# Image Inpainting using Partial Convolution

Amey Kulkarni  Harsh Patel  Shivam Sahni  Udit Vyas  Indian Institute of
Technology Gandhinagar, India

###### Abstract

Image Inpainting is one of the very popular tasks in the field of image
processing with broad applications in computer vision. In various practical
applications, images are often deteriorated by noise due to the presence of
corrupted, lost or undesirable information. There have been various
restoration techniques used in the past with both classical [1] [2] and deep
learning approach for handling such issues [3] . Some traditional methods
include image restoration by filling gap pixels using the nearby known pixels
or using the moving average over the same. The aim of this project is to
perform image inpainting using robust deep learning methods that use partial
convolution layers.

Partial Convolution, Image Inpainting, Image Reconstruction

##  1 Problem Statement

Image Inpainting is an image reconstruction technique, where missing sections
of an image(holes) are filled or ”predicted”. This technique finds several
applications, some of them include removing unwanted objects from an image,
restore damaged portions of old images, or removing unwanted text. Classical
vision approaches in Image Inpainting deal with the propagation of unmasked
image parameters onto the holes. These methods have shown varied success,
however, they all lack the ability to semantically reconstruct the holes.
Recent work on Image Inpainting has predominantly been based on learning-based
approaches which can effectively learn the semantics of the image. Generative
Adversarial Networks (GANs) have also been used in this domain to generate
satisfactory results. [3]

This project is inspired from the original paper ”Image Inpainting for
Irregular Holes Using Partial Convolutions” [4]. This approach of image
inpainting is capable of restoring image with any arbitrary shaped holes. The
central idea of this work involves in its Partial Convolution layers, and
specific high-level feature loss networks.

The upcoming section of the report contain sections describing the Approach,
Quantitative and Qualitative results, and Key Observations.

##  2 Approach

In this project, we make use of Partial Convolution Layers with mask update
method to perform the inpainting operation. The following section describes
the approach in detail.

Figure 1: Pipeline of the approach

###  2.1 Dataset

We use the CelebA-HQ/256 [5], a large-scale dataset that has 30,000 human face
images each of size 256x256. We divided the dataset into train, validation and
tests datasets in the proportion of 70%, 15%, and 15%. For applications
described in the later sections, we make use of samples of image size 256x256
from Places2 dataset [6].

###  2.2 Partial Convolution Layers

A partial convolution, as the name suggests, is very similar to a usual
Convolution layer except that it only performs the convolution operations on
the pixels where the input pixel are not currently masked or ”available”. More
formally, if W is the convolution filter, b is its corresponding filter, M is
the binary mask corresponding to the convolution filter, and X are the image
features, then the partial convolution is defined as-

$x^{\prime}=\begin{cases}W^{T}(X\circledast
M)\frac{sum(\textbf{1}))}{sum(M)}+b,&\text{ if }sum(M)>0\\\ 0,&\text{ if
}otherwise\end{cases}$ (1)

Note that 1 is a vector with same shape as M, containing all the elements as
one.

###  2.3 Mask Update Step

Every Partial Convolution layer is followed by a mask update step. As seen in
Section 2.2, only the available pixels are used for calculating the
convolution output. The mask update step is about updating a mask value from 0
to 1, if there is at least one valid input corresponding to the pixel under
consideration. Mathematically, the updated mask $m^{\prime}$ can be defined as
-

$m^{\prime}=\begin{cases}1,&\text{ if }sum(M)\\\ 0,&\text{ if
}otherwise\end{cases}$ (2)

It is important to observe that after successive iterations of the masked
image through the network, all values of the mask will eventually be filled
with ones, if the input contained at least one valid pixel.

###  2.4 UNet Architecture

Figure 2: Skeleton of typical UNET

The UNet architecture, as can be seen from figure 2, has a ’U’ shape. In image
2 there are only two layers, however a typical UNet contains four.

The two limbs of figure 2 represent the contraction(left portion, also called
encoders) and expansion(right portion, also called decoders) paths
respectively.

