.057$
EquiMod[8] | $0.734\pm 0.035$ | $0.153\pm 0.038$ | $0.652\pm 0.048$ | $0.718\pm 0.059$ | $0.226\pm 0.055$ | $0.657\pm 0.031$
TC w/o DM | $0.761\pm 0.032$ | $0.163\pm 0.027$ | $0.658\pm 0.034$ | $0.748\pm 0.033$ | $0.220\pm 0.026$ | $0.653\pm 0.031$
TC w/o $\ell_{\text{reg}}$ | $0.732\pm 0.021$ | $0.137\pm 0.024$ | $0.661\pm 0.009$ | $0.721\pm 0.035$ | $0.195\pm 0.025$ | $0.655\pm 0.028$
TC | $\mathbf{0.769\pm 0.027}$ | $0.185\pm 0.032$ | $\mathbf{0.677\pm 0.026}$ | $\mathbf{0.752\pm 0.034}$ | $\mathbf{0.265\pm 0.043}$ | $\mathbf{0.671\pm 0.032}$
TCsyn | $\underline{0.784\pm 0.022}$ | $\underline{0.188\pm 0.036}$ | $\underline{0.683\pm 0.024}$ | $\underline{0.775\pm 0.030}$ | $\underline{0.304\pm 0.052}$ | $\underline{0.695\pm 0.023}$

####  3.2.1 Downstream prediction performance

After pretraining, we evaluated the extracted representations with a linear
classifier on the downstream tasks (Sec. 3.1.1), as it is common practice in
contrastive SSL [5]. A scan from a single time-point is used for future
conversion prediction. The linear classifier is trained for 50 epochs with
Adam optimizer and a learning rate of 1E-4. We reported the results (Table 1)
with AUROC, PRAUC and Balanced Accuracy (BAcc) due to a large class imbalance
(6-month 1:20, 12-month 1:10). Accordingly, the random prediction PRAUC for
6-month is 0.05, and for 12-month is 0.11.

Our TC model surpassed all the other ones in AUROC and BAcc in both tasks. In
terms of 12-month PRAUC, TC outperformed all while achieving competitive
6-month PRAUC performance. Interestingly, AugSelf achieved better results
against the comparison methods, which can be attributed to its enforcement of
the equivariance in the representation space rather than the projection space.
A decline in 12-month prediction accuracy was noted for all the models, likely
due to the extended time frame and the fact that the predictive biomarkers
were more subtle. Finally, the performance gains of equivariant contrastive
methods over the others (Table 1 line 1-3) underscore the importance of
including temporal equivariance in disease progression prediction.

####  3.2.2 Evaluation of the equivariance module

Each scan representation is synthetically propagated in time by 6 months using
equivariance module $h_{\psi}$ without accessing the future scans. Then they
were combined with the initial representations by averaging. Their predictive
capability was assessed using the same linear evaluation, with the improved
performances in Table 1 \- TCsyn demonstrating the equivariance module’s
ability to discern significant disease progression from a single scan.
Notably, this performance increase was more distinct in the 12-month task,
where the biomarkers are less distinguishable due to the longer conversion
time-frame. Similar improvement is not possible with the other equivariant
contrastive methods since they cannot manipulate the representation space.

####  3.2.3 Ablation study

We investigated the impact of the explicit DM prediction (Table 1 \- TC w/o
DM) and the regularization loss term (Table 1 \- TC w/o $\ell_{\text{reg}}$)
components of TC by removing each at the time. In both cases, the downstream
task performance degraded considerably underscoring the importance of both
components. When compared against the non-equivariant methods, the ablated
models still perform better highlighting the importance of time component.
When TC w/o DM compared against TC w/o $\ell_{\text{reg}}$, it performs
considerably better. This can be attributed to TC w/o DM directly predicting
the future representation without relying on an unregularized DM. This
highlights the significance of $\ell_{\text{reg}}$ to predict meaningful DMs.

##  4 Conclusion

Predicting late-stage disease onset is challenging due to the varying
progression speeds among patients and subtle prognostic biomarkers. In this
study, we proposed exploiting temporal information in unlabeled longitudinal
OCT datasets. We introduced Time-equivariant Contrastive (TC) SSL with a
learnable equivariance module for directly propagating image representations
in time. The module retains characteristics of disease progression, such as
irreversible tissue loss or degeneration, over time. A novel regularization
loss was introduced to avoid the trivial solution for the equivariance induced
by the contrastive loss with minimal computational overhead, unlike other
equivariant models. We tested TC for disease progression on a longitudinal
dataset consisting of eye scans with different stages of AMD. The TC
pretraining consistently outperformed other equivariant contrastive methods as
well as non-equivariant ones. Results highlight the importance of temporal
sensitivity in representations for accurately assessing a patient’s risk of
conversion to a late disease stage. Furthermore, we found that integrating the
synthetically propagated representations with the training data enhanced the
conversion prediction performance, illustrating not only that TC captured
relevant longitudinal information but also provided a functional prediction
module, demonstrating its ability to generate a proper time-equivariant
representation space.

####  4.0.1 Acknowledgements

***

## References

  * [1] Azizi, S., Mustafa, B., Ryan, F., Beaver, Z., Freyberg, J., Deaton, J., Loh, A., Karthikesalingam, A., Kornblith, S., Chen, T., et al.: Big self-supervised models advance medical image classification. In: Proceedings of the IEEE/CVF international conference on computer vision. pp. 3478–3488 (2021)
  * [2] Bardes, A., Ponce, J., LeCun, Y.: Vicreg: Variance-invariance-covariance regularization for self-supervised learning. In: International Conference on Learning Representations (2022)
  * [3] Bressler, N.M.: Age-Related Macular Degeneration Is the Leading Cause of Blindness . . . JAMA 291(15), 1900–1901 (04 2004). https://doi.org/10.1001/jama.291.15.1900, https://doi.org/10.1001/jama.291.15.1900
  * [4] Burges, C., Shaked, T., Renshaw, E., Lazier, A., Deeds, M., Hamilton, N., Hullender, G.: Learning to rank using gradient descent. In: International Conference on Machine Learning. pp. 89– 96 (2005)
  * [5] Chen, X., Xie, S., He, K.: An empirical study of training self-supervised vision transformers. In: Proceedings of the IEEE/CVF international conference on computer vision. pp. 9620–9629 (2021)
  * [6] Cohen, T.S., Welling, M.: Group equivariant convolutional networks. In: International Conference on Machine Learning. PMLR, JMLR.org (2016)
  * [7] Dangovski, R., Jing, L., Loh, C., Han, S., Srivastava, A., Cheung, B., Agrawal, P., Soljacic, M.: Equivariant self-supervised learning: Encouraging equivariance in representations. In: International Conference on Learning Representations (2022), https://openreview.net/forum?id=gKLAAfiytI
  * [8] Devillers, A., Lefort, M.: Equimod: An equivariance module to improve visual instance discrimination. In: International Conference on Learning Representations (2023), https://openreview.net/forum?id=eDLwjKmtYFt
  * [9] Emre, T., Chakravarty, A., Rivail, A., Riedl, S., Schmidt-Erfurth, U., Bogunović, H.: Tinc: Temporally informed non-contrastive learning for disease progression modeling in retinal oct volumes. In: International Conference on Medical Image Computing and Computer-Assisted Intervention. pp. 625–634. Springer (2022)
  * [10] Garrido, Q., Najman, L., Lecun, Y.: Self-supervised learning of split invariant equivariant representations. In: International Conference on Machine Learning. PMLR (2023)
  * [11] Gidaris, S., Singh, P., Komodakis, N.: Unsupervised representation learning by predicting image rotations. In: International Conference on Learning Representations (2018)
  * [12] Holland, R., et al.: Metadata-enhanced contrastive learning from retinal optical coherence tomography images. CoRR abs/2208.02529 (2022)
  * [13] Jayaraman, D., Grauman, K.: Slow and steady feature analysis: higher order temporal coherence in video. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. pp. 3852–3861 (2016)
  * [14] Jenni, S., Jin, H.: Time-equivariant contrastive video representation learning. In: Proceedings of the IEEE/CVF International Conference on Computer Vision. pp. 9970–9980 (2021)
  * [15] Jing, L., Vincent, P., LeCun, Y., Tian, Y.: Understanding dimensional collapse in contrastive self-supervised learning. In: International Conference on Learning Representations (2022), https://openreview.net/forum?id=YevsQ05DEN7
  * [16] Kim, H., Sabuncu, M.R.: Learning to compare longitudinal images. In: Medical Imaging with Deep Learning (2023)
  * [17] Lee, H., Lee, K., Lee, K., Lee, H., Shin, J.: Improving transferability of representations via augmentation-aware self-supervision. In: Beygelzimer, A., Dauphin, Y., Liang, P., Vaughan, J.W. (eds.) Advances in Neural Information Processing Systems (2021), https://openreview.net/forum?id=U34rQjnImpM
  * [18] Lin, A.C., Lee, C.S., Blazes, M., Lee, A.Y., Gorin, M.B.: Assessing the clinical utility of expanded macular octs using machine learning. Translational vision science & technology 10(6), 32–32 (2021)
  * [19] Rivail, A., Schmidt-Erfurth, U., Vogl, W.D., Waldstein, S.M., Riedl, S., Grechenig, C., Wu, Z., Bogunovic, H.: Modeling disease progression in retinal octs with longitudinal self-supervised learning. In: International Workshop on PRedictive Intelligence In MEdicine. pp. 44–52. Springer (2019)
  * [20] Russakoff, D.B., Lamin, A., Oakley, J.D., Dubis, A.M., Sivaprasad, S.: Deep learning for prediction of amd progression: a pilot study. Investigative ophthalmology & visual science 60(2), 712–722 (2019)
  * [21] Veeling, B.S., Linmans, J., Winkens, J., Cohen, T., Welling, M.: Rotation equivariant cnns for digital pathology. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2018: 21st International Conference, Granada, Spain, September 16-20, 2018, Proceedings, Part II 11. pp. 210–218. Springer (2018)
  * [22] Xiao, T., Wang, X., Efros, A.A., Darrell, T.: What should not be contrastive in contrastive learning. In: International Conference on Learning Representations (2020)
  * [23] Xu, D., Xiao, J., Zhao, Z., Shao, J., Xie, D., Zhuang, Y.: Self-supervised spatiotemporal learning via video clip order prediction. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. pp. 10334–10343 (2019)
  * [24] Yan, Q., Weeks, D.E., Xin, H., Swaroop, A., Chew, E.Y., Huang, H., Ding, Y., Chen, W.: Deep-learning-based prediction of late age-related macular degeneration progression. Nature machine intelligence 2(2), 141–150 (2020)
  * [25] Yim, J., Chopra, R., Spitz, T., Winkens, J., Obika, A., Kelly, C., Askham, H., Lukic, M., Huemer, J., Fasler, K., et al.: Predicting conversion to wet age-related macular degeneration using deep learning. Nature Medicine 26(6), 892–899 (2020)
  * [26] Zbontar, J., Jing, L., Misra, I., LeCun, Y., Deny, S.: Barlow twins: Self-supervised learning via redundancy reduction. In: International Conference on Machine Learning. pp. 12310–12320. PMLR (2021)
  * [27] Zhao, Q., Liu, Z., Adeli, E., Pohl, K.M.: Longitudinal self-supervised learning. Medical Image Analysis 71, 102051 (2021). https://doi.org/https://doi.org/10.1016/j.media.2021.102051, https://www.sciencedirect.com/science/article/pii/S1361841521000979

