# CAT: Exploiting Inter-Class Dynamics for Domain Adaptive Object Detection

Mikhail Kennerley  National University of Singapore, Department of Electrical
and Computer Engineering  Institute for Infocomm Research, A*STAR  Jian-Gang
Wang  Institute for Infocomm Research, A*STAR  Bharadwaj Veeravalli  National
University of Singapore, Department of Electrical and Computer Engineering
Robby T. Tan  ASUS Intelligent Cloud Services  National University of
Singapore, Department of Electrical and Computer Engineering

###### Abstract

Domain adaptive object detection aims to adapt detection models to domains
where annotated data is unavailable. Existing methods have been proposed to
address the domain gap using the semi-supervised student-teacher framework.
However, a fundamental issue arises from the class imbalance in the labelled
training set, which can result in inaccurate pseudo-labels. The relationship
between classes, especially where one class is a majority and the other
minority, has a large impact on class bias. We propose Class-Aware Teacher
(CAT) to address the class bias issue in the domain adaptation setting. In our
work, we approximate the class relationships with our Inter-Class Relation
module (ICRm) and exploit it to reduce the bias within the model. In this way,
we are able to apply augmentations to highly related classes, both inter- and
intra-domain, to boost the performance of minority classes while having
minimal impact on majority classes. We further reduce the bias by implementing
a class-relation weight to our classification loss. Experiments conducted on
various datasets and ablation studies show that our method is able to address
the class bias in the domain adaptation setting. On the Cityscapes $\to$ Foggy
Cityscapes dataset, we attained a 52.5 mAP, a substantial improvement over the
51.2 mAP achieved by the state-of-the-art method.
111www.github.com/mecarill/cat

##  1 Introduction

Figure 1: Performance of Class-Aware Teacher (CAT). AT [29] (top left), with
Inter-Class Loss, ICL, (top-right), with Class Relation Augmentation, CRA,
(bottom-left), and CAT (bottom-right). CAT is able to address
misclassification and false positives, blue and red boxes, respectively, in
minority classes such as ’train’. The combination of ICL and CRA further
boosts performance by reducing the number of false positives shown as pink
boxes.

Domain adaptive object detection (DAOD) has been proposed as a solution for
object detection in domains where no annotated data is available. This need is
due to the increasing demands of data tied with annotation being both cost-
prohibitive and potentially inaccurate in challenging domains. DAOD has been
progressing with the introduction of adversarial learning [5, 33, 43, 48],
style transfer [51, 50], and notably, student-teacher frameworks [37, 8, 29,
16, 1, 24, 26, 49]. Yet, these methods ignore the critical issue of class
imbalance, which is a problem in many real-life scenarios, such as autonomous
driving. For instance, in the Cityscapes dataset [7], the ’car’ class
dominates the dataset with 26,963 instances while classes such as ’train’
contain only 168 instances.

Previous work to mitigate class imbalance in DAOD has applied class-specific
discriminators [48] to align classes in distinct domains. Additionally, class
weights has been proposed to boost minority categories while aligning domain
features [3]. Recently, many DAOD methods employ the student-teacher
framework, leading to improved performance. Despite their effectiveness, these
student-teacher methods suffer from the class imbalance problem, resulting in
poor performance for minority classes.

In the student-teacher framework, class-specific thresholding, which provides
more lenient thresholds for minority classes, has been proposed [27, 23, 46].
Yet, this approach does not address the fundamental class imbalance. Even with
perfectly accurate pseudo-labels guiding the student, the model’s bias would
at best align with the biases present in the dataset, rather than providing an
unbiased view. Moreover, inter-class dynamics play a crucial role in
addressing class imbalance, especially when minority classes share high
similarities with majority classes, increasing the likelihood of
misclassification.

To address these challenges, in this paper, we introduce our Class-Aware
Teacher (CAT), specifically designed to tackle class imbalance in the DAOD
setting. CAT implements an Inter-Class Relation module (ICRm) that
approximates the model’s existing class biases as well as inter-class
dynamics. With the knowledge of these biases, CAT applies Class-Relation
Augmentation (CRA) to the training images. CRA increases the representation of
minority classes by blending them with highly similar majority classes at the
instance-level. To aid in this augmentation, a Cropbank [46] is used to store
a collection of cropped instances. Furthermore, this augmentation is not just
confined to the source domain but is also applied across domains. By allowing
cross-domain augmentation, we are able to address the domain gap more
holistically. To further address the inter-class bias, we propose an Inter-
Class Loss (ICL). ICL utilises the insights from the ICRm to prioritise the
model’s attention towards minority classes. This priority is particularly
focused on cases where minority classes are prone to being misclassified as
majority classes.

By integrating these methods, our results indicate an improvement in the
accuracy of minority class predictions, with a quantifiable increase in
performance on benchmarks such as Cityscapes $\to$ Foggy Cityscapes by +1.3
mAP. Figure 1 demonstrates the performance of our method. We summarise the
contributions of this paper as follows:

  * •

We propose our Class-Aware Teacher (CAT) model, supported by our inter-class
relation module (ICRm), which is able to map the model’s existing class
biases.

  * •

We present Class-Relation Augmentation which emphasises augmentation between
related classes across domains, coupled with Inter-Class Loss to further
prioritise the performance of minority classes.

  * •

Thorough experimental analysis that confirms the capabilities of CAT. Our
experiments demonstrate significant improvements in performance compared with
the state-of-the-art methods in DAOD benchmarks.

##  2 Previous Work

#### UDA for Object Detection

Unsupervised Domain Adaptation (UDA) is designed to adapt a model trained on a
labelled source domain to an unlabelled target domain. In object detection
tasks, methods like adversarial training coupled with domain classifiers [5,
33, 43, 48] are prevalent for cultivating domain-invariant image feature
representations. Other strategies, such as image-to-image translation, use
generative models [51, 50] or frequency-based methods [44] to bridge the gap
between domains. Recent approaches have applied the mean-teacher (MT)
framework [37, 8, 29, 16, 1, 24, 26], initially conceived for semi-supervised
learning, to UDA challenges. For instance, the UMT [8] leverages CycleGAN-
generated images to train the student-teacher model, aiming to diminish domain
bias. AT [29] employs strong-weak image augmentation, intentionally degrading
the student’s input compared to the teacher’s, and incorporates adversarial
training to further reduce the domain gap. 2PCNet [24] takes a two-stage
approach to provide more diverse pseudo-labels with domain specific
augmentation. Despite significant improvements over their predecessors, these
methods often overlook the class imbalance issue prevalent in benchmark
datasets. This oversight can lead to suboptimal performance on minority
classes, some of which may appear up to 20 times less frequently than majority
classes [7].

#### Class-Imbalanced Object Detection

The issue of imbalance in object detection largely stems from an
overrepresentation of background over foreground classes in predictions [31].
Our research, however, addresses the imbalance among foreground classes
themselves, which often suffer from unequal frequency within datasets. A
challenge here is the risk of overfitting to minority classes, particularly
when their instances are sparse [14]. Over-sampling of minority classes, a
strategy borrowed from classification tasks, has been adapted for object
detection as well [46]. In student-teacher frameworks, a static hard threshold
for pseudo-label generation has evolved into a dynamic, class-specific
threshold to mitigate teacher bias [27, 23]. Although this approach can
enhance the quality of pseudo-labels, it does not necessarily balance the
sample distribution between majority and minority classes. In the DAOD,
methods such as class-specific discriminators [48] and weighted losses [3]
have been proposed to address class imbalance alongside domain adaptation. A
critical aspect that remains underexplored is the inter-class relationship,
particularly between majority and minority classes with similar features. In
this paper, we aim to explore inter-class dynamics to effectively tackle class
imbalance.

Figure 2: (a) Class-Aware Teacher (CAT) consists of: a student-teacher
network; Inter-Class Relation module (ICRm), which estimates inter-class
biases; Class-Relation Augmentation, which augments images to reduce the
inter-class biases by mixing the cropped instances of related classes; and
Inter-Class Loss, which emphasises the loss on highly misclassified minority
classes. (b) Class-Relation Augmentation demonstrated on majority (Car) and
minority (Bus) classes.

##  3 Preliminaries

#### Problem Definition

In this paper, we propose a method for class balanced domain adaptive object
detection, employing a labelled source dataset $D_{s}=\\{I_{s};Y_{s}\\}$ and
an unlabelled target dataset $D_{t}=\\{I_{t}\\}$. $I_{s}$ and
$Y_{s}=\\{b_{s};c_{s}\\}$ denote the images and their corresponding ground-
truth labels, which include bounding box and class information, indicated as
$b_{s}$ and $c_{s}$, respectively.

#### Mean Teacher

We utilise the mean-teacher framework, comprising of a student and teacher
network that shares identical architectures and network parameters. The
teacher’s network parameters, denoted as $\theta_{t}$, are not updated through
backpropagation but are instead updated using the Exponential Moving Average
(EMA) of the student’s parameters $\theta_{s}$, following:

$\theta_{t}\leftarrow\alpha\theta_{t}+(1-\alpha)\theta_{s},$ (1)

where $\alpha$ is the decay rate that controls the update momentum.

The teacher network generates pseudo-labels , $Y_{t}=\\{b_{t},c_{t}\\}$ from
weakly augmented unlabelled images. These pseudo-labels are utilised by the
student network to calculate the unsupervised loss in conjunction with the
strongly augmented inputs. The student’s inputs are intentionally degraded
compared to the teacher’s inputs to challenge the student network further. The
supervised loss is consistent with the Faster R-CNN framework [32], while the
unsupervised loss is formulated as:

$\mathcal{L}_{\rm unsup}=\mathcal{L}_{\rm obj}^{\rm
rpn}(I_{t},b_{t})+\mathcal{L}_{\rm cls}^{\rm roi}(I_{t},b_{t},c_{t}).$ (2)

notably excluding the regression losses in the unsupervised context. We
implement a hard threshold, $\tau$, on the classification scores generated by
the teacher to ensure that only pseudo-labels with high confidence are
utilised by the student network, thereby encouraging more reliable learning
outcomes. Following [29], a discriminator is added to encourage domain
invariant feature representations with an associated loss, $\mathcal{L}_{\rm
dis}$.

##  4 Proposed Method

Our method, Class-Aware Teacher (CAT), as depicted in Figure 2.a, builds on
the mean-teacher framework. Central to CAT is our Inter-Class Relation module
(ICRm), designed to quantify the class biases inherent in the model. Unlike
traditional approaches that address class bias in a broad sense, ICRm maps the
dynamic relationships between classes. It particularly focuses on minority
classes that are disproportionately misclassified as dominant majority
classes. This mapping is achieved by constructing a confusion matrix for each
batch, which is normalised against the ground truth, allowing for real-time
bias estimation. A global matrix, updated continually with batch-level data,
serves as a robust representation of the model’s class biases. The ICRm is
integral to our methodology, underpinning two components: the Class-Relation
Augmentation and the Inter-Class Loss.