The contraction path is a stack of convolutional neural networks and max
pooling layers applied successively. Every layer contains two CNNs, each
followed by an activation function. Typically, the activation function for all
layers is ReLu. The output of these two CNNs is passed through a max pooling
layer with a kernel size that is typically equal to 2. The convolution
operations increase the number of channels(say from 3 for an RGB image) to a
greater number of channels in a latent space(say 256 to 512). Since we apply
max pooling(kernel size $\geq$2), the height and width reduce.

Functionally, the contraction path is used to capture information from images
that helps us identify people, places or other entities relevant to us. In
other words, it determines the ”context” of the image. This can be understood
by the fact that as the image goes through the contraction path, its size
reduces, but the number of channels increase. This means that information
regarding spatial locality is being lost, however information regarding the
features in the image is being amplified.

The expansion path is also a stack of CNNs, but here instead of the max
pooling layers, we have upsampling layers. Every decoding layer consists of
two CNNs, each followed by an activation function(the last layer usually has
no activation function).

An important property of UNet is the presence of skip connections. These refer
to the connections that run across the pipeline (the ”copy and crop”
connections in image 2). These separate it from regular encoder-decoder nets,
that just use the output from the previous layer. The output from the
corresponding layer(same depth) of the contraction path is concatenated with
the output of the previous decoding layer. Therefore the size of the image is
gradually restored while the number of channels are reduced through the CNN
layers.

The function of the expansion path is recovering the spatial locality
information that was lost in the contraction path. Not only does this happen
by upsampling, but also by the skip connections. This is because, the feature
map from the encoder at the same depth level contains locality information.

All in all, the UNet architecture is one arranged in a U-shape, with a
contraction path and an expansion path containing encoders and decoders
respectively, and where shallow layers(encoders) are connected to deeper
layers(decoders) through skip links.

###  2.5 Modified UNet (Our Contribution)

For our task of image inpainting, we have used a total of seven layers in our
model. In other words, we have seven encoders and seven decoders. We have the
same Module List inside each encoder and decoder as described in the previous
section. However there are a few changes.

We have used the following activation functions for different layers-

  * •

ReLu for all the encoding layers.

  * •

Leaky ReLu for all decoding layers except the first layer(depth = 1).

  * •

No activation for the first decoding layer.

Note that the first decoding layer is the topmost layer in the architecture
and the final layer that is used in the sequence followed by the pipeline.

For the upsampling step we have used the bilinear mode of Pytorch since that
is recommended for 4D data.

We have not required to crop the image from the encoder in any skip
connection, since we have maintained the invariant that the image size at the
$i^{th}$ encoder and decoder is
$\frac{H}{2^{(}i-1)}\times\frac{W}{2^{(}i-1)}$. We ensure this by having
padding of size $n$ for a kernel size of $2\times n+1$, with no stride or
dilation for every convolution operation. This ensures that the output image
size of any CNN is the same as its input image size. The Max Pooling layer
with kernel dimension of 2x2 is responsible for contraction by 2 while the
Upsampling layer with scale factor = 2 is responsible for expansion.

###  2.6 Loss Functions

Choice of Loss functions plays a significant role in helping us to achieve
both good predictions and faster convergence in any deep learning framework.
In this image restoration problem, the following loss functions take charge in
improving both features and their compositions in the restored image. For the
subsequent sections please consider the following notation.

  * •

$I_{gt}$: Ground Truth Image

  * •

$I_{out}$: Output Restored image generated by the network

  * •

$I_{comp}$: Output Restored image with non-hole regions replaced with the
ground truth.

  * •

$\psi_{i}$: Activation map of the ith layer of the network

  * •

$\xi(I)$ = Gram matrix of I

  * •

$K_{h*w}$: Gram matrix normalising factor, dependent on the size of the input
(h*w)

####  2.6.1 Total Variation Loss

The total variation loss is quite analogous to the nature of regularization
losses. It is responsible for maintaining spatial continuity along with
smoothness in the restored image. It is widely used in similar computer vision
applications like Style Transfer and also, in digital signal processing for
removal of noise.