##  Appendix 0.A Appendix

Table 2: Contrastive Augmentations and OCT Details Contrastive Transformations | Parameter
---|---
Random Crop & Resize (percentage) | 0.4 - 0.8
Random Horizontal Flip (probability) | 0.5
Random Color Jittering (probability | 0.8
Random Gaussian Blur (kernel size) | 21
Random Solarize (threshold) | 0.5
Random Rotation (degrees) | $\pm 5$
Random Translation (percentage) |  $\pm$0.05
Input Time Difference | 1-12 Months
Normalization Mean & std | (0.202, 0.113)
OCT Scanner | Cirrus OCT
Resolution |  $6\times 6\times 2$ mm3
Projector MLP dimensions | 4096-4096-4096
Predictor $\rho_{\psi}$ MLP dimensions | 2049-2048
GPU Used | Nvidia A100 80GB

###  0.A.1 Non-Contrastive VicReg Loss

VICReg loss with invariance, variance, and covariance terms, as follows:

$\displaystyle\ell_{\mathrm{contr}}(Z_{t},Z_{t+\Delta t})$
$\displaystyle=\lambda_{S}\cdot S(Z_{t},Z_{t+\Delta
t})+\lambda_{V}\cdot(V(Z_{t})+V(Z_{t+\Delta t}))$ (6)
$\displaystyle+\lambda_{C}\cdot(C(Z_{t})+C(Z_{t+\Delta t}))$ $\displaystyle
S(Z,Z^{\prime})$
$\displaystyle=\frac{1}{n}\sum_{i}\|z_{i}-z^{\prime}_{i}\|_{2}^{2}\quad
V(Z)=\frac{1}{d}\sum_{j=1}^{d}max(0,1-\mathrm{std}(z^{j},\epsilon))$
$\displaystyle C(Z)$ $\displaystyle=\frac{1}{d}\sum_{i\neq
j}[Cov(Z)]_{i,j}^{2},$ $\displaystyle\textrm{where}\ \ \ Cov(Z)$
$\displaystyle=\frac{1}{n-1}\sum_{i=1}^{n}(z_{i}-\bar{z})(z_{i}-\bar{z})^{T}.$

$\lambda_{S}$, $\lambda_{V}$, and $\lambda_{C}$ are set to 15, 25, 5 to bring
their magnitude in the same range.

###  0.A.2 Derivation of Displacement Map Regularization Term

The pairwise RankNet [4] loss is based on cross-entropy loss for calculating
the score-based ranking a pair of elements. Let $f$ be a scoring function for
the items $x_{i}$ and $x_{j}$, such that their respective scores are
$s_{i}=f(x_{i})$ and $s_{j}=f(x_{j})$. Concordantly the score difference is
defined as $s_{ij}=s_{i}-s_{j}$. Then, the probability of ranking $x_{i}$
greater than $x_{j}$ is defined using the logistic function:

$\mathcal{P}_{ij}=\frac{e^{s_{ij}}}{1+e^{s_{ij}}}=\frac{1}{1+e^{-s_{ij}}}$ (7)

When ranking the items, there are 3 values for ground truth $Y$; 1 when
$x_{i}$ has higher rank than $x_{j}$, 0 when the relation is reversed, and
$\frac{1}{2}$ when both items have the same rank. Accordingly, cross-entropy
loss for correctly ranking the items is:

$\mathcal{L}_{ce}=-Y\cdot\log(\mathcal{P}_{ij})-(1-Y)\cdot\log(1-\mathcal{P}_{ij})$
(8)

In TC, $s_{i}$ becomes $r_{t+\Delta t}$ and $s_{j}$ becomes $r_{t}$, hence
$s_{ij}$ is the DM. For the cross-entropy loss, we implemented $s_{ij}$ as the
norm of the DM. Also the ranking class is always $1$ because the time
difference is kept positive, thus the Eq. 8 becomes:

$\displaystyle\mathcal{L}_{reg}$
$\displaystyle=-1\cdot\log(\mathcal{P}_{ij})-(0)\cdot\log(1-\mathcal{P}_{ij})=-\log\left(\frac{1}{1+e^{-s_{ij}}}\right)$
(9)
$\displaystyle=\log(1+e^{-s_{ij}})=\log\left(1+\exp(-||\rho_{\psi}(r_{t},\Delta
t)||_{2})\right)$

Table 3: Computational Costs of Each Model Model | Batch Update (Seconds) | # Trainable Parameters
---|---|---
VICReg | 9.3 | 65.4M
ESSL | 16.3 | 78.0M
AugSelf | 14.2 | 78.0M
EquiMod | 12.6 | 124.2M
TC | 10.4 | 82.2M
Figure 3: Distance rankings between the original reprensentation $r_{0}$ and
its propoageted prediction $r^{\prime}_{t+\Delta t}$ for 9 consecutive months
for 40 patients.


††*Corresponding author

# Zero order meromorphic solutions of $q$-difference equations of Malmquist
type

Risto Korhonen  Department of Physics and Mathematics, University of Eastern
Finland, P.O. Box 111, FI-80101 Joensuu, Finland,
<EMAIL_ADDRESS>and  Yueyang Zhang∗ School of Mathematics and Physics,
University of Science and Technology Beijing, No. 30 Xueyuan Road, Haidian,
Beijing, 100083, P.R. China<EMAIL_ADDRESS>
###### Abstract.

We consider the first order $q$-difference equation

(†) $f(qz)^{n}=R(z,f),$

where $q\not=0,1$ is a constant and $R(z,f)$ is rational in both arguments.
When $|q|\not=1$, we show that, if $({\dagger})$ has a zero order
transcendental meromorphic solution, then $({\dagger})$ reduces to a
$q$-difference linear or Riccati equation, or to an equation that can be
transformed to a $q$-difference Riccati equation. In the autonomous case,
explicit meromorphic solutions of $({\dagger})$ are presented. Given that
$({\dagger})$ can be transformed into a difference equation, we proceed to
discuss the growth of the composite function $f(\omega(z))$, where $\omega(z)$
is an entire function satisfying $\omega(z+1)=q\omega(z)$, and demonstrate how
the proposed difference Painlevé property, as discussed in the literature,
applies for $q$-difference equations.