Class-Relation Augmentation addresses class imbalance at the image-level.
Minority classes that share a high similarity to majority classes in an image
are identified using ICRm. MixUp [47], which has been shown to address the
imbalanced class problem [11, 6], is then used to merge the related minority
and majority classes thus increasing the representation of these minority
classes. Our process not only boosts the number of minority class samples but
also encourages the model to distinguish between closely associated classes,
as illustrated in Figure 2.b.

Additionally, ICRm informs the distribution of weights within our Inter-Class
Loss. This weighted loss function emphasises the loss on minority classes,
especially those frequently mislabelled as majority classes. By doing so, we
provide a counterbalance to the model’s learned biases, nudging it towards a
more balanced classification performance.

###  4.1 Inter-Class Relation module

Prior research addressing class imbalance within domain adaptation [17, 21,
36] has significantly advanced the performance on imbalanced classes. However,
these methods often overlook the inter-class relationships and their impact on
class imbalance. Our experimental observations suggest that the likelihood of
misclassification between minority and majority classes is heavily influenced
by their similarity. For instance, minority vehicle classes are more prone to
be misclassified as ’car’, a majority class, rather than as ’person’, another
majority class, due to their inherent resemblance.

Our approach aims to leverage these observed relationships through the Inter-
Class Relation module (ICRm). Distinct from general class bias, inter-class
dynamics cannot be directly inferred from the dataset but must be extrapolated
from the model during training. We achieve this by generating a confusion
matrix at each training batch that cross-references the ground-truth labels
with the model’s predictions. This matrix is normalised with respect to the
ground truth to estimate the bias between classes.

Subsequently, we employ EMA to iteratively update a global matrix, which
represents a more stable and comprehensive approximation of the model’s class
biases. The EMA’s utility extends beyond smoothing; it removes the need for
each class’s presence in every batch, simplifying the training process. The
process for constructing this matrix is outlined in Algorithm 1. The ICRm is
formulated separately for both source images, referencing actual ground-truth
labels, and target images, utilising pseudo-labels to mirror the ground-truth.

###  4.2 Class-Relation Augmentation

In classification task, oversampling is a common technique to counter class
imbalance by increasing the presence of minority images. However, this
approach presents challenges in object detection, where images often contain a
mix of multiple object classes. Our analysis on the Cityscapes [7] dataset
indicates that most images include at least one instance of a majority class,
rendering image-level resampling ineffective. This complexity demands more
nuanced augmentation strategies for object detection.

Similar to Zhang et al. [46], we employ instance-level oversampling. Instances
are cropped from their images using bounding box annotations and are then
strategically inserted into other images.

Images are randomly selected from each batch and utilising the ICRm, we
differentiate classes as majority or minority based on their likelihood of
correct classification. We then derive the mean probability :

${\rm ICRm}_{\rm avg}=\frac{1}{C}\sum^{C}_{c=0}{\rm ICRm}(c,c),$ (3)

where $C$ is the number of classes. Classes with a probability, ${\rm
ICRm}(c,c)$, above and below the mean probability are designated as majority
and minority classes, respectively.

Instead of random overlaying, which has been adopted by other previous
approaches, our method matches highly related minority and majority instances
and uses MixUp [47] to blend them, allowing the model to have better
generalisation towards minority classes.

We achieve this by pairing each base instance in an image with a sampled
instance chosen through weighted random sampling, using the ICRm class
probabilities as weights. For majority base instances, the corresponding
column in ICRm, namely, ${\rm ICRm}(:,c)$ is used, discounting the class’s own
probability by setting ${\rm ICRm}(c,c)$ to zero to avoid self-augmentation.
This allows us to select classes that are frequently misclassified as the
majority class. Conversely, for minority base instances, we use the
corresponding row from ICRm as weights without zeroing out ${\rm ICRm}(c,c)$,
allowing for the possibility of self-augmentation, which can be beneficial for
minority classes. This is demonstrated in Figure 2.b.

Sampled instances are resized to match the base instance dimensions for
bounding box consistency. The MixUp augmentation is then applied as per the
following formulation, where a beta distribution determines the mixing ratio:

$\displaystyle\hat{I}$ $\displaystyle=\beta\cdot I_{\rm base}+(1-\beta)\cdot
I_{\rm mix},$ (4) $\displaystyle\hat{c}$ $\displaystyle=\beta\cdot c_{\rm
base}+(1-\beta)\cdot c_{\rm mix},$

where $I_{\rm base}$ and $I_{\rm mix}$ represent the cropped images of the
base and mixed instances, respectively, with $\hat{I}$ denoting the resulting
augmented image. Similarly, $c_{\rm base}$ and $c_{\rm mix}$ refer to the
classes of the base and mixed instances, while $\hat{c}$ indicates the class
vector of the augmented instance.

For source domain images, we incorporate instances from both domains to
leverage accurate source labels and to aid domain adaptation with target
domain samples. Whereas for target domain images, we prioritise target
instances, using source instances only when no target instances are available
for a specific class. This ensures that more focus is put onto the target
domain for stronger domain adaptation. Additionally, we do not apply
augmentations to minority base instances in the target domain to preserve
their integrity. This ensures that the model is able to focus on the target
domain and does not drift to an intermediate domain.

Algorithm 1 Inter-Class Relation module (ICRm)

0: Global class-relation matrix $ICRm$ with shape $(C,C)$, where each element
equals $0$.

while $training$ do

Create batch matrix ${\rm ICRm}_{l}$ with shape $(C,C)$, where each element
equals $0$.

for ground-truth, $c_{i}$, and prediction,$x_{i}$, in $Y,X$ do

${\rm ICRm}_{l}[c_{i},x_{i}]\leftarrow{\rm ICRm}_{l}[c_{i},x_{i}]+1$

end for

for each class $c$ in $C$ do

Normalise class matrix ${\rm ICRm}_{l}[c]$

if global class matrix ${\rm ICRm}[c]$ is empty then

//Copy batch class matrix to global

${\rm ICRm}\leftarrow{\rm ICRm}_{l}$

else

//Update global class matrix with EMA

${\rm ICRm}\leftarrow\beta*{\rm ICRm}+1(1-\beta)*{\rm ICRm}_{l}$

end if

end for

end while

To implement Class-Relation Augmentation, we store class-specific instance
crops from each batch, which we term a Cropbank [46]. These crops are
extracted from bounding box annotations of labelled source images and pseudo-
labelled target images. Separate Cropbanks are maintained for both the source
and target datasets, allowing for more targeted augmentation. To ensure a
diverse range of samples, we update the class instances on a first-in-first-
out basis. This is particularly beneficial for the target Cropbank, where
earlier samples may less accurate due to the early pseudo-labels’ robustness.

###  4.3 Inter-Class Loss

To further mitigate class bias, we introduce a weighted parameter to the
classification loss, informed by the Inter-Class Relation module (ICRm) for
foreground classes. This weighting prioritises classes that are frequently
misclassified as majority classes, allowing the model to concentrate on
refining their performance. To emphasise the focus on underperforming classes,
we employ a non-linear transformation on the ICRm values:

$w_{i}=\begin{cases}\sqrt{1-{\rm ICRm}(c_{i},x_{i})},&\text{if }c_{i}=x_{i}\\\
\sqrt{{\rm ICRm}(c_{i},x_{i})/{\rm
ICRm}(c_{i},c_{i})},&\text{otherwise,}\end{cases}$ (5)

where $w_{i}$ is the $i$th weight in $W$, and $c_{i}$ and $x_{i}$ are the
$i$th ground-truth and predicted class, respectively. We normalise on the
diagonal when $c_{i}\neq x_{i}$ as our primary objective is to prioritise low
performing classes. Weights for background classes are uniformly set to 1 to
avoid biasing the model against them. To reconcile the disparity between
foreground and background class weights, the weights of the foreground
instances are normalised so that their mean equals the background class
weight:

$W_{f}=W_{f}/{\rm mean}(W_{f}),$ (6)

where $W_{f}$ denotes the collection of foreground instance weights. Moreover,
we integrate an additional regularisation term, $\lambda_{l}$, across all
class-relation weights to prevent extreme weight values from distorting the
loss:

$W=\frac{(W+\lambda_{l})}{(1+\lambda_{l})}$ (7)

This regularisation ensures a moderated, balanced impact on the classification
loss, which is now defined as:

$\mathcal{L}_{\rm cls}=\frac{1}{N}\sum^{N}_{i=0}w_{i}*{\rm CE}(c_{i},x_{i})$
(8)

where $N$ is the number of instances and $\rm CE$ is the cross-entropy loss.

The overall loss is then:

$\mathcal{L}=\mathcal{L}_{\rm sup}+\lambda_{u}\mathcal{L}_{\rm
unsup}+\lambda_{d}\mathcal{L}_{\rm dis},$ (9)

where $\lambda_{u}$ and $\lambda_{d}$ represent the weights for the
unsupervised and discriminator losses, respectively.