$\centering L_{tv}=\sum_{(i,j),(i,j+1)\epsilon
I}\left|I_{comp}^{[i,j+1]}\right|+\sum_{(i,j),(i+1,j)\epsilon
I}\left|I_{comp}^{[i+1,j]}\right|\@add@centering$ (3)

####  2.6.2 Perceptual Loss

Perceptual Loss also known as the feature reconstruction loss, encourages the
pixels in the restored image to have similar feature representations rather
than exactly matching the ground truth pixels [7].

These feature representation are obtained by passing the image through a pre-
trained Convolutional Neural Network (VGG-16 [8] in our experiments) which is
a series of few convolution layers followed by some fully connected layers.
The layers from the input to the last max pooling layer are used for the
feature extraction.

This loss function tries to minimize the semantic differences between the
features that are activated for the input image and for the reconstructed
image, at the different available layers in the network.

$L_{perceptual}=\sum_{i=0}^{N-1}\left|\psi_{i}(I_{out})-\psi_{i}(I_{gt})\right|+\sum_{i=0}^{N-1}\left|\psi_{i}(I_{comp})-\psi_{i}(I_{gt})\right|$
(4)

####  2.6.3 Style Loss

Style loss is used to incorporate texture like feature details of the ground
truth image rather than its global arrangement into the restored image. It
helps to capture general appearance of the ground truth image in terms of
colors and localised compositions [9].

These style represented as gram matrices which are obtained by computing the
correlations between the feature maps (same as in Section 2.6.2) obtained from
the activations of the convolutional neural network at different layers of the
network.

This loss function tries to minimize the L1 distance between the entries of
the Gram matrix from the style of the input image and both the Gram matrix of
the reconstructed image and $I_{comp}$ generated after every iteration.

$\centering
L_{style}=\sum_{i=0}^{N-1}\left|\xi(\psi_{i}(I_{out}))-\xi(\psi_{i}(I_{gt}))\right|+\sum_{i=0}^{N-1}\left|\xi(\psi_{i}(I_{comp}))-\xi(\psi_{i}(I_{gt}))\right|\@add@centering$
(5) $\centering\xi(I_{h*w})=K_{h*w}(I^{T}I)\@add@centering$ (6)

####  2.6.4 Pixel losses

The pixel losses for both hole and non-hole regions target to improve the per-
pixel reconstruction accuracy.

$L_{hole}=\left|(1-M)\cdot(I_{out}-I_{gt})\right|$ (7)
$L_{valid}=\left|(M)\cdot(I_{out}-I_{gt})\right|$ (8)

The aggregate loss function is shown in the equation -9. For our initial
experiments we use the values of the hyperparameters ($\lambda$s) according to
the following equation:

$L_{total}=\lambda_{1}L_{tv}+\lambda_{2}L_{valid}+\lambda_{3}L_{hole}+\lambda_{4}L_{perceptual}+\lambda_{5}(L_{style})$
(9)

###  2.7 Training Details

The above described architecture, along with the loss functions was trained
for 34500 iterations, with a batch size of 3. The totally approximate training
time was 20 hours, depending on the GPU system available.

##  3 Applications

We demonstrate the applications of this project in following two different
ways.

###  3.1 Automated Segmented Object Removal

In this section, we make use of automatic Image Segmentation Technique with
state of the art FCN-ResNet101 Model [10], to generate masks of the segmented
regions. The Segmentation model can identify large number of classes (20 in
our case), and thus can generate masks corresponding to different objects in
the image. We use the pre-trained Pytorch version of this model trained on the
Pascal VOC dataset (20 classes) to create this pipeline. For the demonstration
of this application, we use the pre-trained Image Inpainting model trained on
the Places-2 Dataset. The user just needs to give the input image and the
automated pipeline firstly detects the humans in the image (any of the 20
classes from the Pascal VOC dataset) and then performs the image inpainting
using those detected objects regions as masks. Here we given an example for
the same: In this demonstration, we have chosen the object to be a human
figure in the image. The segmentation model creates a Segmented mask as shown
in Figure 3.