###### Key words and phrases:

$q$-difference equations; Zero order meromorphic solutions; Growth

###### 2010 Mathematics Subject Classification:

Primary 39A13; Secondary 30D35 and 39A12

The second author is supported by the Fundamental Research Funds for the
Central Universities (FRF-TP-19-055A1) and a Project supported by the National
Natural Science Foundation of China (12301091)

##  1\. Introduction

An ordinary differential equation is said to possess the Painlevé property if
all of its solutions are single-valued about all movable singularities. It is
widely believed that all ordinary differential equations that possess the
Painlevé property are integrable. In [1], Ablowitz, Halburd and Herbst
introduced Nevanlinna theory (see, e.g., [16]) to study the integrability of
difference equations and suggested that the existence of sufficiently many
finite-order meromorphic solutions of a difference equation is a good
candidate for a difference analogue of the Painlevé property. Halburd and
Korhonen [14] implemented this idea to the second order difference equation

(1.1) $f(z+1)+f(z-1)=R(z,f),$

where $R(z,f)$ is rational in $f$ with meromorphic coefficients having slower
growth than $f(z)$ in terms of Nevanlinna theory, and showed that if equation
(1.1) has a meromorphic solution $f$ of finite order, then either $f$
satisfies a difference Riccati equation, or a linear transformation of (1.1)
reduces it to one in a short list of difference equations which consists
solely of difference Painlevé equations and equations related to them, linear
equations and linearizable equations. It was shown that the finite-order
condition of the proposed difference Painlevé property can be relaxed to
hyper-order strictly less than one in [15], and later to minimal hyper-type
(i.e. $\log T(r,f)=o(r)$ as $r\to\infty$) in [18, 30].

When detecting the integrability of $q$-difference equation, it is natural to
consider the zero order meromorphic solutions. One of the main purposes of
this paper is to display how the proposed difference Painlevé property works
for $q$-difference equations. We shall consider the first order $q$-difference
equation

(1.2) $f(qz)^{n}=R(z,f),$

where $n\in\mathbb{N}$, $q\not=0,1$ is a constant and $R(z,f)$ is rational in
both arguments. Equation (1.2) is the $q$-difference version of the first
order difference equation

(1.3) $g(z+1)^{n}=R(z,g),$

where $n\in\mathbb{N}$ and $R(z,g)$ is rational in $g$ with meromorphic
coefficients having slower growth than $g(z)$. In fact, we may perform the
transformation $z\to\omega(z)$ to equation (1.2) with an entire function
$\omega(z)$ satisfying $\omega(z+1)=q\omega(z)$. We may choose
$\omega(z)=\kappa(z)e^{z\log q}$ with an arbitrary entire function $\kappa(z)$
such that $\kappa(qz)=\kappa(z)$. Denoting $g(z)=f(\omega(z))$, then
$g(z+1)=f(q\omega(z))$, and $g(z)$ is a meromorphic solution of the difference
equation

(1.4) $g(z+1)^{n}=\hat{R}(z,g),$

where $\hat{R}(z,g)$ is now rational in $g$ with coefficients that are
rational in $\omega(z)$. According to a result of Edrei and Fuchs [9] (see
also [16, Lemma 2.6]), we have

(1.5)
$\begin{split}\lim_{r\to\infty}\frac{T(r,f(\omega(z)))}{T(r,\omega(z))}=\infty.\end{split}$

Since, by the Valiron–Mohon’ko identity [24, 22] (see also [21]), the
characteristic function of each coefficient of $\hat{R}(z,g)$, say
$\hat{a}(z)$, satisfies $T(r,\hat{a}(z))=\hat{n}T(r,\omega(z))+O(\log r)$,
where $\hat{n}$ is some integer, all coefficients of (1.4) have growth of type
$o(T(r,g))$ when the solution $f(z)$ of (1.2) is transcendental. Under the
condition that each coefficient of (1.3), say $\tilde{a}(z)$, satisfies
$T(r,\tilde{a}(z+k))=o(T(r,g))$ for any finite positive integer $k$, where
$r\to\infty$ outside an exceptional set of finite logarithmic measure (i.e.
$\int_{E}dr/r<\infty$), the present authors [19, 29, 20] used Yamanoi’s Second
Main Theorem [25, 26] to provide a classification for equation (1.3),
including a list of $33$ equations of canonical form. In particular, in the
autonomous case all these equations can be solved in terms of elliptic
functions, exponential type functions or functions which are solutions to a
certain autonomous first-order difference equation. These results provide a
complete difference analogue of the results on Malmquist type differential
equations $f^{\prime}(z)^{n}=R(z,f)$ due to Steinmetz [23] and Bank and
Kaufman [3]; see also [21, Chapter 10].

Now, if equation (1.2) has a zero order transcendental meromorphic solution
$f$, then it follows from an estimate for the proximity function
$m(r,f(qz)/f(z))$ in [4] (see also [28]) and the Valiron–Mohon’ko identity
that

$\begin{split}\deg_{f}(R(z,f))T(r,f)&=T(r,R(z,f))+O(\log r)\\\
&=T(r,f(qz)^{n})+O(\log r)\\\ &=nT(r,f(qz))+O(\log r)\\\
&=nT(r,f)+o(T(r,f)),\end{split}$

where $r\to\infty$ on a set of logarithmic density 1 (i.e.
$\lim_{r\to\infty}(\int_{E\cap[0,r]}dt/t)/\log r=1$). This implies that
$\deg_{f}(R(z,f))=n$. Together with previous discussions, we now discard the
assumption that $f(z)$ is of zero order and conclude from the results in [19,
20] for equation (1.3) with $\deg_{f}(R(z,f))=n$ that: If the $q$-difference
equation (1.2) has a transcendental meromorphic solution $f$, then either $f$
satisfies a $q$-difference linear or Riccati equation

(1.6) $\displaystyle f(qz)$ $\displaystyle=$ $\displaystyle
a_{1}(z)f(z)+a_{2}(z),$ (1.7) $\displaystyle f(qz)$ $\displaystyle=$
$\displaystyle\frac{b_{1}(z)f(z)+b_{2}(z)}{f(z)+b_{3}(z)},$

where $a_{i}(z)$ and $b_{j}(z)$ are rational functions, or, by implementing a
transformation $f\rightarrow\alpha f$ or $f\rightarrow 1/(\alpha f)$ with an
algebraic function $\alpha$ of degree at most $3$, (1.2) reduces to one of the
following equations

(1.8) $\displaystyle f(qz)^{2}$ $\displaystyle=$ $\displaystyle 1-f(z)^{2},$
(1.9) $\displaystyle f(qz)^{2}$ $\displaystyle=$ $\displaystyle
1-\left(\frac{\delta(z)f(z)-1}{f(z)-\delta(z)}\right)^{2},$ (1.10)
$\displaystyle f(qz)^{2}$ $\displaystyle=$ $\displaystyle
1-\left(\frac{f(z)+3}{f(z)-1}\right)^{2},$ (1.11) $\displaystyle f(qz)^{2}$
$\displaystyle=$ $\displaystyle\frac{f(z)^{2}-\kappa(z)}{f(z)^{2}-1},$ (1.12)
$\displaystyle f(qz)^{3}$ $\displaystyle=$ $\displaystyle 1-f(z)^{-3},$

where $\delta(z)\not\equiv\pm 1$ is an algebraic function of degree at most
$2$ and $\kappa(z)\not\equiv 0,1$ is a rational function such that
$\kappa(qz)=\kappa(z)$, or to one of the following equations

(1.13) $\displaystyle f(qz)^{2}$ $\displaystyle=$
$\displaystyle\delta_{1}(z)(f(z)^{2}-1),$ (1.14) $\displaystyle f(qz)^{2}$
$\displaystyle=$ $\displaystyle\delta_{2}(z)(1-f(z)^{-2}),$ (1.15)
$\displaystyle f(qz)^{2}$ $\displaystyle=$
$\displaystyle\frac{\delta_{3}(qz)f(z)^{2}-1}{f(z)^{2}-1},$ (1.16)
$\displaystyle f(qz)^{2}$ $\displaystyle=$
$\displaystyle\theta\frac{f(z)^{2}-\delta_{4}(z)f(z)+1}{f(z)^{2}+\delta_{4}(z)f(z)+1},$
(1.17) $\displaystyle f(qz)^{3}$ $\displaystyle=$ $\displaystyle 1-f(z)^{3},$
(1.18) $\displaystyle f(qz)^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\frac{(1+\delta_{5}(z))^{2}}{1+\delta_{5}(z)^{2}}\frac{(f(z)-1)(f(z)-\delta_{5}(z)^{2})}{(f(z)-\delta_{5}(z))^{2}},$

where $\theta=\pm 1$ and $\delta_{1}$, $\delta_{2}$, $\delta_{3}$,
$\delta_{4}$ and $\delta_{5}$ are rational or algebraic functions, each of
which satisfying

(1.19) $\displaystyle\delta_{1}(qz)(\delta_{1}(z)+1)+1$ $\displaystyle=$
$\displaystyle 0,$ (1.20) $\displaystyle\delta_{2}(qz)\delta_{2}(z)$
$\displaystyle=$ $\displaystyle\delta_{2}(qz)+\delta_{2}(z),$ (1.21)
$\displaystyle\delta_{3}(qz)\delta_{3}(z)$ $\displaystyle=$ $\displaystyle 1,$
(1.22) $\displaystyle\delta_{4}(qz)(\delta_{4}(z)-4)$ $\displaystyle=$
$\displaystyle 2(1-\theta)\delta_{4}(z)-8(1+\theta),$ (1.23) $\displaystyle
8\delta_{5}^{4}(qz)(\delta_{5}^{2}(z)+1)\delta_{5}(z)$ $\displaystyle=$
$\displaystyle(\delta_{5}(z)+1)^{4},$

respectively. From the proof in [19, 20] we know that solutions of (1.10) are
still meromorphic after the transformation $f\to\alpha f$ or $f\to 1/(\alpha
f)$. Concerning zero order meromorphic solutions of (1.2), we shall prove the
following

######  Theorem 1.1.

Suppose that $|q|\not=1$ in (1.2). If equation (1.2) has a zero order
transcendental meromorphic solution, then either $f$ satisfies (1.6) or (1.7)
with rational coefficients, or, by a transformation $f\rightarrow\alpha f$ or
$f\rightarrow 1/(\alpha f)$ with an algebraic function $\alpha$ of degree at
most $2$, (1.2) reduces to (1.9).