##  5 Experiments

Method |  Detector |  person |  rider |  car |  truck |  bus |  train |  mcycle |  bicycle |  mAP
---|---|---|---|---|---|---|---|---|---|---
Source [38] |  FCOS |  36.9 |  36.3 |  44.1 |  18.6 |  29.3 |  8.4 |  20.3 |  31.9 |  28.2
SIGMA [28] |  FCOS |  46.9 |  48.4 |  63.7 |  27.1 |  50.7 |  35.9 |  34.7 |  41.4 |  43.5
OADA [45] |  FCOS |  47.8 |  46.5 |  62.9 |  32.1 |  48.5 |  50.9 |  34.3 |  39.8 |  45.4
HT [9] |  FCOS |  52.1 |  55.8 |  67.5 |  32.7 |  55.9 |  49.1 |  40.1 |  50.3 |  50.4
Source [52] |  Def DETR |  37.7 |  39.1 |  44.2 |  17.2 |  26.8 |  5.8 |  21.6 |  35.5 |  28.5
AQT [19] |  Def DETR |  49.3 |  52.3 |  64.4 |  27.7 |  53.7 |  46.5 |  36.0 |  46.4 |  47.1
MRT [49] |  Def DETR |  52.8 |  51.7 |  68.7 |  35.9 |  58.1 |  54.5 |  41.0 |  47.1 |  51.2
Source [32] |  FRCNN |  22.4 |  26.6 |  28.5 |  9.0 |  16.0 |  4.3 |  15.2 |  25.3 |  18.4
Oracle |  FRCNN |  39.5 |  47.3 |  59.1 |  33.1 |  47.3 |  42.9 |  38.1 |  40.8 |  43.5
MeGA [39] |  FRCNN |  37.7 |  49.0 |  52.4 |  25.4 |  49.2 |  46.9 |  34.5 |  39.0 |  41.8
TIA [48] |  FRCNN |  34.8 |  46.3 |  49.7 |  31.1 |  52.1 |  48.6 |  37.7 |  38.1 |  42.3
UMT [8] |  FRCNN |  33.0 |  46.7 |  48.6 |  34.1 |  56.5 |  46.8 |  30.4 |  37.3 |  41.7
TDD [16] |  FRCNN |  39.6 |  47.5 |  55.7 |  33.8 |  47.6 |  42.1 |  37.0 |  41.4 |  43.1
PT [4] |  FRCNN |  40.2 |  48.8 |  59.7 |  30.7 |  51.8 |  30.6 |  35.4 |  44.5 |  42.7
AT‡ [29] |  FRCNN |  45.3 |  55.7 |  63.6 |  36.8 |  64.9 |  34.9 |  42.1 |  51.3 |  49.3
CMT [1] |  FRCNN |  45.9 |  55.7 |  63.7 |  39.6 |  66.0 |  38.8 |  41.4 |  51.2 |  50.3
MILA [26] |  FRCNN |  45.6 |  52.8 |  64.8 |  34.7 |  61.4 |  54.1 |  39.7 |  51.5 |  50.6
CAT (Ours) |  FRCNN |  44.6 |  57.1 |  63.7 |  40.8 |  66.0 |  49.7 |  44.9 |  53.0 |  52.5
Table 1: Object detection results on the Foggy Cityscapes test set for
Cityscapes $\to$ Foggy Cityscapes (0.02) domain adaptation. We group methods
based on their detector frameworks (FCOS/Def DETR/FRCNN) and highlight the
best performing method. CAT is able to outperform the previous state-of-the-
art, MRT, by 1.3 mAP and improve on AT by 3.2 mAP. The mean average precision
at .50 IoU (mAP) is reported for all classes. ‡ AT performance is reproduced
on Foggy Cityscapes (0.02) with publicly available code for fairness.

###  5.1 Datasets

We assess the performance of Class-Aware Teacher (CAT) using benchmarks in
domain adaptive object detection (DAOD) following prior work [29].

#### Cityscapes $\to$ Foggy Cityscapes:

The Cityscapes dataset [7] is a road-centric dataset with 2,975 training and
500 validation images from various urban settings under clear weather,
annotated across 8 classes. Foggy Cityscapes [34] is a synthesised dataset
generated on Cityscapes to simulate foggy weather, using the same base images
and annotations. We conduct our experiment on the most severe fog level (0.02)
where Foggy Cityscapes is used as the target domain.

#### PASCAL VOC $\to$ Clipart1K:

We use PASCAL VOC 2012 [10], which comprises of 11,540 real-world images
across 20 categories, for training. The Clipart1k dataset [20] consists of 20
corresponding clipart object categories. Following [29], we split Clipart1k
into 500 training and 500 testing images.

The benchmarks of Sim10K [22] $\to$ Cityscapes and KITTI [12] $\to$ Cityscapes
are excluded from our evaluation. Despite their popularity in DAOD research,
they focus solely on the ’Car’ category, which does not align with our class
imbalance setting.

###  5.2 Experimental Setup

Following previous works in DAOD, we adopt the Faster R-CNN object detector
with VGG-16 [35] and ResNet-101 [15] as backbones for our detection model. Our
hyperparameters: EMA decay rate $\alpha=0.9996$, beta-distribution hyper
parameters [0.5,0.5], adversarial loss weight, $\lambda_{d}$ $0.1$, and
unsupervised loss weight, $\lambda_{u}$ $1.0$, regularisation term,
$\lambda_{l}$ 1.0. We employ a hard threshold $\tau$ of 0.8 for pseudo-
labelling. Weak-strong augmentation [30] is applied to both source and target
images. We train our student model for 20,000 iterations on the labelled
source data. The parameters of the student is copied to the teacher which is
then updated via EMA of the student at each iteration. We continue training
for 60,000 iterations with both labelled source and unlabelled target data.
Our framework is developed on top of the publicly available Detectron2 [41].
Experiments are performed using a batch size of 8 source and 8 target images,
distributed across 4 NVIDIA RTX3090 GPUs. Additional details regarding our
experimental setup are provided in the supplementary materials.

Method |  aero |  bike |  bird |  boat |  bottle |  bus |  car |  cat |  chair |  cow |  table |  dog |  horse |  mtr |  prsn |  plant |  shp |  sofa |  train |  tv |  mAP
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Source [13] |  23.0 |  39.6 |  20.1 |  23.6 |  25.7 |  42.6 |  25.2 |  0.9 |  41.2 |  25.6 |  23.7 |  11.2 |  28.2 |  49.5 |  45.2 |  46.9 |  9.1 |  22.3 |  38.9 |  31.5 |  28.8
Oracle |  33.3 |  47.6 |  43.1 |  38.0 |  24.5 |  82.0 |  57.4 |  22.9 |  48.4 |  49.2 |  37.9 |  46.4 |  41.1 |  54.0 |  73.7 |  39.5 |  36.7 |  19.1 |  53.2 |  52.9 |  45.0
I3Net [3] |  30.0 |  67.0 |  32.5 |  21.8 |  29.2 |  62.5 |  41.3 |  11.6 |  37.1 |  39.4 |  27.4 |  19.3 |  25.0 |  67.4 |  55.2 |  42.9 |  19.5 |  36.2 |  50.7 |  39.3 |  37.8
ICR-CCR [42] |  28.7 |  55.3 |  31.8 |  26.0 |  40.1 |  63.6 |  36.6 |  9.4 |  38.7 |  49.3 |  17.6 |  14.1 |  33.3 |  74.3 |  61.3 |  46.3 |  22.3 |  24.3 |  49.1 |  44.3 |  38.3
HTCN [2] |  33.6 |  58.9 |  34.0 |  23.4 |  45.6 |  57.0 |  39.8 |  12.0 |  39.7 |  51.3 |  20.1 |  20.1 |  39.1 |  72.8 |  61.3 |  43.1 |  19.3 |  30.1 |  50.2 |  51.8 |  40.3
DM [25] |  25.8 |  63.2 |  24.5 |  42.4 |  47.9 |  43.1 |  37.5 |  9.1 |  47.0 |  46.7 |  26.8 |  24.9 |  48.1 |  78.7 |  63.0 |  45.0 |  21.3 |  36.1 |  52.3 |  53.4 |  41.8
UMT [8] |  39.6 |  59.1 |  32.4 |  35.0 |  45.1 |  61.9 |  48.4 |  7.5 |  46.0 |  67.6 |  21.4 |  29.5 |  48.2 |  75.9 |  70.5 |  56.7 |  25.9 |  28.9 |  39.4 |  43.6 |  44.1
TIA [48] |  42.2 |  66.0 |  36.9 |  37.3 |  43.7 |  71.8 |  49.7 |  18.2 |  44.9 |  58.9 |  18.2 |  29.1 |  40.7 |  87.8 |  67.4 |  49.7 |  27.4 |  27.8 |  57.1 |  50.6 |  46.3
AT‡ [29] |  33.1 |  66.1 |  35.3 |  44.9 |  57.5 |  44.9 |  51.0 |  5.8 |  59.5 |  54.9 |  34.6 |  23.5 |  64.3 |  84.0 |  75.4 |  51.5 |  17.1 |  30.3 |  43.3 |  37.2 |  45.7
CMT [1] |  39.8 |  56.3 |  38.7 |  39.7 |  60.4 |  35.0 |  56.0 |  7.1 |  60.1 |  60.4 |  35.8 |  28.1 |  67.8 |  84.5 |  80.1 |  55.5 |  20.3 |  32.8 |  42.3 |  38.2 |  47.0
CAT (Ours) |  40.5 |  64.1 |  38.8 |  41.0 |  60.7 |  55.5 |  55.6 |  14.3 |  54.7 |  59.6 |  46.2 |  20.3 |  58.7 |  92.9 |  62.6 |  57.5 |  22.4 |  40.9 |  49.5 |  46.0 |  49.1
Table 2: Object detection results on the Clipart1k test set for PASCAL VOC
$\to$ Clipart1k domain adaptation. CAT improves on the previous state-of-the-
art, CMT, by 2.1 mAP, achieving the new best of 49.1 mAP. The mean average
precision at .50 IoU (mAP) is reported for all classes.‡ AT performance is
reproduced following [1].

###  5.3 Comparison with SOTA methods

We compare our method with the state-of-the-art in DAOD as well as reporting a
source only FCOS/Def DETR/Faster RCNN for a baseline comparison. We
additionally include an oracle upper bound, which is trained on only the
target domain and its ground truth annotations.

#### Foggy Weather Adaptation

When object detectors are deployed in real-world scenarios, the performance
could drop significantly under sub-optimal conditions, e.g. adverse weather.
This is because that the samples in adverse weather do not present in the
training of the model resulting in a domain shift. The domain adaptation task
is designed to overcome this gap between normal and adverse conditions. To
demonstrate this, we conduct an experiment on the commonly used Cityscapes
$\to$ Foggy Cityscapes benchmark.

Our results are shown in Table 1. We can observe that methods utilising the
student-teacher framework (HT, UMT, TDD, PT, AT, CMT, MILA, MRT) outperform
non-student-teacher frameworks by a large margin. CAT, which is built on
existing SOTA mean teacher frameworks, significantly improves the performance
at 52.5 mAP. Additionally, our method is able to improve minority classes
while not impacting majority classes.

#### Real to Artistic Adaptation

Adapting object detection from real to artistic domains is particularly
challenging due to the significant differences between these domains. In our
experiment, detailed in Table 2, we observe CAT achieves a mAP of 49.1,
outperforming the previous best, by 2.1 mAP and AT by 3.4 mAP.

Notably, CAT shows substantial improvements in minority classes, such as
’motorbike’ which in the Clipart1k training set consists of only 7 images. The
results of this experiment demonstrate CAT’s effectiveness in addressing class
imbalances even in dissimilar domains.

Method |  ICRm |  CRA |  ICL |  mAP |  $\sigma\downarrow$
---|---|---|---|---|---
Base (AT [29]) |  |  |  |  49.3 |  10.8
|  ✓ |  ✓ |  |  51.0 |  8.8
CAT (Ours) |  ✓ |  |  ✓ |  51.6 |  10.3
|  ✓ |  ✓ |  ✓ |  52.5 |  8.6
Table 3: Ablation studies on CAT components. ICRm is included in all studies
as it forms the basis of CRA and ICL. We report the mean average precision at
.50 IoU (mAP) and the standard deviation of class-mAP ($\sigma$). Our
contributions are not included in the base framework (AT).