Figure 3: Automated Segmented Object Removal, The rightmost image is the
ground truth that is provided as input to the automated pipeline. The Image
segmentation part of the pipeline detects all the humans and generates a mask
for the same as can be seen in the second image from the left. The leftmost
image goes as input into the Image Inpainting pipeline which eventually
generates reconstruced outputs as seen in the middle and the second last
image.

This mask, along with the original image, is fed as an input to our inpainting
model. The results are compared with other alternate inpainting algorithms.

###  3.2 Manual Mask Generation and Inpainting

In this demonstration, we facilitate the user with a custom board where one
can create a mask manually. The created mask is then inpainted using our
trained model. As with the previous application, this method is also compared
with other Classical approaches to compare the results. The results of this
experiment are shown in Figure 4.

Figure 4: Manual Mask Generation and Inpainting. Part A: The user can create a
manual mask in the input image. Part B: The resulting output generated
consists of the masked input, the mask, output of our work, output from Navier
stokes algorithm, and the ground truth from left to right respectively.

##  4 Observations

The state of the art models for image feature extraction break the images into
primarily two types of information. The first kind of features are the
features that are essential for the shape (edges, blobs, corners etc.) of the
image, and the other kind of features, are essential for the style (color
patterns) of the image. In this project, we tried to understand the
contribution of each of these types of features, and study how removing one of
them can affect the performance of the architecture in generating the
resulting images. Our qualitative and quantitative observations are described
in the following subsections.

###  4.1 Ablation Study

An Ablation Study typically involves removing some component of the model, to
understand the contribution of the component to the overall system. As
described in the previous sections, this work focuses significantly on the
complex loss functions. Therefore, we performed the following two experiments.
In the first experiment, we removed perceptual loss (described in Section 2.5)
component from the loss function of the model, and trained the model for 20000
epochs. In the second experiment, we removed the Style loss
component(described in Section 2.5) and similarly trained the model. These
experiments are a clear proof of concept for the losses described in Section
2.

####  4.1.1 Perceptual Loss Ablation

Figure 5 shows the results of this experiment. The results of the Perceptual
loss ablation clearly reflect the lack of sharp image features, in terms of
the edges, corners etc. Although the generated images capture the overall
structure of the input image via several other loss functions involved, the
results are clearly seem lacking.

####  4.1.2 Style Loss Ablation

Figure 5 shows the results of this experiment. As described in Section 2.5.3,
Style loss is responsible for capturing colors and localised compositions. As
we remove the effect of Style loss, we can clearly see that output images
contain structural feature details, but are unable to match the ground truth
in terms of color compositions.

Figure 5: Results of Ablation Study UNET

###  4.2 Quantitative Study

In this section, we make use of the quantitative metric such as L1 Norm, Mean
Squared Error(MSE), Peak Signal to Noise Ratio(PSNR), and Structural
Similarity Index(SSIM) to measure the performance of our model. The reader
must carefully note that there could be multiple possible inpainting results
for the same image and mask provided as input. Therefore, although the
quantitative analysis gives a measure for comparison different methods, this
analysis is not the most robust way for analysing the performance. Previous
works in the same field agree on this idea, however, have been reporting these
performance metric for comparisons. Therefore, we follow the convention and
report our results in terms of the above mentioned metrics.

We compare our outputs with a classical approach named Navier Stokes Image
Inpainting Technique. This approach uses ideas from classical fluid dynamics
to propagate isophote lines continuously from the exterior into the region to
be inpainted. We repeat our study for various mask size to image size ratios.
The results are displayed in the Table 1 below.