By combining the results in the first two parts of Section 2, we have actually
shown that, if $|q|\not=1$, then equation (1.2) can have transcendental
meromorphic solutions only in the case of the four equations (1.6), (1.7),
(1.9) and (1.18).

The rather simple $q$-difference equation $f(z)-a(z)f(qz)=0$, where $a(z)$ is
a nonconstant polynomial, has a zero order transcendental entire solution [7,
Theorem 1]. In fact, Bergweiler, Ishizaki and Yanagihara [6] have previously
shown that all meromorphic solutions of the $q$-difference equation
$\sum_{j=0}^{m}a_{j}(z)g(q^{i}z)=Q(z)$, where $q\in\mathbb{\mathbb{C}}$,
$0<|q|<1$, and $Q$ and the $a_{j}$ are polynomials, are of zero order. They
also showed that, under a suitable assumption on $a_{0}(z)$ and $a_{1}(z)$,
the $q$-difference equation $g(q^{2}z)+a_{1}(z)g(qz)+a_{0}(z)g(z)=0$ has a
transcendental meromorphic solution $g(z)$. Letting $f(z)=g(qz)/g(z)$, then
$f(z)$ satisfies the $q$-difference Riccati equation
$f(qz)=-[a_{1}(z)f(z)+a_{0}(z)]/f(z)$. With this result we may construct zero
order meromorphic solutions of equation (1.9) for nonconstant $\delta$. From
[19] we know that the solution $f(z)$ of equation (1.9) is represented as
$f(z)=(\gamma(z)+\gamma(z)^{-1})/2$, where $\gamma(z)$ satisfies a
$q$-difference Riccati equation

(1.24) $\gamma(qz)=\left\\{-\theta\frac{(\pm
i\delta(z)-\sqrt{1-\delta(z)^{2}})\gamma(z)\pm i}{\gamma(z)-\delta(z)\pm
i\sqrt{1-\delta(z)^{2}}}\right\\}^{\theta},\quad\theta=\pm 1.$

Consider the case $\theta=-1$ and $\pm$ is chosen to be $-$. We choose
$\delta=\frac{1}{2}(\alpha(z)+\alpha(z)^{-1})$ with a suitable polynomial
$\alpha(z)$ so that $a_{1}(z)=-(\alpha(qz)+i\alpha(z))$ and
$a_{0}(z)=2i\alpha(z)$. Then, by letting $\Gamma(z)=\gamma(z)+\alpha(z)$, we
have

(1.25)
$\Gamma(qz)=\frac{(\alpha(qz)+i\alpha(z))\Gamma(z)-2i\alpha(z)^{2}}{\Gamma(z)}.$

In the autonomous case, the meromorphic solutions of the $q$-difference linear
and Riccati equations (1.6) and (1.7) have been clearly characterised in [13].
In Section 2 we shall present some explicit meromorphic solutions for these
two equations.

In the proof of Theorem 1.1, we show that $\delta_{5}(z)$ in (1.18) must be a
constant. Then we can determine all nonconstant meromorphic solutions of
equation (1.18), which are actually of positive even order. However, in the
case of the seven equations (1.11)–(1.17), the existence of transcendental
meromorphic solutions does not necessarily imply that the coefficients of
(1.2) are constants. Nonetheless, when all coefficients of equation (1.2) are
constants, we may remove the condition $|q|\not=1$ in Theorem 1.1 and obtain
the following

######  Corollary 1.2.

If equation (1.2) with constant coefficients has a nonconstant zero order
meromorphic solution, then either $f$ satisfies (1.6) or (1.7) with constant
coefficients, or, by a transformation $f\rightarrow\alpha f$ or $f\rightarrow
1/(\alpha f)$ with a constant $\alpha$, (1.2) reduces to equations (1.8),
(1.9), (1.10) with constant coefficients.

We remark that, when $|q|=1$, equation (1.2) with nonconstant rational
coefficients can also have zero order transcendental meromorphic solutions in
the case of equation (1.8). From [19] we know that the solution $f(z)$ of
(1.8) is represented as $f(z)=(h(z)+h(z)^{-1})/2$, where $h(z)$ satisfies
$h(qz)=ih(z)^{\theta}$, and $\theta=\pm 1$. If $f(z)$ is written as
$f(z)=g(z)/\alpha(z)$ for an algebraic function $\alpha(z)$ such that
$\alpha(z)^{2}$ is a rational function and $\alpha(qz)^{2}=\alpha(z)^{2}$,
then we see that $h(z)$ takes the form $h(z)=w(z)/\alpha(z)$ for a meromorphic
function $w(z)$ such that $w(qz)/\alpha(z)=i(w(z)/\alpha(z))^{\theta}$. This
equation can have zero order transcendental meromorphic solutions when
$|q|=1$.

This paper is structured as follows. In Section 2, we first present
meromorphic solutions for equation (1.2) in the autonomous case. The $11$
equations (1.8)–(1.18) are solved in terms of rational functions or elliptic
functions. Together with these solutions, we then provide a proof for Theorem
1.1. Since equation (1.2) can be transformed to an equation of the form in
(1.3), in the final part of Section 2, we discuss the relationship between
solutions of equation (1.2) and (1.3) by analysing the growth of the composite
function $g(z)=f(\omega(z))$, where $\omega(z)$ is an entire function such
that $\omega(z+1)=q\omega(z)$. We point out that the existence of sufficiently
many zero order meromorphic solutions of the $q$-difference equation (1.2)
implies the existence of sufficiently many minimal hyper-type meromorphic
solutions of the difference equation (1.3). Finally, in Section 3 we make two
comments on the integrability of $q$-difference equations.

##  2\. Zero order meromorphic solutions of equation (1.2)

###  2.1. Meromorphic solutions of (1.2) in the autonomous case

In this section we present meromorphic solutions of equation (1.2) in the
autonomous case. In particular, we will show that the five equations
(1.6)–(1.10) indeed have zero order meromorphic solutions and nonconstant
meromorphic solutions of the eight equations (1.11)–(1.18) have even positive
order or infinite order.

For the autonomous $q$-difference linear equation (1.6): $f(qz)=af(z)+b$,
where $a,b$ are constants, the general solution is $f(z)=z^{\log a/\log
q}+b/(1-a)$ when $|a|\not=1$; or $f(z)=\kappa(z)z^{\log a/\log q}+b/(1-a)$
when $|a|=1$ but $a\not=1$, where $\kappa(z)$ is an arbitrary function such
that $\kappa(qz)=\kappa(z)$; or $f(z)=\kappa(z)+b\log z/\log q$ when $a=1$,
where $\kappa(z)$ is an arbitrary function such that $\kappa(qz)=\kappa(z)$.
When $a\not=1$, if the solution $f(z)$ is meromorphic, then the constant $q$
should satisfy $\log a/\log q=k$ for an integer $k$; when $a=1$, if the
solution $f(z)$ is meromorphic, then the coefficient $b$ should vanish. In
particular, we may choose $\kappa(z)$ to be a zero order transcendental
meromorphic (entire) function. We remark that, if $|q|\not=1$ or $|q|=1$ and
$q=e^{2\pi i\theta}$ for an irrational number $\theta\in[0,1)$, then the
equation $\kappa(qz)=\kappa(z)$ does not have any non-rational zero order
meromorphic solutions. This is due to the fact that any possible zeros or
poles of the solution accumulate (so they must be Picard exceptional) and a
nonconstant zero order meromorphic function has at most one Picard exceptional
value.

For the autonomous $q$-difference Riccati equation (1.7), by a result of
Ishizaki [17], after doing certain transformations we obtain the
$q$-difference Riccati equation

(2.1) $\begin{split}f(qz)=\frac{f(z)+A}{1-f(z)}\end{split}$

or

(2.2) $\begin{split}f(qz)=\frac{B}{f(z)},\end{split}$

where $A$ and $B\not=0$ are two constants. For equation (2.1), let $a_{1}$ and
$a_{2}$ be two nonzero constants and $b=-2a_{1}a_{2}/(a_{1}^{2}+a_{2}^{2})$.
When $A=-b^{2}$ and $a$ is a constant such that
$e^{ia}=(a_{1}-a_{2})^{2}/(a_{1}+a_{2})^{2}$, we have the solution

(2.3)
$f(z)=-\left(\frac{2a_{1}a_{2}}{a_{1}^{2}+a_{2}^{2}}\right)\frac{\kappa(z)z^{a/\log
q}-a_{1}}{\kappa(z)z^{a/\log q}+a_{2}},$

where $\kappa(z)$ is an arbitrary function such that $\kappa(qz)=\kappa(z)$.
For equation (2.2), letting $a_{3}$ be a nonzero constant, then we have the
solution $f(z)=a_{3}Bw(z)/w(qz)$, where $w(z)$ is an arbitrary entire function
such that $w(q^{2}z)=a_{3}^{2}B^{2}w(z)$. Obviously, we may choose $\kappa(z)$
or $w(z)$ to be a transcendental meromorphic function of zero order, as
mentioned in previous discussions on equation (1.6).