###  5.4 Ablation Studies

To verify the significance of our contributions, we conducted an ablation
study. All experiments within this study were performed on the Cityscapes
$\to$ Foggy Cityscapes benchmark using the VGG16 backbone.

#### Quantitative Ablation

Table 3 quantitatively showcases the efficacy of each contribution within our
framework. The base framework, prior to integrating our modules, corresponds
to the AT model as described in [29]. Since our Inter-Class Relation module
(ICRm) is pivotal for both the class-relation augmentation and loss, it is a
constant across all experimental variations. To highlight our method’s
capability in addressing class imbalance, we introduce $\sigma$. This value
represents the standard deviation of the mAP across different classes and
serves as an indicator of performance equity among classes.

Inclusion of our Inter-Class Loss (ICL) yields a notable 2.3 mAP gain over the
base AT model and a decreased $\sigma$, indicating a more balanced performance
across classes. The Class Relation Augmentation (CRA) also benefits AT, though
to a lesser extent than ICL in terms of mAP. Notably, CRA significantly
narrows the performance gap between minority and majority classes, as
reflected by a reduced $\sigma$ of 8.8. Employing both ICL and CRA not only
enhances overall performance but also achieves a lower $\sigma$ compared to
the base model, reinforcing our method’s effectiveness in managing class
imbalance.

Figure 3: Qualitative results of CAT. We show the results of AT and CAT on the top and bottom, respectively. CAT is able to address misclassification (col 1,2,4), false negatives (col 1,3), and false positives (col 1,3,4). Box colour represents: Green $\to$ true positives, Blue $\to$ misclassified, Red $\to$ false negatives, Pink $\to$ false positives. Selection Method |  mAP |  $\sigma\downarrow$
---|---|---
Random (0.5) |  51.8 |  8.5
CRA (1.0) |  50.2 |  9.2
CRA (0.5) |  52.5 |  8.6
Table 4: Comparison of selecting class instances randomly and via CRA. Values
in brackets refer to the likelihood of an instance being augmented.

#### Impact of Augmentation

We demonstrate the impact of augmentation in terms of ratio and selection
criteria. Augmenting images is a key part of our approach, enriching the
dataset with additional representations of minority classes. Yet, if images
are augmented excessively, the model may fail to learn an accurate
representation of the classes. To strike a balance, we selectively augment a
random subset of the images. Additionally, how class instances are paired is
key to improve the quality of augmentation. Compared to randomly selecting
class pairs, CRA is able to prioritise pairing highly related minority and
majority class instances. This ensures that the Mixup output is more
meaningful for minority class performance.

The experimental results of this approach are given in Table 4. We compare
randomly selecting class instances for MixUp with our Class-Relation
Augmentation at different values. These values represent the likelihood of a
instance in the base image being augmented. Randomly applying MixUp improves
the overall performance by +0.2 mAP. However, by using CRA we can further
increase performance by +0.9 mAP. This shows that pairing highly-related
classes for MixUp strengthens the performance of the minority class while
minimally affecting the majority class. In addition, our experiments show that
too much augmentation can have a negative effect on the performance of the
model.

#### Weighing Strategy for Class Loss

We introduce a weighted classification loss to improve the performance of
minority class performance in Eq. 5. Class-level loss is a common strategy
that has been adopted to address the class imbalance in a dataset [18, 40]. We
compare our Inter-Class Loss (ICL) with a variant of previous class-level
losses where only the diagonal of the Inter-Class Relation module is used. The
diagonal corresponds to the ground-truth class likelihood of accurate
classification. In contrast, ICL uses the likelihood of the ground-truth being
classified as the predicted class to influence its loss. We show in Table 5
that by using this inter-class relationship, we are able to improve the
performance by +0.5 mAP. However, there may be cases where ICL overly
penalises well-performing classes. This is addressed by applying a
regularisation term as seen in Eq. 7. We can see that if this regularisation
term is removed, performance drops significantly as the weight of certain
classes gets too small during training.

Class Weight |  mAP |  $\sigma\downarrow$
---|---|---
Class-Level |  52.0 |  9.0
ICL w/o Reg. |  51.3 |  9.5
ICL |  52.5 |  8.6
Table 5: Class Loss Weighing Strategies. Class-level only uses the diagonal
values in our ICRm along with regularisation, ICL refers to our Inter-Class
Loss.

#### Qualitative Results

Figure 3 illustrates the qualitative results of our method, with the top and
bottom row displaying predictions from AT and CAT, respectively.

CAT is able to correct misclassifications, represented by blue boxes.
Additionally, CAT can bring a reduction in both false positives and false
negatives, represented by pink and red boxes, showcasing improved detection
accuracy across various scales and classes.

##  6 Conclusion

In this paper, we propose Class-Aware Teacher (CAT) for Domain Adaptive Object
Detection. We demonstrate that CAT, by leveraging our Inter-Class Relation
module, effectively approximates and mitigates class biases, leading to more
equitable performance across classes. Furthermore, Class-Relation Augmentation
and Inter-Class Loss were shown to be effective in enhancing minority class
representation. Our experimental results on Cityscapes → Foggy Cityscapes and
PASCAL VOC → Clipart1K have demonstrated the effectiveness of our method
achieving SOTA performance at 52.5 mAP and 49.1 mAP, respectively. Based on
our findings, we believe that further investigation into inter-class dynamics
is a promising direction for advancing class imbalance in the DAOD setting.

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Supplementary Material

##  7 Additional Details on Methods

###  7.1 Additional Details on Class-Relation Augmentation

We further describe the details of our Class-Relation Augmentation (CRA)
approach below. CRA augments random images in a batch based on the source and
target augmentation ratio. For each selected image, we identify class
instances using labels or pseudo-labels for source and target images,
respectively, termed ’base instances.’

Following the methodology outlined in Section 4.2, we select ’mix instances’
that exhibit a strong relationship with the base instances, determined by our
Inter-Class Relation module (ICRm). A ’mixed instance’ is then randomly chosen
from a predefined crop bank. To mitigate the effects of upsampling
degradation, we ensure the mixed instances is at least 0.25 of the base
instance’s size.

We resize the mixed instance to the base instance’s dimensions, allowing the
aspect ratio of the mixed instance to be adjusted. This resizing allows us to
use a single bounding box to represent both the base and mixed instance after
augmentation. Experimental results, presented in Table 1, demonstrate that
this resizing strategy not only maintains but enhances model performance
compared to maintaining the mixed instance aspect ratio. This is because the
ambiguity of labelling when two bounding boxes are used is complex, especially
when employing mixup.

Once the mixed instance has been resized, mixup [47] is then applied to
combine the two instances and their labels. Given the distinct class
representations, we employ one-hot encoding to support multi-class labelling.
This process is repeated across all objects in the selected image.

Maintain Aspect Ratio |  mAP
---|---
✗ |  52.5
✓ |  51.1
Table 6: Performance of Class-Aware Teacher (CAT) with and without maintaining
the aspect ratio during CRA. We can see that disregarding the aspect ratio
during resizing improves performance while being a simpler resizing strategy.

##  8 Experiments

###  8.1 Additional Details on Experimental Setup

In this section, we provide additional details on the experimental setup.
Consistent with prior research in the domain of adaptive object detection, our
experiments are conducted using the Faster R-CNN detection framework. VGG-16
[35] and ResNet-101 [15] are used as the backbones for our detection model
depending on the benchmark used. PASCAL VOC $\to$ Cliapart1K utilises the
ResNet-101 backbone. Both Cityscapes $\to$ Foggy Cityscapes and Cityscapes
$\to$ BDD100K utilises the VGG-16 backbone.

Across all experiments, we maintain consistent hyperparameter settings, which
are detailed in Table 2.

###  8.2 Additional Details on Dataset Class Distributions

The distribution of classes in our datasets plays an important role during
training. Minority classes tend to under perform, especially when there is a
distribution shift between training and validation datasets. To validate the
effectiveness of our method, we show the class distributions of the evaluation
datasets and how our method is able to address minority class performance.

Figure 4 shows the class distribution for the Cityscapes $\to$ Foggy
Cityscapes task. Car and person forms the majority in all the datasets used
for this task and truck, bus, and train form the minority. This is to be
expected as the datasets are from the same source and would share similar
distributions. This forms a simpler task as we do not need to account for a
distribution shift during testing. Our method matches or outperforms SOTA for
the truck and bus class, as well as strongly outperforming our base method
[29] for all three minority classes.

The class distribution of the PASCAL VOC $\to$ Clipart1k task is shown in
Figure 5. The PASCAL VOC dataset is fairly balanced with the only outlier
being the person class. This ensures that the initial training has less bias
towards specific classes, however, Clipart1k exhibits stronger class
imbalance. This results in a distribution shift during unsupervised training
and evaluation which may result in suboptimal performance. CAT is able to have
strong performance on the motorbike minority class and is able to outperform
its base on the bus class.

The Cityscapes $\to$ BDD100K (Daytime) task contains two road-centric datasets
taken in different locations which would result in both imbalanced data as
well as a distribution shift as seen in Figure 6 . This would be a harder task
as a minority class in one dataset may not be the same minority the other. For
example, truck and bus are the minority for Cityscapes but motorcycle and
bicycle are the minority for BDD100K. CAT is able to outperform SOTA for
truck, bus, and bicycle and is only 0.1 mAP lower for the motorcycle minority
class.

Figure 4: Class distribution of datasets used for the Cityscapes $\to$ Foggy
Cityscapes task. We can see that person and car classes form the majority of
all classes. The distribution of classes for the labeled dataset and
validation set is similar which makes for an simpler task. Figure 5: Class
distribution of datasets used for the PASCAL VOC $\to$ Clipart1k task. Person
is a majority class for all datasets, however other classes for PASCAL VOC
have a similar number of instance. The imbalance is stronger in the Clipart1K
dataset with classes such as motorbike and bus being a minority. Figure 6:
Class distribution of datasets used for the Cityscapes $\to$ BDD100K (Daytime)
task. We can see that the car class form the majority of all classes,
especially for the BDD100K dataset. Note that the class distribution of
labeled and validation set differs, especially for minority classes which can
make the task more difficult.

###  8.3 Cityscapes $\to$ BDD100K

In addition to experiments performed in Section 5.3 of the main paper, we
include the Cityscapes $\to$ BDD100K-Daytime benchmark.

The BDD100K [bdd100k] dataset is a large-scale dataset containing 100,000
images. For this experiment, we use the day-time split which contains 36,728
training and 5,258 testing images. We remove the train, traffic light and
traffic sign categories following previous work. The Cityscapes $\to$ BDD100K
benchmark covers scene adaptation as well as small-to-large dataset
adaptation.