Trained Model | r=0.05 | r=0.1 | r =0.2
---|---|---|---
| l1 | mse | psnr | ssim | l1 | mse | psnr | ssim | l1 | mse | psnr | ssim
PConv | $6.05\mathrm{e}{+4}$ | $1.92\mathrm{e}{-1}$ | $5.55\mathrm{e}{+1}$ | $4.94\mathrm{e}{-1}$ | $6.81\mathrm{e}{+4}$ | $2.88\mathrm{e}{-1}$ | $5.40\mathrm{e}{+1}$ | $4.66\mathrm{e}{-1}$ | $8.46\mathrm{e}{+4}$ | $9.31\mathrm{e}{-1}$ | $5.09\mathrm{e}{+1}$ | $4.23\mathrm{e}{-1}$
PConv, no Style Loss | $8.43\mathrm{e}{+4}$ | $3.35\mathrm{e}{-1}$ | $5.33\mathrm{e}{+1}$ | $3.98\mathrm{e}{-1}$ | $1.08\mathrm{e}{+5}$ | $1.43\mathrm{e}{0}$ | $5.02\mathrm{e}{+1}$ | $3.74\mathrm{e}{-1}$ | $2.58\mathrm{e}{+5}$ | $1.29\mathrm{e}{+1}$ | $4.10\mathrm{e}{+1}$ | $3.21\mathrm{e}{-1}$
PConv, no Perceptual Loss | $1.56\mathrm{e}{+5}$ | $9.83\mathrm{e}{-1}$ | $4.85\mathrm{e}{+1}$ | $1.51\mathrm{e}{-1}$ | $1.70\mathrm{e}{+5}$ | $1.35\mathrm{e}{0}$ | $4.73\mathrm{e}{+1}$ | $1.36\mathrm{e}{-1}$ | $2.29\mathrm{e}{+5}$ | $4.27\mathrm{e}{0}$ | $4.33\mathrm{e}{+1}$ | $1.17\mathrm{e}{-1}$
Classical Method | $1.47\mathrm{e}{+7}$ | $1.90\mathrm{e}{+4}$ | $6.34\mathrm{e}{0}$ | $1.43\mathrm{e}{-1}$ | $1.40\mathrm{e}{+7}$ | $1.81\mathrm{e}{+4}$ | $6.55\mathrm{e}{0}$ | $1.35\mathrm{e}{-1}$ | $1.26\mathrm{e}{+7}$ | $1.64\mathrm{e}{+4}$ | $7.04\mathrm{e}{0}$ | $1.20\mathrm{e}{-1}$
Table 1: Quantitative Measure

###  4.3 Key Observations

Some of our key observations are-

  1. 1.

An image is made of structural features and features that correspond to
colours.

  2. 2.

Classical techniques are not as effective as the learning based approaches for
the problem of image inpainting.

  3. 3.

Capturing the semantics of the image rather than per pixel information, is a
much more efficient method for image reconstruction.

##  5 Conclusion and Future Work

This project was an attempt to reproduce and innovate the Deep Learning based
Image Inpainting Methodology by making use of the Partial Convolution Layers
and complex Loss Networks. In this project we performed an in-detail analysis
of the UNet inspired network architecture, performed ablation studies, and
implemented two methodologies to show the application of this technique.

This current project can be extended as a part of other larger Deep learing
frameworks such Image Deocclusion, or can be further implemented for automatic
modifications to Videos.

## References

  * [1] M. Bertalmio, A. Bertozzi, and G. Sapiro, “Navier-stokes, fluid dynamics, and image and video inpainting,” in _Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition. CVPR 2001_ , vol. 1, 2001, pp. I–I.
  * [2] C. A. Z. Barcelos and M. A. Batista, “Image restoration using digital inpainting and noise removal,” _Image and Vision Computing_ , vol. 25, no. 1, pp. 61–69, 2007.
  * [3] R. V. Marinescu, D. Moyer, and P. Golland, “Bayesian image reconstruction using deep generative models,” 2021.
  * [4] G. Liu, F. A. Reda, K. J. Shih, T. Wang, A. Tao, and B. Catanzaro, “Image inpainting for irregular holes using partial convolutions,” _CoRR_ , vol. abs/1804.07723, 2018. [Online]. Available: http://arxiv.org/abs/1804.07723
  * [5] Z. Liu, P. Luo, X. Wang, and X. Tang, “Deep learning face attributes in the wild,” in _Proceedings of International Conference on Computer Vision (ICCV)_ , December 2015.
  * [6] B. Zhou, A. Lapedriza, A. Khosla, A. Oliva, and A. Torralba, “Places: A 10 million image database for scene recognition,” _IEEE Transactions on Pattern Analysis and Machine Intelligence_ , 2017.
  * [7] J. Johnson, A. Alahi, and L. Fei-Fei, “Perceptual losses for real-time style transfer and super-resolution,” 2016.
  * [8] K. Simonyan and A. Zisserman, “Very deep convolutional networks for large-scale image recognition,” 2015.
  * [9] L. A. Gatys, A. S. Ecker, and M. Bethge, “A neural algorithm of artistic style,” 2015.
  * [10] E. Shelhamer, J. Long, and T. Darrell, “Fully convolutional networks for semantic segmentation,” 2016.