Look at the three equations (1.8), (1.9) and (1.10). They are so-called Fermat
$q$-difference equations. Recall that a Fermat equation is a function analogue
of the Fermat diophantine equation $x^{n}+y^{n}=1$, i.e.,
$x(z)^{n}+y(z)^{n}=1$, where $n\geq 2$ is an integer. Meromorphic solutions to
Fermat equations have been clearly characterized; see [2, 12, 11], for
example. Solutions of the three equations (1.8), (1.9) and (1.10) are solved
in a similar way as in [19]. For equation (1.8), the solution $f(z)$ is
represented as $f(z)=(\beta(z)+\beta(z)^{-1})/2$, where $\beta(z)$ satisfies
$\beta(qz)=i\beta(z)^{\pm 1}$. For equation (1.9), the solution $f(z)$ is
represented as $f(z)=(\gamma(z)+\gamma(z)^{-1})/2$, where $\gamma(z)$
satisfies

(2.4)
$\gamma(qz)=\left\\{-\theta\frac{(i\delta-\sqrt{1-\delta^{2}})\gamma(z)+i}{\gamma(z)-\delta+i\sqrt{1-\delta^{2}}}\right\\}^{\theta},\quad\theta=\pm
1,$

where $\delta$ is a constant. Moreover, based on the discussions in [20] we
know that equation (2.4) can be transformed into equation (2.1) when
$2\delta^{2}\not=1$ and can be transformed into equation (2.2) when
$2\delta^{2}=1$. For equation (1.10), the solution $f(z)$ is represented as
$f(z)=\frac{8\lambda(z)^{2}-(\lambda(z)^{2}+1)^{2}}{(\lambda(z)^{2}+1)^{2}}$,
where $\lambda(z)$ satisfies

(2.5)
$\lambda(qz)=\left\\{-\theta\frac{-(1+\sqrt{2})\lambda(z)+i}{\lambda(z)-i+i\sqrt{2}}\right\\}^{\theta},\quad\theta=\pm
1.$

Equation (2.5) can be transformed into equation (2.1). Thus meromorphic
solutions of the three equations (1.8), (1.9) and (1.10) are solved in terms
of functions which are solutions of certain $q$-difference linear or Riccati
equation.

Now look at the six equations (1.11), (1.13), (1.14), (1.15), (1.16) and
(1.18). The nonconstant solutions of these equation are Jacobi elliptic
functions composed with entire functions.

We first take the equation (1.18) as an example and show that all nonconstant
meromorphic solutions of this equation have even positive order. From the
proof in [20] we know that a meromorphic solution $f(z)$ of equation (1.18) is
twofold ramified over each of $\pm 1,\pm\delta_{5}^{2}$. Then there exists an
entire function $\phi(z)$ such that $f(z)$ is written as
$f(z)=\text{sn}(\phi(z))$, where
$\text{sn}(\phi)=\text{sn}(\phi,1/\delta_{5}^{2})$ is the Jacobi elliptic
function with modulus $1/\delta_{5}^{2}$ and satisfies the first order
differential equation
$\text{sn}^{\prime}(\phi)^{2}=(1-\text{sn}(\phi)^{2})(1-\text{sn}(\phi)^{2}/\delta_{5}^{4})$.
Let $z_{0}$ be a point such that $f(z_{0})=\text{sn}(\phi(z_{0}))=1$. It
follows from (1.18) that $f(qz_{0})=\text{sn}(\phi(qz_{0}))=0$. Using the
differential equation
$\text{sn}^{\prime}(\phi)^{2}=(1-\text{sn}(\phi)^{2})(1-\text{sn}(\phi)^{2}/\delta_{5}^{4})$
to compute the Taylor series for $\text{sn}(\phi)$ and $\text{sn}(\phi(qz))$
around the point $z_{0}$, respectively, we get

(2.6)
$\begin{split}\text{sn}(\phi(z))&=1-\frac{1}{2}\frac{\delta_{5}^{4}-1}{\delta_{5}^{4}}(\phi(z)-\phi(z_{0}))^{2}+\cdots\\\
&=1-\frac{1}{2}\frac{\delta_{5}^{4}-1}{\delta_{5}^{4}}\phi^{\prime}(z_{0})^{2}(z-z_{0})^{2}+\cdots\end{split}$

and

(2.7)
$\begin{split}\text{sn}(\phi(qz))=\phi(qz)-\phi(qz_{0})+\cdots=q\phi^{\prime}(qz_{0})(z-z_{0})+\cdots.\end{split}$

By substituting the above two expressions into (1.18) and then comparing the
second-degree terms on both sides of the resulting equation, we find

(2.8)
$\begin{split}q^{2}\phi^{\prime}(qz_{0})^{2}=\frac{1}{4}\frac{(1+\delta_{5})^{4}}{\delta_{5}^{4}}\phi^{\prime}(z_{0})^{2}.\end{split}$

Define
$G(z):=q^{2}\phi^{\prime}(qz)^{2}-\frac{1}{4}\frac{(1+\delta_{5})^{4}}{\delta_{5}^{4}}\phi^{\prime}(z)^{2}$.
Similarly as in [20], we can use (1.5) to prove that equation (2.8) holds for
all $z\in\mathbb{C}$, so that $G(z)\equiv 0$. It follows by integration that

(2.9) $\begin{split}\phi(qz)=b_{1}\phi(z)+b_{2},\ \ \
b_{1}=\pm\frac{1}{2}\frac{(1+\delta_{5})^{2}}{\delta_{5}^{2}}.\end{split}$

where $b_{2}$ is a constant. (The coefficient $b_{1}$ is different from the
one in equation (2.34) in [20]. The calculation there contains an error.)
Since $\delta_{5}\not=\pm i$, we see that $|b_{1}|\not=1$ and thus $\phi$ is a
polynomial of the form $B_{1}z^{k}+B_{2}$ with $k$ being a positive integer
such that $q^{k}=b_{1}$. Since $\text{sn}(z)$ has order 2, then by a result of
Bergweiler [8, Satz 3.2] we have that the order of $f(z)=\text{sn}(\phi(z))$
is equal to $2k$.

Similarly, for each of the five equations (1.11), (1.13), (1.14), (1.15) and
(1.16), we can show that the nonconstant solutions are written as
$f(z)=\text{sn}(\phi_{1})=\text{sn}(\phi_{1},\delta)$, where
$\text{sn}(\phi_{1})$ is the Jacobi elliptic function with a certain modulus
$\delta$ and an entire function $\phi_{1}$ satisfying a $q$-difference linear
equation

(2.10) $\phi_{1}(qz)=b_{3}\phi_{1}(z)+b_{4},$

where $b_{3}$ and $b_{4}$ are two constants such that $|b_{3}|=1$. In
particular, for equation (1.11), we have $b_{3}=\pm 1$. We show that $b_{3}=1$
is impossible. Otherwise, from the previous discussions on equation (1.6) we
see that $b_{4}=0$. Recall the Maclaurin series for $\text{sn}\ \varepsilon$:

(2.11) $\begin{split}\text{sn}\
\varepsilon&=\varepsilon-(1+\delta_{0}^{2})\frac{\varepsilon^{3}}{3!}+(1+14\delta_{0}^{2}+\delta_{0}^{4})\frac{\varepsilon^{5}}{5!}+\cdots.\end{split}$

Since $\kappa\not=0,1$ and $\text{sn}(\phi_{1}(qz))=\text{sn}(\phi_{1}(z))$,
then substitution into equation (1.11) yields a contradiction. If $\phi_{1}$
is a polynomial, then $f(z)$ has finite order $2k_{1}$ for some positive
integer $k_{1}$; if $\phi_{1}$ is transcendental, then $f(z)$ has infinite
order by a result of [16, Theorem 2.9].

Then look at the third degree Fermat $q$-difference equations (1.12) and
(1.17). From [2, 12, 11] we know that all meromorphic solutions of the
equation $x(z)^{3}+y(z)^{3}=1$ can be represented as: $x=H(\varphi)$, $y=\eta
G(\varphi)=\eta H(-\varphi)=H(-\eta^{2}\varphi)$, where $\varphi=\varphi(z)$
is an entire function and $\eta$ is a cubic root of 1, and

(2.12) $H(z)=\frac{1+\wp^{\prime}(z)/\sqrt{3}}{2\wp(z)},\quad
G(z)=\frac{1-\wp^{\prime}(z)/\sqrt{3}}{2\wp(z)},$

where $\wp(z)$ is the particular Weierstrass elliptic function satisfying
$\wp^{\prime}(z)^{2}=4\wp(z)^{3}-1$. For equation (1.12), we have
$f(qz)=H(\varphi_{1})$ and $f(z)^{-1}=\eta G(\varphi_{1})$ with an entire
function $\varphi_{1}(z)$. An elementary series analysis on $f(z)$ as for
equation (1.18) shows that $\varphi_{1}(z)$ satisfies a first order
$q$-difference linear equation

(2.13) $\varphi_{1}(qz)=c_{1}\varphi_{1}(z)+c_{2},$

where $c_{1}$ and $c_{2}$ are constants such that $c_{1}^{3}=1$. Moreover, by
the same arguments as for equation (1.11), we may show that $c_{1}\not=1$. For
equation (1.17), we have $f(qz)=H(\varphi_{2})$ and $f(z)=\eta G(\varphi_{2})$
with an entire function $\varphi_{2}$. An elementary series analysis on $f(z)$
as for equation (1.18) shows that $\varphi_{2}(z)$ satisfies a first order
$q$-difference linear equation

(2.14) $\varphi_{2}(qz)=c_{3}\varphi_{2}(z)+c_{4},$

where $c_{3}$ and $c_{4}$ are constants such that $c_{2}^{3}=-1$. Note that
the Weierstrass elliptic function $\wp(z)$ has order $2$. If $\varphi_{1}$ (or
$\varphi_{2}$) is a polynomial, then the solution $f(z)$ of (1.12) (or (1.17))
has finite order $2m$ for some positive integer $m$; if $\varphi_{1}$ (or
$\varphi_{2}$) is transcendental, then the solution $f(z)$ of (1.12) (or
(1.17)) has infinite order.