Table 7 shows the results of our experiment. We can observe that CAT has
stronger performance compared to the previous SOTA at 38.5 mAP. Minority
classes such as rider, truck, bus, and bicycle also show a significant
improvement. This shows that our strategy to address inter-class dynamics
provides a viable solution to address class imbalance for domain adaptive
object detection.

Method |  person |  rider |  car |  truck |  bus |  mcycle |  bicycle |  mAP
---|---|---|---|---|---|---|---|---
Faster RCNN [32] |  28.8 |  25.4 |  44.1 |  17.9 |  16.1 |  13.9 |  22.4 |  24.1
SIGMA [28] |  46.9 |  29.6 |  64.1 |  20.2 |  23.6 |  17.9 |  26.3 |  32.7
TDD [16] |  39.6 |  38.9 |  53.9 |  24.1 |  25.5 |  24.5 |  28.8 |  33.6
PT [4] |  40.5 |  39.9 |  52.7 |  25.8 |  33.8 |  23.0 |  28.8 |  34.9
CAT (Ours) |  44.6 |  41.5 |  61.2 |  31.4 |  34.6 |  24.4 |  31.7 |  38.5
Table 7: Object detection results on the BDD100k-Daytime test set for Cityscapes $\to$ BDD100k-Daytime domain adaptation. The mean average precision at .50 IoU (mAP) is reported for all classes. Hyperparameter | Description |  C$\to$F |  PV$\to$CA |  C$\to$B
---|---|---|---|---
- | Detector |  FRCNN |  FRCNN |  FRCNN
- | Backbone |  VGG |  ResNet-101 |  VGG
- | BatchNorm |  True |  True |  False
$\alpha$ | Decay rate for student-teacher EMA |  0.9996 |  0.9996 |  0.9996
$\beta$ | Beta-distribution parameters for mixup |  [0.5,0.5] |  [0.5,0.5] |  [0.5,0.5]
$\lambda_{d}$ | Weight for Adverserial Loss |  0.1 |  0.1 |  0.1
$\lambda_{u}$ | Weight for Unsupervised Loss |  1.0 |  1.0 |  1.0
$\tau$ | Threshold value for pseudo-label confidence |  0.8 |  0.8 |  0.8
$\lambda_{l}$ | Regularization term for Inter-Class Loss |  1.0 |  1.0 |  1.0
- | Source Augmentation Ratio |  0.5 |  0.5 |  0.5
- | Target Augmentation Ratio |  0.5 |  0.5 |  0.5
- | Burn-Up Step Iterations |  20000 |  20000 |  20000
- | Total Training Iterations |  80000 |  80000 |  80000
- | Learning Rate |  0.2 |  0.2 |  0.2
Table 8: Model Hyperparameters for Experiments.From left to right, Cityscapes
$\to$ Foggy Cityscapes, PASCAL VOC $\to$ Clipart1K, and Cityscapes $\to$
BDD100K-Day.


#  $q$-Supercongruences on triple and quadruple sums

Xiaoxia Wang  DEPARTMENT OF MATHEMATICS, SHANGHAI UNIVERSITY, SHANGHAI 200444,
P. R. CHINA  ∗ Corresponding author<EMAIL_ADDRESS>(X. Wang),
<EMAIL_ADDRESS>(C. Xu). and  Chang Xu∗

###### Abstract.

Inspired by the recent work of El Bachraoui, we present some new
$q$-supercongruences on triple and quadruple sums of basic hypergeometric
series. In particular, we give a $q$-supercongruence modulo the fifth power of
a cyclotomic polynomial, which is a $q$-analogue of the quadruple sum of Van
Hamme’s supercongruence (G.2).

###### Key words and phrases:

basic hypergeometric series; $q$-supercongruences; creative microscoping;
Chinese remainder theorem

###### 2010 Mathematics Subject Classification:

Primary 33D15; Secondary 11A07, 11B65

This work is supported by Natural Science Foundations of Shanghai
(22ZR1424100).

##  1\. Introduction

In 1913, Ramanujan (see [3]) proposed some well-known hypergeometric series
without any proof, including

$\sum_{k=0}^{\infty}(8k+1)\frac{\left(\frac{1}{4}\right)^{4}_{k}}{k!^{4}}=\frac{2\sqrt{2}}{\sqrt{\pi}\Gamma\left(\frac{3}{4}\right)^{2}}.$

Here $(a)_{n}=a(a+1)\cdots(a+n-1)$ is the Pochhammer symbol and $\Gamma(x)$ is
the gamma function. In 1997, Van Hamme [15] studied partial sums of
Ramanujan’s summation formular for $1/\pi$ and conjectured $13$ Ramanujan-type
supercongruences, involving

$\displaystyle\mathrm{(G.2)}\quad\sum_{k=0}^{\frac{p-1}{4}}(8k+1)\frac{\left(\frac{1}{4}\right)^{4}_{k}}{k!^{4}}\equiv\frac{p\Gamma_{p}\left(\frac{1}{4}\right)\Gamma_{p}\left(\frac{1}{2}\right)}{\Gamma_{p}\left(\frac{3}{4}\right)}\pmod{p^{3}},\quad\mathrm{if}\quad
p\equiv 1\pmod{4}.$

Here and in what follows, $p$ is an odd prime and the $p$-adic gamma function
is defined as

$\Gamma_{p}(x)=\lim_{m\rightarrow
x}\left(-1\right)^{m}\prod_{\begin{subarray}{c}0<k<m\\\
(k,p)=1\end{subarray}}k,$

where the limit is for $m$ tending to $x$ $p$-adically in $\mathbb{Z}_{\geq
0}$.

The Ramanujan-type supercongruences and their $q$-analogues have attracted a
number of experts. The reader interested in $q$-analogues of supercongruences
is referred to [4, 6, 13, 16, 17, 18, 20]. Especially, a $q$-analogue of Van
Hamme’s (G.2) was proposed by Liu and Wang [11] as follows: for integers
$n\equiv 1\pmod{4}$,

$\sum_{k=0}^{\frac{n-1}{4}}[8k+1]\frac{(q;q^{4})^{4}_{k}}{(q^{4};q^{4})^{4}_{k}}q^{2k}\equiv\frac{(q^{2};q^{4})_{\frac{n-1}{4}}}{(q^{4};q^{4})_{\frac{n-1}{4}}}[n]q^{\frac{1-n}{4}}\pmod{[n]\Phi_{n}(q)^{2}}.$
(1.1)

We now need to introduce some necessary definitions and concepts. Let $a$, $q$
be complex numbers with $\left|q\right|<1$ and $n$ a nonnegative integer. The
$q$-integer is defined as $[n]=[n]_{q}=1+q+\cdots+q^{n-1}$. And the
$q$-shifted factorial is defined by

$(a;q)_{0}=1\quad\mathrm{and}\quad(a;q)_{n}=(1-a)(1-aq)\cdots(1-aq^{n-1}).$

Usually, the multiple $q$-shifted factorial is directly written as

$(a_{1},a_{2},\ldots,a_{r};q)_{m}=(a_{1};q)_{m}(a_{2};q)_{m}\cdots(a_{r};q)_{m},$

where $r\in\mathbb{N}$ and $m\in\mathbb{Z}\cup\left\\{\infty\right\\}$.
Moreover, the $n$-th cyclotomic polynomial in $q$ is represented by
$\Phi_{n}(q)$:

$\displaystyle\Phi_{n}(q)=\prod_{\begin{subarray}{c}1\leqslant k\leqslant n\\\
\gcd(n,k)=1\end{subarray}}\left(q-\zeta^{k}\right),$

where $\zeta$ is an $n$-th primitive root of unity.

By transforming $q$-supercongruences on double sums into the ones from squares
of truncated basic hypergeometric series, El Bachraoui [1] obtained the
following result: for any positive odd integer $n$ with $\gcd(n,6)=1$,

$\sum_{k=0}^{n-1}\sum_{j=0}^{k}c_{q}(j)c_{q}(k-j)\equiv
q^{1-n}[n]^{2}\pmod{[n]\Phi_{n}(q)^{2}},$

where
$c_{q}(k)=[8k+1](q;q^{2})^{2}_{k}(q;q^{2})_{2k}q^{2k^{2}}/((q^{6};q^{6})^{2}_{k}(q^{2};q^{2})_{2k})$.
Recently, many experts paid attention to El Bachraoui’s work and obtained some
new results (see, for example, [5, 14, 10]).

Inspired by the above work, in this paper, by some essential tools including
the ‘creative microscoping’ method introduced by Guo and Zudilin [8] and the
Chinese remainder theorem for coprime polynomials, we shall investigate
several $q$-supercongruences on triple and quadruple sums.

######  Theorem 1.1.

Let $d$ and $n$ be positive integers with $d\geq 3$ and $n\equiv 1\pmod{d}$.
For $k\geq 0$, let

$c_{q}(k)=[2dk+1]\frac{(q;q^{d})^{4}_{k}}{(q^{d};q^{d})^{4}_{k}}q^{(d-2)k}.$

Then, we have

$\displaystyle\sum_{{i+j+k\leq
n-1}}c_{q}(i)c_{q}(j)c_{q}(k)\equiv[n]^{3}q^{\frac{3(1-n)}{d}}\frac{(q^{2};q^{d})^{3}_{\frac{n-1}{d}}}{(q^{d};q^{d})^{3}_{\frac{n-1}{d}}}\pmod{[n]\Phi_{n}(q)^{3}}.$
(1.2)

Letting $d=3$ in Theorem 1.1, we get the following result.

######  Corollary 1.2.

Let $n$ be a positive integer with $n\equiv 1\pmod{3}$. For $k\geq 0$, let

$c_{q}(k)=[6k+1]\frac{(q;q^{3})^{4}_{k}}{(q^{3};q^{3})^{4}_{k}}q^{k}.$

Then, modulo $[n]\Phi_{n}(q)^{3}$,

$\displaystyle\sum_{{i+j+k\leq
n-1}}c_{q}(i)c_{q}(j)c_{q}(k)\equiv[n]^{3}q^{1-n}\frac{(q^{2};q^{3})^{3}_{\frac{n-1}{3}}}{(q^{3};q^{3})^{3}_{\frac{n-1}{3}}}.$

Note that He [9] obtained the following Ramanujan-type supercongruence: for
primes $p\geq 5$,

$\sum_{k=0}^{\frac{p-1}{3}}(6k+1)\frac{\left(\frac{1}{3}\right)^{4}_{k}}{k!^{4}}\equiv-p\Gamma_{p}\left(\tfrac{1}{3}\right)^{3}\pmod{p^{4}},\quad\mathrm{if}\
p\equiv 1\pmod{3},$

and Liu and Wang [12] presented several different $q$-analogues of He’s
supercongruence.