Determinant approach to the 2-Iterated $q$-Appell and
mixed type $q$-Appell polynomials⋆

Subuhi Khan⋆††⋆Corresponding author;E-mail<EMAIL_ADDRESS>(Subuhi Khan)
Second author ;E-mail<EMAIL_ADDRESS>(Mumtaz Riyasat) and Mumtaz Riyasat
Department of Mathematics

Aligarh Muslim University

Aligarh, India

###### Abstract

In this article, the 2-iterated $q$-Appell family is introduced. Certain
2-iterated $q$-Appell and mixed type $q$-special polynomials are considered as
members of this family. The numbers related to these polynomials are obtained.
The determinant definitions for the 2-iterated $q$-Appell family and for the
2-iterated and mixed type $q$-Appell polynomials are established. The graphs
of some 2-iterated $q$-Appell and mixed type $q$-Appell polynomials are drawn
for different values of indices and the roots of these polynomials are also
investigated for certain values of index $n$ by using Matlab. Finally, the
approximate solutions of the real zeros of these polynomials are given.

Keywords: $q$-Appell polynomials; 2-iterated $q$-Appell polynomials;
$q$-Bernoulli polynomials; $q$-Euler polynomials.

MSC 2010: 33D45, 33D99, 33E20.

##  1  Introduction and preliminaries

The area of $q$-calculus has in the last twenty years served as a bridge
between mathematics and physics. Recently, there is a significant increase of
activities in the area of $q$-calculus due to its applications in various
fields such as mechanics, mathematics and physics. The definitions and
notations of $q$-calculus reviewed here are taken from [3].

The $q$-analogue of the shifted factorial $(a)_{n}$ is defined by

$(a;q)_{0}=1,(a;q)_{n}=\prod\limits_{m=0}^{n-1}(1-q^{m}a),~{}n\in\mathbb{N}.$
(1.1)

The $q$-analogues of a complex number $a$ and of the factorial function are
defined by

$[a]_{q}=\frac{1-q^{a}}{1-q},~{}~{}q\in\mathbb{C}-\\{1\\};~{}~{}~{}a\in\mathbb{C},$
(1.2)
$[n]_{q}!=\prod_{m=1}^{n}[m]_{q}=[1]_{q}[2]_{q}\cdots[n]_{q}=\frac{(q;q)_{n}}{(1-q)^{n}},~{}q\neq
1;~{}n\in\mathbb{N},~{}~{}[0]_{q}!=1,~{}q\in\mathbb{C};~{}0<q<1.$ (1.3)

The Gauss $q$-binomial coefficient ${n\brack k}_{q}$ is defined by

${n\brack
k}_{q}=\frac{[n]_{q}!}{[k]_{q}![n-k]_{q}!}=\frac{(q;q)_{n}}{(q;q)_{k}(q;q)_{n-k}},~{}~{}~{}k=0,1,\ldots,n.$
(1.4)

The $q$-exponential function is defined as:

$e_{q}(x)=\sum\limits_{n=0}^{\infty}\frac{x^{n}}{[n]_{q}!},~{}0<|q|<1.$ (1.5)