Finally, we show how to determine the nonconstant rational solutions of the
five equations (1.6)–(1.10) in the autonomous case. From the remarks in [10,
Section 2] we know that the autonomous $q$-difference linear and Riccati
equations (1.6) and (1.7) have only rational solutions when $|q|\not=1$. We
see that the autonomous $q$-difference linear equation (1.6) can have
nonconstant rational solutions in both of the two cases $|q|\not=1$ and
$|q|=1$. Below we shall show that equation (1.7) has no nonconstant rational
solutions when $|q|\not=1$.

We only need to consider the two $q$-difference equations (2.1) and (2.2).
Suppose that equation (2.1) has a nonconstant rational solution $f(z)$. We may
write $f(z)$ as $f(z)=P(z)/Q(z)$, where $P(z)$ and $Q(z)$ are two coprime
polynomials of degrees $s_{1}$ and $s_{2}$ respectively. Denote the leading
coefficients of $P(z)$ and $Q(z)$ by $a$ and $1$, respectively. Note that
$A\not=-1$. When $A\not=0$, it is easy to see that $P(z)$ and $Q(z)$ have the
same degrees, say $s$, and also that $P(0)\not=0$ and $Q(0)\not=0$. Since
$a\not=1$, then by substituting $f(z)=P(z)/Q(z)$ into (2.1), we may have

(2.15) $\begin{split}q^{-s}P(qz)&=(1-a)^{-1}[P(z)+AQ(z)],\\\
q^{-s}Q(qz)&=(1-a)^{-1}[-P(z)+Q(z)].\end{split}$

From the above two equations we get

(2.16)
$\begin{split}P(q^{2}z)=2(1-a)^{-1}q^{s}P(qz)-(1-a)^{-2}q^{2s}(1+A)P(z).\end{split}$

We write the formal form of $P(z)$:

(2.17) $\begin{split}P(z)=a_{s}z^{s}+a_{s-1}z^{s-1}+\cdots+a_{0},\end{split}$

where $a_{s}=a$ and $a_{0}\not=0$. By substituting $P(z)$ into (2.16) and then
comparing the coefficients of the resulting polynomials on both sides, we get

(2.18) $\begin{split}a_{s-i}=[2(1-a)^{-1}q^{i}-(1-a)^{-2}(1+A)q^{2i}]a_{s-i},\
\ i=0,1,\cdots,s.\end{split}$

By the two equations in (2.18) in the case $i=0$ and $i=s$, we obtain
$q^{s}=1$. We see that $a_{s}=a$ is determined by the equation
$2(1-a)^{-1}-(1-a)^{-2}(1+A)=1$. Further, denoting by $\hat{s}$ the largest
factor of $s$ such that $q^{\hat{s}}=1$, say $s/\hat{s}=\mathfrak{m}$, then
the coefficients $a_{\hat{s}},a_{2\hat{s}},\cdots,a_{\mathfrak{m}\hat{s}}$ can
be chosen arbitrarily while the other coefficients vanish. Then we can also
give the form of $Q(z)$ using the identities in (2.15). On the other hand,
when $A=0$, it is easy to see that $s_{1}\leq s_{2}$. If $s_{1}=s_{2}$, then
we have equations in (2.15) with $A=0$ and by comparing the leading
coefficients of the polynomials on both sides of the first equation in (2.15)
we get $a=0$, a contradiction. If $s_{1}<s_{2}$, then we have the following
two equations

(2.19) $\begin{split}q^{-s}P(qz)&=P(z),\\\
q^{-s}Q(qz)&=-P(z)+Q(z),\end{split}$

from which we get $P(qz)=q^{s}P(z)$ and
$Q(q^{2}z)=q^{s}Q(qz)-(q^{2s}-q^{s})Q(z)$. Since $Q(0)\not=0$, then by the
same arguments as in the case $A\not=0$ we get $q^{s}=1$. However, it follows
that $Q(qz)=Q(z)$, which is impossible.

Similarly, for equation (2.2), substitution of $f(z)=P(z)/Q(z)$, where $P(z)$
and $Q(z)$ are two coprime polynomials of the same degree $s$ and that
$P(0)\not=0$ and $Q(0)\not=0$, gives

(2.20) $\begin{split}q^{-s}P(qz)&=a^{-1}BQ(z),\\\
q^{-s}Q(qz)&=a^{-1}P(z).\end{split}$

It follows that $P(q^{2}z)=a^{-2}q^{2s}BP(z)$, which yields that $q^{2s}=1$.
Similarly as for equation (2.1), explicit forms for $P(z)$ and $Q(z)$ can be
obtained from the system of two equations in (2.20). Then all nonconstant
rational solutions of the five equations (1.8)–(1.10) with constant
coefficients can be obtained after doing some transformations.

###  2.2. Proof of Theorem 1.1

From the discussions in the introduction, we only need to consider the $13$
equations (1.6)–(1.18). Below we first consider the four equations (1.13),
(1.14), (1.15) and (1.16).

From the discussions in [19] we know that, if we let $w=f+1/f$ for a solution
$f$ of equation (1.16), then $w$ satisfies either an equation of the form in
(1.13) when $\theta=1$ or an equation of the form in (1.14) when $\theta=-1$.
Moreover, by the Valiron–Mohon’ko identity the characteristic function of $w$
satisfies $T(r,w)=2T(r,f)+O(1)$. Further, for equation (1.14), if we define a
function $g(z)$ to be such that $-g(z)^{2}=f^{2}(z)-1$, then $g(z)$ satisfies

(2.21)
$\begin{split}g(qz)^{2}=[\hat{\delta}_{2}(qz)g(z)^{2}-1]/(g(z)^{2}-1),\end{split}$

where $\hat{\delta}_{2}(z)$ satisfies
$\hat{\delta}_{2}(qz)\hat{\delta}_{2}(z)=1$. This is just equation (1.15).
Moreover, by the Valiron–Mohon’ko identity the characteristic function of
$f(z)^{2}$ satisfies $T(r,f^{2})=T(r,g^{2})+O(1)$. Therefore, we only need to
consider the two equations (1.13) and (1.14).

Consider equation (1.13). In general, under the assumptions of Theorem 1.1 the
solution $f(z)$ may have some algebraically branched points, but $f(z)^{2}$ is
a meromorphic function. By iterating equation (1.13) together with the
relation in (1.19) we find that

(2.22)
$\begin{split}f(q^{2}z)^{2}&=-\frac{\delta_{1}}{1+\delta_{1}}f^{2}(z)+1,\\\
f(q^{3}z)^{2}&=f(z)^{2}.\end{split}$

Then, at the origin $f(z)^{2}$ has a series expansion

(2.23)
$\begin{split}f(z)^{2}=a_{-k}z^{-k}+\cdots+a_{0}+a_{1}z+\cdots.\end{split}$

It follows that

(2.24)
$\begin{split}f(q^{3}z)^{2}=a_{-k}q^{-3k}z^{-k}+\cdots+a_{0}+a_{1}q^{3}z+\cdots\end{split}$

at the origin. From the above two series we conclude that $q^{3k}=1$ for some
integer $k$, which is a contradiction to our assumption. Also, for equation
(1.14), by iteration together with the relation in (1.20) we easily find that
$f(q^{4}z)^{2}=f(z)^{2}$. Then by the same arguments as for equation (1.13) we
have a contradiction. From the above reasoning, we conclude that in the case
of equations (1.13), (1.14), (1.15) and (1.16), equation (1.2) cannot have any
zero order transcendental meromorphic solutions when $|q|\not=1$. In fact,
when $|q|\not=1$, we see that these equations cannot have any transcendental
meromorphic solutions.

By the the same arguments as for equation (1.13) we can show that, in the case
of the four equations (1.8), (1.11), (1.12) and (1.17), equation (1.2) cannot
have any transcendental meromorphic solutions when $|q|\not=1$.