Similarly, setting $d=4$ in Theorem 1.1 yields a new $q$-supercongruence
related to Van Hamme’s (G.2) supercongruence.

######  Corollary 1.3.

Let $n$ be a positive integer with $n\equiv 1\pmod{4}$. For $k\geq 0$, let

$c_{q}(k)=[8k+1]\frac{(q;q^{4})^{4}_{k}}{(q^{4};q^{4})^{4}_{k}}q^{2k}.$

Then, modulo $[n]\Phi_{n}(q)^{3}$,

$\displaystyle\sum_{{i+j+k\leq
n-1}}c_{q}(i)c_{q}(j)c_{q}(k)\equiv[n]^{3}q^{\frac{3(1-n)}{4}}\frac{(q^{2};q^{4})^{3}_{\frac{n-1}{4}}}{(q^{4};q^{4})^{3}_{\frac{n-1}{4}}}.$
(1.3)

Moreover, we shall give a $q$-analogue of the quadruple sum of (G.2) as
follows.

######  Theorem 1.4.

Let $n$ be a positive integer with $n\equiv 1\pmod{4}$. For $k\geq 0$, let

$c_{q}(k)=[8k+1]\frac{(q;q^{4})^{4}_{k}}{(q^{4};q^{4})^{4}_{k}}q^{2k}.$

Modulo $[n]\Phi_{n}(q)^{4}$, we have

$\displaystyle\sum_{{j_{1}+j_{2}+j_{3}+j_{4}\leq
n-1}}c_{q}(j_{1})c_{q}(j_{2})c_{q}(j_{3})c_{q}(j_{4})\equiv[n]^{8}q^{4(1-n)}\left(\sum_{k=0}^{\frac{n-1}{4}}\frac{(q,q,q^{3};q^{4})_{k}}{(q^{4},q^{4},q^{5};q^{4})_{k}}q^{4k}\right)^{4}.$
(1.4)

The rest of the paper is organized as follows. We shall prove Theorem 1.1 in
the next section by making use of the ‘creative microscoping’ method and the
Chinese remainder theorem for coprime polynomials. Finally, in Section 3, we
give a proof of Theorem 1.4 through establishing its parametric
generalization.

##  2\. Proof of Theorem 1.1

We first give an important result due to El Bachraoui [2], which plays an
important role in our proof.

######  Lemma 2.1.

Let $d$, $n$, $t$ be positive integers with $d\geq t\geq 2$ and $n\equiv
1\pmod{d}$. Let $\\{c(k)\\}^{\infty}_{k=0}$ be a sequence of complex numbers.
If $c(k)=0$ for $(n-1)/d<k<n$, then

$\sum_{{k_{1}+\cdots+k_{t}\leq n-1}}c(k_{1})\cdots
c(k_{t})=\left(\sum_{k=0}^{\frac{n-1}{d}}c(k)\right)^{t},$ (2.1)

Furthermore, if $c(ln+k)/c(ln)=c(k)$ for all nonnegative integers $k$ and $l$
such that $0\leq k\leq n-1$, then,

$\displaystyle\sum_{{k_{1}+\cdots+k_{t}=ln+k}}c(k_{1})c(k_{2})\cdots c(k_{t})$
$\displaystyle\quad\quad=\sum_{{k_{1}+\cdots+k_{t}=k}}c(k_{1})c(k_{2})\cdots
c(k_{t})$
$\displaystyle\quad\quad\times\sum_{s_{1}=0}^{l}c(s_{1}n)\sum_{s_{2}=0}^{l-s_{1}}c(s_{2}n)\cdots\sum_{s_{t-1}=0}^{l-s_{1}-\cdots-
s_{t-2}}c(s_{t-1}n)c\left((l-s_{1}-\cdots-s_{t-1})n\right).$ (2.2)

In order to prove Theorem 1.1, we need to confirm the following two lemmas
based on the result presented in [7].

######  Lemma 2.2.

Let $n$, $d$ be positive integers with $d\geq 3$ and $n\equiv 1\pmod{d}$, $a$
and $b$ indeterminates. For $k\geq 0$, let

$z_{q}(k)=[2dk+1]\frac{\left(q,aq,q/a,q/b;q^{d}\right)_{k}}{\left(q^{d},aq^{d},q^{d}/a,bq^{d};q^{d}\right)_{k}}b^{k}q^{(d-2)k}.$

Then, we have, modulo $[n](1-aq^{n})(a-q^{n})$,

$\displaystyle\sum_{{i+j+k\leq
n-1}}z_{q}(i)z_{q}(j)z_{q}(k)\equiv[n]^{3}\frac{\left(b/q\right)^{\frac{3(n-1)}{d}}\left(q^{2}/b;q^{d}\right)^{3}_{\frac{n-1}{d}}}{\left(bq^{d};q^{d}\right)^{3}_{\frac{n-1}{d}}}.$
(2.3)

###### Proof.

For $n=1$, the result is clearly true. We now assume that $n$ is an integer
with $n>1$. Recall that Guo and Schlosser [7, Equation (4.3)] gave the
following result: modulo ${[n](1-aq^{n})(a-q^{n})(b-q^{n})}$,

$\displaystyle\sum_{k=0}^{M}z_{q}(k)$
$\displaystyle\equiv[n]\frac{\left(b/q\right)^{\frac{n-1}{d}}\left(q^{2}/b;q^{d}\right)_{\frac{n-1}{d}}}{\left(bq^{d};q^{d}\right)_{\frac{n-1}{d}}}\frac{(b-q^{n})(ab-1-a^{2}+aq^{n})}{(a-b)(1-ab)}$
$\displaystyle\quad+[n]\frac{(q,q^{d-1};q^{d})_{\frac{n-1}{d}}}{(aq^{d},q^{d}/a;q^{d})_{\frac{n-1}{d}}}\frac{(1-aq^{n})(a-q^{n})}{(a-b)(1-ab)},$
(2.4)

where $M=(n-1)/d$ or $n-1$. It is clear to see that
$\left(q;q^{d}\right)_{k}\equiv 0\pmod{\Phi_{n}(q)}$ for $(n-1)/d<k<n$. This
means that $z_{q}(k)\equiv 0\pmod{\Phi_{n}(q)}$ for $(n-1)/d<k<n$. Therefore,
employing Lemma 2.1 with $t=3$ produces

$\sum_{{i+j+k\leq n-1}}z_{q}(i)z_{q}(j)z_{q}(k)\equiv 0\pmod{\Phi_{n}(q)}.$
(2.5)

Let $\zeta\neq 1$ be an $n$-th root of unity, not necessarily primitive.
Namely, $\zeta$ is a primitive $m$-th root of unity with $m\mid n$ and $m>1$.
From (2.5), we get

$\sum_{{i+j+k\leq m-1}}z_{\zeta}(i)z_{\zeta}(j)z_{\zeta}(k)=0.$ (2.6)

It is not difficult to check that $z_{\zeta}(lm+s)/z_{\zeta}(lm)=z_{\zeta}(s)$
for nonnegative integers $l$ and $s$ with $0\leq s\leq m-1$. Applying Lemma
2.1 with $t=3$ again, we have

$\displaystyle\sum_{{i+j+k\leq n-1}}z_{\zeta}(i)z_{\zeta}(j)z_{\zeta}(k)$
$\displaystyle\quad=\sum_{r=0}^{n-1}\sum_{i+j+k=r}z_{\zeta}(i)z_{\zeta}(j)z_{\zeta}(k)$
$\displaystyle\quad=\sum_{l=0}^{\frac{n}{m}-1}\sum_{s=0}^{m-1}\sum_{i+j+k=lm+s}z_{\zeta}(i)z_{\zeta}(j)z_{\zeta}(k)$
$\displaystyle\quad=\sum_{l=0}^{\frac{n}{m}-1}\sum_{a=0}^{l}z_{\zeta}(am)\sum_{t=0}^{l-a}z_{\zeta}(tm)z_{\zeta}((l-a-t)m)\sum_{s=0}^{m-1}\sum_{i+j+k=s}z_{\zeta}(i)z_{\zeta}(j)z_{\zeta}(k)$
$\displaystyle\quad=0.$

Since the above equality is true for any $n$-th root of unity $\zeta\neq 1$,
we conclude that

$\sum_{{i+j+k\leq n-1}}z_{q}(i)z_{q}(j)z_{q}(k)\equiv 0\pmod{[n]}.$ (2.7)

On the other hand, letting $a=q^{n}$ or $q^{-n}$ in (2.4), we have

$\sum_{k=0}^{M}\tilde{z}_{q}(k)=[n]\frac{\left(b/q\right)^{\frac{n-1}{d}}\left(q^{2}/b;q^{d}\right)_{\frac{n-1}{d}}}{\left(bq^{d};q^{d}\right)_{\frac{n-1}{d}}},$

where

$\tilde{z}_{q}(k)=[2dk+1]\frac{\left(q,q^{1+n},q^{1-n},q/b;q^{d}\right)_{k}}{\left(q^{d},q^{d+n},q^{d-n},bq^{d};q^{d}\right)_{k}}b^{k}q^{(d-2)k}.$

Note the fact that $\tilde{z}_{q}(k)=0$ for $(n-1)/d<k<n$. Using Lemma 2.1
with $t=3$ again, we arrive at

$\sum_{{i+j+k\leq
n-1}}\tilde{z}_{q}(i)\tilde{z}_{q}(j)\tilde{z}_{q}(k)=[n]^{3}\frac{\left(b/q\right)^{\frac{3(n-1)}{d}}\left(q^{2}/b;q^{d}\right)^{3}_{\frac{n-1}{d}}}{\left(bq^{d};q^{d}\right)^{3}_{\frac{n-1}{d}}}.$

Then, we have the following $q$-congruence involving a variable $a$: modulo
$\left(1-aq^{n}\right)\left(a-q^{n}\right)$,

$\sum_{{i+j+k\leq
n-1}}z_{q}(i)z_{q}(j)z_{q}(k)\equiv[n]^{3}\frac{\left(b/q\right)^{\frac{3(n-1)}{d}}\left(q^{2}/b;q^{d}\right)^{3}_{\frac{n-1}{d}}}{\left(bq^{d};q^{d}\right)^{3}_{\frac{n-1}{d}}}.$
(2.8)

From (2.7) and (2.8), we conclude that (2.8) is true modulo
$[n]\left(1-aq^{n}\right)\left(a-q^{n}\right)$, as desired. ∎

######  Lemma 2.3.

Let $n$, $d$ be positive integers with $d\geq 3$ and $n\equiv 1\pmod{d}$, $a$
and $b$ indeterminates. For $k\geq 0$, let

$z_{q}(k)=[2dk+1]\frac{\left(q,aq,q/a,q/b;q^{d}\right)_{k}}{\left(q^{d},aq^{d},q^{d}/a,bq^{d};q^{d}\right)_{k}}b^{k}q^{(d-2)k}.$

Then, we have

$\displaystyle\sum_{{i+j+k\leq
n-1}}z_{q}(i)z_{q}(j)z_{q}(k)\equiv[n]^{3}\frac{\left(q,q^{d-1};q^{d}\right)^{3}_{\frac{n-1}{d}}}{\left(aq^{d},q^{d}/a;q^{d}\right)^{3}_{\frac{n-1}{d}}}\pmod{b-q^{n}}.$
(2.9)

###### Proof.

Letting $b=q^{n}$ in (2.4), we get

$\sum_{k=0}^{M}\hat{z}_{q}(k)=[n]\frac{\left(q,q^{d-1};q^{d}\right)_{\frac{n-1}{d}}}{\left(aq^{d},q^{d}/a;q^{d}\right)_{\frac{n-1}{d}}},$

where

$\hat{z}_{q}(k)=[2dk+1]\frac{\left(q,aq,q/a,q^{1-n};q^{d}\right)_{k}}{\left(q^{d},aq^{d},q^{d}/a,q^{d+n};q^{d}\right)_{k}}q^{(n+d-2)k}.$

Note that $\hat{z}_{q}(k)=0$ for $(n-1)/d<k<n$. From Lemma 2.1 with $t=3$, we
obtain

$\sum_{{i+j+k\leq
n-1}}\hat{z}_{q}(i)\hat{z}_{q}(j)\hat{z}_{q}(k)=[n]^{3}\frac{\left(q,q^{d-1};q^{d}\right)^{3}_{\frac{n-1}{d}}}{\left(aq^{d},q^{d}/a;q^{d}\right)^{3}_{\frac{n-1}{d}}},$

which means that

$\sum_{{i+j+k\leq
n-1}}z_{q}(i)z_{q}(j)z_{q}(k)\equiv[n]^{3}\frac{\left(q,q^{d-1};q^{d}\right)^{3}_{\frac{n-1}{d}}}{\left(aq^{d},q^{d}/a;q^{d}\right)^{3}_{\frac{n-1}{d}}}\pmod{b-q^{n}}$

is true. ∎

###### Proof of Theorem 1.1.

It is easy to see that $[n]\left(1-aq^{n}\right)\left(a-q^{n}\right)$ and
$b-q^{n}$ are relatively prime polynomials. Noting the relations

$\displaystyle\frac{\left(b-q^{n}\right)(ab-1-a^{2}+aq^{n})}{(a-b)(1-ab)}$
$\displaystyle\equiv 1\pmod{\left(1-aq^{n}\right)(a-q^{n})},$
$\displaystyle\frac{(1-aq^{n})(a-q^{n})}{(a-b)(1-ab)}$ $\displaystyle\equiv
1\pmod{b-q^{n}},$

and employing the Chinese remainder theorem for coprime polynomials, we derive
the following result from (2.3) and (2.9): modulo
$[n](1-aq^{n})(a-q^{n})(b-q^{n})$,

$\displaystyle\sum_{{i+j+k\leq n-1}}z_{q}(i)z_{q}(j)z_{q}(k)$
$\displaystyle\quad\quad\equiv[n]^{3}\left\\{\frac{(b-q^{n})(ab-1-a^{2}+aq^{n})}{(a-b)(1-ab)}\frac{\left(b/q\right)^{\frac{3(n-1)}{d}}\left(q^{2}/b;q^{d}\right)^{3}_{\frac{n-1}{d}}}{\left(bq^{d};q^{d}\right)^{3}_{\frac{n-1}{d}}}\right.$
$\displaystyle\quad\quad\quad\quad\left.+\frac{(1-aq^{n})(a-q^{n})}{(a-b)(1-ab)}\frac{\left(q,q^{d-1};q^{d}\right)^{3}_{\frac{n-1}{d}}}{\left(aq^{d},q^{d}/a;q^{d}\right)^{3}_{\frac{n-1}{d}}}\right\\}.$
(2.10)

Taking $b\rightarrow 1$ in (2.10), we deduce that, modulo
$\Phi_{n}(q)^{2}(1-aq^{n})(a-q^{n})$,

$\displaystyle\sum_{{i+j+k\leq
n-1}}z_{q}(i)z_{q}(j)z_{q}(k)\equiv[n]^{3}q^{\frac{3(1-n)}{d}}\frac{(q^{2};q^{d})^{3}_{\frac{n-1}{d}}}{(q^{d};q^{d})^{3}_{\frac{n-1}{d}}}.$
(2.11)

Here we have employed the relation:

$(1-q^{n})(1+a^{2}-a-aq^{n})=(1-a)^{2}+(1-aq^{n})(a-q^{n}).$

Finally, letting $a\rightarrow 1$ in (2.11) and noting $1-q^{n}$ has the
factor $\Phi_{n}(q)$, we see that the congruence (1.2) is true modulo
$\Phi_{n}(q)^{4}$. On the other hand, our proof of (2.7) is still valid for
$a=b=1$, which means that (1.2) is also correct modulo $[n]$. Since the least
common multiple of $\Phi_{n}(q)^{4}$ and $[n]$ is $[n]\Phi_{n}(q)^{3}$, we
finish the proof of Theorem 1.1. ∎

##  3\. Proof of Theorem 1.4

We first establish a parametric extension of Theorem 1.4 as follows.

######  Theorem 3.1.

Let $d$ and $n$ be positive integers with $d\geq 4$ and $n\equiv 1\pmod{d}$,
$c$ an indeterminate. For $k\geq 0$, let

$c_{q}(k)=[2dk+1]\frac{(q;q^{d})^{5}_{k}(cq;q^{d})_{k}}{(q^{d};q^{d})^{5}_{k}\left(q^{d}/c;q^{d}\right)_{k}}\left(\frac{q^{2d-3}}{c}\right)^{k}.$

Then, we have, modulo $[n]\Phi_{n}(q)^{4}$,

$\displaystyle\sum_{{j_{1}+j_{2}+j_{3}+j_{4}\leq
n-1}}c_{q}(j_{1})c_{q}(j_{2})c_{q}(j_{3})c_{q}(j_{4})$
$\displaystyle\quad\quad\equiv[n]^{4}\left(cq\right)^{\frac{4(1-n)}{d}}\frac{\left(cq^{2};q^{d}\right)^{4}_{\frac{n-1}{d}}}{(q^{d}/c;q^{d})^{4}_{\frac{n-1}{d}}}\left(\sum_{k=0}^{\frac{n-1}{d}}\frac{\left(q,q,cq,q^{d-1};q^{d}\right)_{k}}{\left(q^{d},q^{d},q^{d},cq^{2};q^{d}\right)_{k}}q^{dk}\right)^{4}.$
(3.1)

Clearly, Theorem 1.4 can be gotten by setting $c=q^{d-1}$ and $d=4$ in Theorem
3.1.

In order to confirm Theorem 3.1, we need to prove the following parametric
extension.

######  Lemma 3.2.

Let $d$ and $n$ be positive integers with $d\geq 4$ and $n\equiv 1\pmod{d}$,
$a$, $b$ and $c$ indeterminates. For $k\geq 0$, let

$z_{q}(k,a,b)=[2dk+1]\frac{\left(aq,q/a,bq,q/b,cq,q;q^{d}\right)_{k}}{\left(q^{d}/q,aq^{d},q^{d}/b,bq^{d},q^{d}/c,q^{d};q^{d}\right)_{k}}\left(\frac{q^{2d-3}}{c}\right)^{k}.$

Then, modulo $[n](1-aq^{n})(a-q^{n})(1-bq^{n})(b-q^{n})$,

$\displaystyle\sum_{{j_{1}+j_{2}+j_{3}+j_{4}\leq
n-1}}z_{q}(j_{1},a,b)z_{q}(j_{2},a,b)z_{q}(j_{3},a,b)z_{q}(j_{4},a,b)$
$\displaystyle\quad\quad\equiv[n]^{4}\left(cq\right)^{\frac{4(1-n)}{d}}\frac{\left(cq^{2};q^{d}\right)^{4}_{\frac{n-1}{d}}}{\left(q^{d}/c;q^{d}\right)_{\frac{n-1}{d}}}$
$\displaystyle\quad\quad\times\left\\{\frac{(1-bq^{n})(b-q^{n})(-1-a^{2}+aq^{n})}{(a-b)(1-ab)}\left(\sum_{k=0}^{\frac{n-1}{d}}\frac{\left(aq,q/a,cq,q^{d-1};q^{d}\right)_{k}}{\left(bq^{d},q^{d}/b,cq^{2},q^{d};q^{d}\right)_{k}}q^{dk}\right)^{4}\right.$
$\displaystyle\quad\quad\left.\quad+\frac{(1-aq^{n})(a-q^{n})(-1-b^{2}+bq^{n})}{(b-a)(1-ab)}\left(\sum_{k=0}^{\frac{n-1}{d}}\frac{\left(bq,q/b,cq,q^{d-1};q^{d}\right)_{k}}{\left(aq^{d},q^{d}/a,cq^{2},q^{d};q^{d}\right)_{k}}q^{dk}\right)^{4}\right\\}.$
(3.2)

###### Proof.

For $n=1$, the result is obviously true. We now assume that $n$ is an integer
with $n>1$. Setting $r=1$ in [19, Theorem 5.7], we have, modulo
$[n](1-aq^{n})(a-q^{n})(1-bq^{n})(b-q^{n})$,

$\displaystyle\sum_{k=0}^{M}z_{q}(k,a,b)$
$\displaystyle\>\equiv[n](cq)^{\frac{1-n}{d}}\frac{(cq^{2};q^{d})_{\frac{n-1}{d}}}{(q^{d}/c;q^{d})_{\frac{n-1}{d}}}$
$\displaystyle\quad\times\left\\{\frac{(1-bq^{n})(b-q^{n})(-1-a^{2}+aq^{n})}{(a-b)(1-ab)}\sum_{k=0}^{\frac{n-1}{d}}\frac{\left(aq,q/a,cq,q^{d-1};q^{d}\right)_{k}}{\left(aq^{d},q^{d}/a,cq^{2},q^{d};q^{d}\right)_{k}}q^{dk}\right.$
$\displaystyle\quad\left.\quad+\frac{(1-aq^{n})(a-q^{n})(-1-b^{2}+bq^{n})}{(b-a)(1-ab)}\sum_{k=0}^{\frac{n-1}{d}}\frac{\left(bq,q/b,cq,q^{d-1};q^{d}\right)_{k}}{\left(aq^{d},q^{d}/a,cq^{2},q^{d};q^{d}\right)_{k}}q^{dk}\right\\},$
(3.3)

where $M=(n-1)/d$ or $n-1$. Then, from (3.3), we can deduce that

$\displaystyle\sum_{k=0}^{M}z_{q}(k,a,b)\equiv 0\pmod{[n]}.$ (3.4)