For equation (1.18), we recall from [20] that $\delta_{5}(z)\not\equiv 0,\pm
1,\pm i$. We shall show that $\delta_{5}(z)$ is a constant. Otherwise, from
the proof in [20] we know that $\delta_{5}(z)$ may be written as
$\delta_{5}(z)=\beta(z)/\alpha_{1}(z)$, where $\beta(z)$ is a rational
function and $\alpha_{1}(z)$ is in general an algebraic function, namely
$\alpha_{1}(z)$ satisfies an equation of the form
$a_{2}(z)\alpha(z)^{2}+a_{1}(z)\alpha(z)+a_{0}(z)=0$ with some polynomials
$a_{0}(z),a_{1}(z),a_{2}(z)$. Moreover, letting $\alpha_{2}(z)$ also satisfy
this equation, we have $\alpha_{2}(z)/\alpha_{1}(z)=\delta_{5}(z)^{2}$ and
$\delta_{5}(z)$ satisfies an equation of the form
$2a_{p}(z)/\alpha_{1}(qz)^{2}=(1+\delta(z))^{2}/(1+\delta(z)^{2})$, where
$a_{p}(z)$ is a nonzero rational function. Together with the relation in
(1.23), we find that $\beta(z)^{2}=\alpha_{1}(z)\alpha_{2}(z)$ and

(2.25)
$\begin{split}H_{2}(z)\delta_{5}(z)^{2}-H_{1}(z)\delta_{5}(z)+H_{2}(z)=0,\end{split}$

where $H_{1}(z)$ and $H_{2}(z)$ are two relatively prime polynomials such that
$H_{1}(z)/H_{2}(z)=2a_{p}(z)^{2}/\beta(qz)^{4}$. Then $\delta_{5}(z)$ is in
general an algebraic function defined on two-sheeted Riemann surface. From the
relation in (1.23) we see that $\delta_{5}(z)$ does not tend to $0$, $\pm i$
and also does not tend to $\infty$ as $z\to\infty$. This implies that the
degrees of $H_{1}(z)$ and $H_{2}(z)$ are the same. We may write
$\delta_{5}(z)$ as

(2.26)
$\begin{split}\delta_{5}(z)=\frac{G_{1}(z)}{G_{2}(z)}=\frac{H_{1}(z)+\sqrt{H_{1}(z)^{2}-4H_{2}(z)^{2}}}{2H_{2}(z)}.\end{split}$

Then we have from the relation in (1.23) that

(2.27)
$\begin{split}8\frac{G_{1}(qz)^{4}}{G_{2}(qz)^{4}}\cdot\frac{G_{1}(z)^{2}+G_{2}(z)^{2}}{G_{2}(z)^{2}}\cdot\frac{G_{1}(z)}{G_{2}(z)}=\frac{[G_{1}(z)+G_{2}(z)]^{4}}{G_{2}(z)^{4}}.\end{split}$

Let $z_{0}$ be a zero of $H_{2}(z)$ with least order. Then $\delta_{5}(z)$ has
an (algebraic) pole at the point $z_{0}$. From equation (2.27) we see that
$qz_{0}$ is also an (algebraic) pole of $\delta_{5}(z)$. However, by comparing
the order of the poles of the functions on both sides of (2.27), we get a
contradiction. Thus $\delta_{5}(z)$ must be a constant. Then from the
discussions in previous section we know that equation (1.18) cannot have zero
order transcendental meromorphic solutions.

Finally, for equation (1.10), by the results in [19] we know that the solution
$f(z)$ of equation (1.10) is meromorphic and is represented as
$f(z)=\frac{8\lambda(z)^{2}-(\lambda(z)^{2}+1)^{2}}{(\lambda(z)^{2}+1)^{2}}$,
where $\lambda(z)$ satisfies the $q$-difference Riccati equation in (2.5),
which can be transformed to equation (2.1). However, from previous discussion
we know that equation (2.1) can have only constant solutions when $|q|\not=1$.
Therefore, equation (1.10) does not have transcendental meromorphic solutions
when $|q|\not=1$. We complete the proof.

###  2.3. Growth of the composite function $f(\omega(z))$

Let $f(z)$ be a nonconstant meromorphic solution of equation (1.2) in the
autonomous case. In this section we discuss the growth of the composition
$g(z)=f(\omega(z))$, where $\omega(z)$ is an entire function such that
$\omega(z+1)=q\omega(z)$.

We first look at the equation (1.18). Then the function $g(z)=f(\omega(z))$
satisfies the difference equation

(2.28)
$g(z+1)^{2}=\frac{1}{2}\frac{(1+\delta_{5})^{2}}{1+\delta_{5}^{2}}\frac{(g(z)-1)(g(z)-\delta_{5}^{2})}{(g(z)-\delta_{5})^{2}},$

where $\delta_{5}$ is a constant such that
$8\delta_{5}^{4}(\delta_{5}^{2}+1)\delta_{5}=(\delta_{5}+1)^{4}$. Moreover,
from the proof in [20] we know that $g(z)$ is written as
$g(z)=\text{sn}(\varphi(z))$, where $\text{sn}(\varphi)$ is the Jacobi
elliptic function with modulus $1/\delta_{5}^{2}$ and $\varphi$ is an entire
function of $z$ satisfying the difference linear equation

(2.29) $\varphi(z+1)=c_{1}\varphi(z)+c_{2},\ \
c_{1}=\pm\frac{1}{2}\frac{(1+\delta_{5})^{2}}{\delta_{5}^{2}},$

where $c_{2}$ is a constant. Then $T(r,\varphi)\geq Lr$ for some $L>0$ and all
$r\geq r_{0}$ with some $r_{0}\geq 0$.

We estimate the characteristic function $T(r,g)$ using sn and $\varphi$ as in
[20, Lemma 2.7]. For the composite function $g(z)=\text{sn}(\varphi(z))$,
since $\text{sn}(z)$ has positive exponent of convergence of zeros, say
$\lambda$, and $\varphi$ has at most one finite Picard’s exceptional value, we
may choose a constant $w_{1}>0$ such that $\varphi$ takes in $|z|<t$ every
value $w$ in the annulus $w_{1}<|w|<M(t,\varphi)$, provided that $t$ is large
enough. Here $M(t,\varphi)$ denotes the maximum modulus of $\varphi$ on the
circle $|z|=t$. Let sn have $\mu(t)$ zeros in this annulus, counted according
to their multiplicity. Then by the definition of $\lambda$, we actually have

(2.30) $\limsup_{r\to\infty}\frac{\log n(r)}{\log
r}=\limsup_{t\to\infty}\frac{\log\mu(t)}{\log M(t,\varphi)}=\lambda>0.$

Hence, for some $\tau>0$, there exists a sequence $(t_{n})$ such that

(2.31)
$\mu(t_{n})>\left(M(t_{n},\varphi)\right)^{\tau}\geq\left(e^{bt_{n}}\right)^{\tau},$

where $b$ is a positive constant. The second inequality above follows by the
fact that $\log M(t,\varphi)\geq T(t,\varphi)$ for all large $t$. Now,
$\text{sn}\circ\varphi$ has at least $\mu(t)$ zeros in $|z|<t$. Making use of
(2.30), we have

(2.32) $\limsup_{r\to\infty}\frac{\log
n(r,1/\text{sn}\circ\varphi)}{r}\geq\limsup_{t_{n}\to\infty}\frac{\log\mu(t_{n},1/\text{sn}\circ\varphi)}{t_{n}}\geq
b\tau.$

By the definitions of $N(r,1/g)$ and $n(r,1/g)$ we can deduce that
$N(2r,1/g)\geq\frac{1}{2}n(r,1/g)$. Then, by the fact that
$T(r,1/\text{sn}\circ\varphi)\geq N(r,1/\text{sn}\circ\varphi)$, we conclude
that there is a sequence $(r_{n})$ such that the characteristic function
$T(r_{n},g)$ satisfies

(2.33) $\log T(r_{n},g)\geq\frac{1}{2}b\tau r_{n}$

for all large $n$. Similarly, from the discussions in previous section we know
that the nonconstant solutions of each of the seven equations (1.11)–(1.17)
have positive even order or infinite order and thus also gives the estimate in
(2.33).

On the other hand, for the composite function $g(z)=f(\omega(z))$, for any
$\varepsilon>0$, we have from [8, Satz 2.2] that

(2.34) $T(r,f(\omega))\leq(1+\varepsilon)T(M(r,\omega)+\omega(0),f),$

for all $r\geq r_{1}$ and some $r_{1}>0$. We may choose the entire function
$\omega(z)$ to be such that $T(r,\omega)\leq lr$ for some constant $l>0$ and
all $r\geq 0$, say $\omega(z)=e^{z\log q}$. Then $\log M(r,\omega)\leq
3T(2r,\omega)$ by an elementary inequality for the relationship between
$T(r,\omega)$ and $M(r,\omega)$ (see [16]). If $f(z)$ is a zero order
transcendental meromorphic solution of equation (1.2), then by the definition
of order, it follows from the inequality in (2.34) that, for any
$\varepsilon>0$,

(2.35) $\log T(r,g)\leq\log T(M(r,\omega)+\omega(0),f)+O(1)\leq\varepsilon r,$

for all $r\geq r_{2}$ and some $r_{2}>0$. This implies that $\log
T(r,g)=o(r)$. Comparing this with the estimate in (2.33), we see that the only
possible equations in (1.2) that can have zero order transcendental
meromorphic solutions are (1.6), (1.7), (1.8), (1.9) or (1.10).