Since $\left(q;q^{d}\right)_{k}\equiv 0\pmod{\Phi_{n}(q)}$ for $(n-1)/d<k<n$,
we deduce that $z_{q}(k,a,b)\equiv 0\pmod{\Phi_{n}(q)}$ for $(n-1)/d<k<n$.
Then, applying Lemma 2.1 with $t=4$, we arrive at

$\sum_{{j_{1}+j_{2}+j_{3}+j_{4}\leq
n-1}}z_{q}(j_{1},a,b)z_{q}(j_{2},a,b)z_{q}(j_{3},a,b)z_{q}(j_{4},a,b)\equiv
0\pmod{\Phi_{n}(q)}.$ (3.5)

Letting $\zeta\neq 1$ be an $n$-th root of unity, not necessarily primitive.
Namely, $\zeta$ is a primitive $m$-th root of unity with $m\mid n$ and $m>1$.
From (3.5), we have

$\sum_{{j_{1}+j_{2}+j_{3}+j_{4}\leq
m-1}}z_{\zeta}(j_{1},a,b)z_{\zeta}(j_{2},a,b)z_{\zeta}(j_{3},a,b)z_{\zeta}(j_{4},a,b)=0.$
(3.6)

Observe that $z_{\zeta}(lm+k)/z_{\zeta}(lm)=z_{\zeta}(k)$ for nonnegative
integers $l$ and $k$ with $0\leq k\leq m-1$. Using Lemma 2.1 with $t=4$ again,
we obtain

$\displaystyle\sum_{{j_{1}+j_{2}+j_{3}+j_{4}\leq
n-1}}z_{\zeta}(j_{1},a,b)z_{\zeta}(j_{2},a,b)z_{\zeta}(j_{3},a,b)z_{\zeta}(j_{4},a,b)$
$\displaystyle\quad=\sum_{r=0}^{n-1}\sum_{{j_{1}+j_{2}+j_{3}+j_{4}=r}}z_{\zeta}(j_{1},a,b)z_{\zeta}(j_{2},a,b)z_{\zeta}(j_{3},a,b)z_{\zeta}(j_{4},a,b)$
$\displaystyle\quad=\sum_{l=0}^{\frac{n}{m}-1}\sum_{k=0}^{m-1}\sum_{{j_{1}+j_{2}+j_{3}+j_{4}=lm+k}}z_{\zeta}(j_{1},a,b)z_{\zeta}(j_{2},a,b)z_{\zeta}(j_{3},a,b)z_{\zeta}(j_{4},a,b)$
$\displaystyle\quad=\sum_{l=0}^{\frac{n}{m}-1}\sum_{k_{1}=0}^{l}z_{\zeta}(k_{1}m,a,b)\sum_{k_{2}=0}^{l-k_{1}}z_{\zeta}(k_{2}m,a,b)\sum_{k_{3}=0}^{l-k_{1}-k_{2}}z_{\zeta}(k_{3}m,a,b)z_{\zeta}\left((l-k_{1}-k_{2}-k_{3})m,a,b\right)$
$\displaystyle\quad\quad\times\sum_{k=0}^{m-1}\sum_{{j_{1}+j_{2}+j_{3}+j_{4}=k}}z_{\zeta}(j_{1},a,b)z_{\zeta}(j_{2},a,b)z_{\zeta}(j_{3},a,b)z_{\zeta}(j_{4},a,b)$
$\displaystyle\quad=0,$

which indicates that

$\displaystyle\sum_{\begin{subarray}{c}j_{1}+j_{2}+j_{3}+j_{4}\leq
n-1\end{subarray}}z_{q}(j_{1},a,b)z_{q}(j_{2},a,b)z_{q}(j_{3},a,b)z_{q}(j_{4},a,b)\equiv
0\pmod{[n]}$ (3.7)

is true.

Moreover, letting $a=q^{n}$ or $a=q^{-n}$ in (3.3) gives

$\displaystyle\sum_{k=0}^{M}z_{q}(k,q^{n},b)$
$\displaystyle=\sum_{k=0}^{M}z_{q}(k,q^{-n},b)$
$\displaystyle=[n](cq)^{\frac{1-n}{d}}\frac{(cq^{2};q^{d})_{\frac{n-1}{d}}}{(q^{d}/c;q^{d})_{\frac{n-1}{d}}}\sum_{k=0}^{\frac{n-1}{d}}\frac{(q^{1-n},q^{1+n},cq,q^{d-1};q^{d})_{k}}{(bq^{d},q^{d}/b,cq^{2},q^{d};q^{d})_{k}}q^{dk}.$
(3.8)

Noticing that $z_{q}(k,q^{n},b)=z_{q}(k,q^{-n},b)=0$ for $(n-1)/d<k<n$. Then,
employing Lemma 2.1 with $t=4$, we obtain, modulo $(1-aq^{n})(a-q^{n})$,

$\displaystyle\sum_{{j_{1}+j_{2}+j_{3}+j_{4}\leq
n-1}}z_{q}(j_{1},a,b)z_{q}(j_{2},a,b)z_{q}(j_{3},a,b)z_{q}(j_{4},a,b)$
$\displaystyle\quad\quad\quad\equiv[n]^{4}(cq)^{\frac{4(1-n)}{d}}\frac{(cq^{2};q^{d})^{4}_{\frac{n-1}{d}}}{(q^{d}/c;q^{d})^{4}_{\frac{n-1}{d}}}\left(\sum_{k=0}^{\frac{n-1}{d}}\frac{(aq,q/a,cq,q^{d-1};q^{d})_{k}}{(bq^{d},q^{d}/b,cq^{2},q^{d};q^{d})_{k}}q^{dk}\right)^{4}.$
(3.9)

Similarly, setting $b=q^{n}$ or $b=q^{-n}$ in (3.3) and utilizing Lemma 2.1
with $t=4$ again, we get, modulo $(1-bq^{n})(b-q^{n})$,

$\displaystyle\sum_{{j_{1}+j_{2}+j_{3}+j_{4}\leq
n-1}}z_{q}(j_{1},a,b)z_{q}(j_{2},a,b)z_{q}(j_{3},a,b)z_{q}(j_{4},a,b)$
$\displaystyle\quad\quad\quad\equiv[n]^{4}(cq)^{\frac{4(1-n)}{d}}\frac{(cq^{2};q^{d})^{4}_{\frac{n-1}{d}}}{(q^{d}/c;q^{d})^{4}_{\frac{n-1}{d}}}\left(\sum_{k=0}^{\frac{n-1}{d}}\frac{(bq,q/b,cq,q^{d-1};q^{d})_{k}}{(aq^{d},q^{d}/a,cq^{2},q^{d};q^{d})_{k}}q^{dk}\right)^{4}.$
(3.10)

Noting that the polynomials $[n]$, $(1-aq^{n})(a-q^{n})$ and
$(1-bq^{n})(b-q^{n})$ are relatively prime. Then, utilizing the Chinese
remainder theorem for coprime polynomials and the relations:

$\displaystyle\frac{(1-bq^{n})(b-q^{n})(-1-a^{2}+aq^{n})}{(a-b)(1-ab)}$
$\displaystyle\equiv 1\pmod{(1-aq^{n})(a-q^{n})},$
$\displaystyle\frac{(1-aq^{n})(a-q^{n})(-1-b^{2}+bq^{n})}{(b-a)(1-ab)}$
$\displaystyle\equiv 1\pmod{(1-bq^{n})(b-q^{n})},$

we can obtain the desired result from (3.7), (3.9) and (3.10). ∎

Subsequently, we start to prove Theorem 3.1.

###### Proof of Theorem 3.1.

Letting $b\rightarrow 1$ in the congruence (3.2) produces, modulo
$\Phi_{n}(q)^{3}(1-aq^{n})(a-q^{n})$,

$\displaystyle\sum_{{j_{1}+j_{2}+j_{3}+j_{4}\leq
n-1}}z_{q}(j_{1},a,1)z_{q}(j_{2},a,1)z_{q}(j_{3},a,1)z_{q}(j_{4},a,1)$
$\displaystyle\quad\quad\equiv[n]^{4}\left(cq\right)^{\frac{4(1-n)}{d}}\frac{(cq^{2};q^{d})^{4}_{\frac{n-1}{d}}}{(q^{d}/c;q^{d})^{4}_{\frac{n-1}{d}}}$
$\displaystyle\quad\quad\times\left\\{\frac{(1-q^{n})^{2}(-1-a^{2}+aq^{n})}{(a-1)(1-a)}\left(\sum_{k=0}^{\frac{n-1}{d}}\frac{(aq,q/a,cq,q^{d-1};q^{d})_{k}}{(q^{d},q^{d},cq^{2},q^{d};q^{d})_{k}}q^{dk}\right)^{4}\right.$
$\displaystyle\quad\left.\quad-\frac{(1-aq^{n})(a-q^{n})(-2+q^{n})}{(a-1)(1-a)}\left(\sum_{k=0}^{\frac{n-1}{d}}\frac{(q,q,cq,q^{d-1};q^{d})_{k}}{(aq^{d},q^{d}/a,cq^{2},q^{d};q^{d})_{k}}q^{dk}\right)^{4}\right\\}.$
(3.11)

Next, setting $a\rightarrow 1$ and utilizing the L’Hospital rule in (3), we
arrive at

$\displaystyle\sum_{{j_{1}+j_{2}+j_{3}+j_{4}\leq
n-1}}c_{q}(j_{1})c_{q}(j_{2})c_{q}(j_{3})c_{q}(j_{4})$
$\displaystyle\quad\equiv[n]^{4}\left(cq\right)^{\frac{4(1-n)}{d}}\frac{\left(cq^{2};q^{d}\right)^{4}_{\frac{n-1}{d}}}{(q^{d}/c;q^{d})^{4}_{\frac{n-1}{d}}}\left(\sum_{k=0}^{\frac{n-1}{d}}\frac{\left(q,q,cq,q^{d-1};q^{d}\right)_{k}}{\left(q^{d},q^{d},q^{d},cq^{2};q^{d}\right)_{k}}q^{dk}\right)^{4}\pmod{\Phi_{n}(q)^{5}}.$
(3.12)

On the other hand, our proof of (3.7) is still valid for $a=b=1$, which means
that (3.12) is also correct modulo $[n]$. Since the least common multiple of
$\Phi_{n}(q)^{5}$ and $[n]$ is $[n]\Phi_{n}(q)^{4}$, we derive Theorem 3.1 ∎

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