##  3\. Concluding remarks

The integrability of difference equations has been demonstrated to be closely
related to the existence of meromorphic solutions with minimal hyper-type, as
is shown in [1, 14, 30]. The natural integrability criterion for
$q$-difference equations is the existence of zero order meromorphic solutions.
This is because the transformation $z\to\omega(z)$ with an entire function
$\omega(z)$, satisfying $\omega(z+1)=q\omega(z)$, sends $q$-difference
equations to difference equations, and the estimate in (2.35) together with
the results in [30] shows that the existence of sufficiently many zero order
meromorphic solutions of a $q$-difference equation implies the proposed
difference Painlevé property for the related difference equation. However, we
should note that the two $q$-difference equations (1.11) and (1.12) do not
have any nonconstant zero order meromorphic solutions that are transformed to
finite order meromorphic solutions of the two difference equations
$f(z+1)^{2}=(f(z)^{2}-\kappa)/(f(z)^{2}-1)$ or $f(z+1)^{3}=1-f(z)^{-3}$.

Finally, we remark that the transformation $z\to e^{z\log q}$ allows us to
consider zero order meromorphic solutions of equation (1.2) in the punctured
complex plane $\mathbb{C}-\\{0\\}$. Recall the Jacobi elliptic function
$\text{sn}(z)=\text{sn}(z,k)$ with the modulus $0<k<1$ and the complete
elliptic integral $K$ and $K^{\prime}$. Then $\text{sn}(z)$ has order $2$ and
satisfies the first order differential equation
$\text{sn}^{\prime}(z)^{2}=(1-\text{sn}(z)^{2})(1-k^{2}\text{sn}(z))$. Let

(3.1) $\begin{split}\tau=iK^{\prime}/K,\ \ \ \ \mathfrak{q}=\exp(\pi
i\tau)=\exp(-\pi K^{\prime}/K).\end{split}$

Then $\text{sn}(z)$ has the following expression

(3.2) $\begin{split}\text{sn}(z)=2\mathfrak{q}^{1/4}k^{-1/2}\sin\frac{\pi
z}{2K}\prod_{n=1}^{\infty}\left(\frac{1-2\mathfrak{q}^{2n}\cos\frac{\pi
z}{K}+\mathfrak{q}^{4n}}{1-2\mathfrak{q}^{2n-1}\cos\frac{\pi
z}{K}+\mathfrak{q}^{4n-2}}\right).\end{split}$

The above expression can be found in [5]. To avoid confusion, in the above
expression we have used the notation $\mathfrak{q}$ instead of the usual $q$
there. Look at the $q$-difference equation (1.11). By the transformation $z\to
e^{z\log q}$ we have $f(e^{z\log q})=\text{sn}(z)$ is a solution of the
equation $f(z+1)^{2}=(f(z)^{2}-\kappa)/(f(z)^{2}-1)$. Thus

(3.3) $\begin{split}f(z)=2\mathfrak{q}^{1/4}k^{-1/2}\sin\frac{\pi\log
z}{2K\log
q}\prod_{n=1}^{\infty}\left(\frac{1-2\mathfrak{q}^{2n}\cos\frac{\pi\log
z}{K\log q}+\mathfrak{q}^{4n}}{1-2\mathfrak{q}^{2n-1}\cos\frac{\pi\log
z}{K\log q}+\mathfrak{q}^{4n-2}}\right).\end{split}$

Recall that $\sin z=\frac{e^{iz}-e^{-iz}}{2i}$ and $\cos
z=\frac{e^{iz}+e^{-iz}}{2}$. If the constant $q$ satisfies $\frac{i\pi}{2K\log
q}=m$ for some nonzero integer $m$, i.e., $q=\exp(i\pi/2mK)$, then the series
in (3.3) represents a zero order meromorphic function in the punctured complex
plane $\mathbb{C}-\\{0\\}$ that solves the $q$-difference equation (1.11).

Acknowledgements. The second author would like to thank professor Rod Halburd
of University College London (UCL) for a helpful discussion on the
relationship between $q$-difference equations and difference equations during
his visit to UCL in 2023.

## References

  * [1] Ablowitz M.J., Halburd R., Herbst B.: On the extension of the Painlevé property to difference equations. Nonlinearity 13(3), 889–905 (2000).
  * [2] Baker I.N.: On a class of meromorphic functions. Proc. Amer. Math. Soc. 17, 819–822 (1966).
  * [3] Bank S.B., Kaufman R.P.: On the growth of meromorphic solutions of the differential equation $(y^{\prime})^{m}=R(z,\,y)$. Acta Math. 144 (3-4), 223–248 (1980).
  * [4] Barnett D.C., Halburd R.G., Morgan W. and Korhonen R.: Nevanlinna theory for the $q$-difference equations. P. Roy. Soc. Edinb. A 137 3, 454–474 (2007).
  * [5] Baxter R.J.: Exactly solved models in statistical mechanics. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London (1982).
  * [6] Bergweiler W., Ishizaki K. and Yanagihara N.: Meromorphic solutions of some functional equations. Methods Appl. Anal. 5(3), 248–258 (1998).
  * [7] Bergweiler W., Hayman W.K.: Zeros of solutions of a functional equation. Comput. Methods Funct. Theory 3(1), 55–78 (2003).
  * [8] Bergweiler W., Jank G. and Volkmann L.: Wachstumsverhalten zusammengesetzter Funktionen. Results in Math., 7(1): 35–53 (1984).
  * [9] Edrei A., Fuchs W.H.: On the zeros of $f(g(z)$ where $f$ and $g$ are entire functions. J. Analyse Math. 12: 243–255 (1964).
  * [10] Gundersen G.G., Heittokangas J., Laine I., Rieppo J. and Yang D.: Meromorphic solutions of generalized Schröder equations. Aequationes Math. 63(1-2), 110–135 (2002).
  * [11] Gross F.: On the equation $f^{n}+g^{n}=1$. Bull. Amer. Math. Soc. 72, 86–88 (1966).
  * [12] Gross F.: Erratum:”On the equation $f^{n}+g^{n}=1$”. Bull. Amer. Math. Soc. 72, 576 (1966).
  * [13] Goldstein R.: On certain compositions of functions of a complex variable, Aequationes Math. 4, 103–126 (1970).
  * [14] Halburd R.G., Korhonen R.J.: Finite-order meromorphic solutions and the discrete Painlevé equations. Proc. Lond. Math. Soc. 94(2), 443–474 (2007).
  * [15] Halburd R.G., Korhonen R.J., Tohge K.: Holomorphic curves with shift-invariant hyperplane preimages. Trans. Amer. Math. Soc. 366(8), 4267–4298 (2014).
  * [16] Hayman W.K.: Meromorphic functions. Oxford Mathematical Monographs, Clarendon Press, Oxford (1964).
  * [17] Ishizaki K.: Meromorphic solutions of difference Riccati equations. Complex Var. Elliptic Equ. 62(1), 10–122 (2017).
  * [18] Korhonen R., Tohge K., Zhang Y.Y., Zheng J.H.: A lemma on the difference quotients. Ann. Acad. Sci. Fenn. Ser. Math., 45(1) 479–491 (2020).
  * [19] Korhonen R., Zhang Y.Y.: Existence of meromorphic solutions of first-order difference equations. Constr. Approx. 51(3), 465–504 (2020).
  * [20] Korhonen R., Zhang Y.Y.: On meromorphic solutions of Malmquist type difference equations. Ann. Fenn. Math. 48 2, 493–514 (2023).
  * [21] Laine I.: Nevanlinna theory and complex differential equations. De Gruyter Studies in Mathematics, vol. 15, Walter de Gruyter & Co., Berlin (1993).
  * [22] Mohon’ko A.Z.: The Nevanlinna characteristics of certain meromorphic functions. Teor. Funktsii Funktsional. Anal. i Prilozhen 14, 83–87 (1971).
  * [23] Steinmetz N.: Eigenschaften eindeutiger Lösungen gewöhnlicher Differentialgleichungen im Komplexen. Dissertation, Karlsruhe Univ., Karlsruhe (1978).
  * [24] Valiron G.: Sur la dérivée des fonctions algébroïdes. Bull. Soc. Math. France 59, 17–39 (1931).
  * [25] Yamanoi K.: The second main theorem for small functions and related problems. Acta Math. 192(2), 225–294 (2004).
  * [26] Yamanoi K.: Defect relation for rational function as targets. Forum Math. 17(2), 169–189 (2005).
  * [27] Yanagihara N.: Meromorphic solutions of some difference equations. Funkcial. Ekvac. 23(3), 309–326 (1980).
  * [28] Zhang J.L, Korhonen R.: On the Nevanlinna characteristic of $f(qz)$ and its applications. J. Math. Anal. Appl. 369(2), 537-544 (2010).
  * [29] Zhang Y.Y, Korhonen R.: A Malmquist–Steinmetz theorem for difference equations. Constr. Approx. (2023), https://doi.org/10.1007/s00365-023-09648-y.
  * [30] Zheng J.H., Korhonen R.: Studies of differences from the point of view of Nevanlinna theory. Trans. Amer. Math. Soc. 373(6), 4285–4318 (2020).