The $q$-derivative $D_{q}f$ of a function $f$ at a point $0\neq
z\in\mathbb{C}$ is defined as:

$D_{q}f(z)=\frac{f(qz)-f(z)}{qz-z},~{}~{}~{}~{}0<|q|<1.$ (1.6)

The $q$-analogue of Taylor series expansion of an arbitrary function $f(z)$
for $0<q<1$ is defined as:

$f(z)=\sum\limits_{n=0}^{\infty}\frac{{(1-q)}^{n}}{(q;q)_{n}}D_{q}^{n}f(a)(z-a)^{n}_{q},$
(1.7)

where $D_{q}^{n}f(a)$ is the $n^{th}$ $q$-derivative of the function $f$ at
point $a$.

The class of Appell polynomials is characterized completely by Appell [4] in
1880. In 1954, Sharma and Chak [13] introduced a $q$-analogue for the family
of Appell polynomials and called this sequence of polynomials as $q$-harmonic.
Further, in 1967 Al-Salaam [1] introduced the family of $q$-Appell polynomials
$\\{A_{n,q}(x)\\}_{n\geq 0}$ and studied some of its properties. The
$n$-degree polynomials $A_{n,q}(x)$ are called $q$-Appell provided they
satisfy the following $q$-differential equation:

$D_{q,x}~{}\\{A_{n,q}(x)\\}=[n]_{q}~{}A_{n-1,q}(x),~{}n=0,1,2,\ldots~{};~{}q\in\mathbb{C};~{}0<q<1.$
(1.8)

The $q$-Appell polynomials $A_{n,q}(x)$ are also defined by means of the
following generating function [1]:

$A_{q}(t)e_{q}(xt)=\sum\limits_{n=0}^{\infty}A_{n,q}(x)\frac{t^{n}}{[n]_{q}!},~{}~{}~{}~{}0<q<1,$
(1.9)

where

$A_{q}(t):=\sum\limits_{n=0}^{\infty}A_{n,q}\frac{t^{n}}{[n]_{q}!},~{}~{}A_{0,q}=1;~{}A_{q}(t)\neq
0.$ (1.10)

It is to be noted that $A_{q}(t)$ is an analytic function at $t=0$ and

$A_{n,q}:=A_{n,q}(0)$ (1.11)

are the $q$-Appell numbers. Also, there exists a sequence of numbers
$\\{A_{n,q}\\}_{n\geq 0}$, such that the sequence $A_{n,q}(x)$ satisfies the
following relation [1]:

$A_{n,q}(x)=A_{n,q}+{n\brack 1}_{q}A_{n-1,q}~{}x+{n\brack
2}_{q}A_{n-2,q}~{}x^{2}+\cdots+A_{0,q}~{}x^{n},~{}n=0,1,2,\ldots~{}.$ (1.12)

Based on appropriate selection for the function $A_{q}(t)$, different members
belonging to the family of $q$-Appell polynomials can be obtained. These
members are listed in Table 1.

Table 1. Certain members belonging to the $q$-Appell family

S. Name of the $A_{q}(t)$ Generating function Series definition No.
$q$-special polynomials and related numbers I. $q$-Bernoulli
$\left(\frac{t}{e_{q}(t)-1}\right)$
$\left(\frac{t}{e_{q}(t)-1}\right)e_{q}(xt)=\sum\limits_{n=0}^{\infty}B_{n,q}(x)\frac{t^{n}}{[n]_{q}!}$
$B_{n,q}(x)=\sum\limits_{k=0}^{n}{n\brack k}_{q}B_{k,q}x^{n-k}$ polynomials
$\left(\frac{t}{e_{q}(t)-1}\right)=\sum\limits_{n=0}^{\infty}B_{n,q}\frac{t^{n}}{[n]_{q}!}$
and numbers $B_{n,q}:=B_{n,q}(0)$ [5, 2] II. $q$-Euler
$\left(\frac{2}{e_{q}(t)+1}\right)$
$\left(\frac{2}{e_{q}(t)+1}\right)e_{q}(xt)=\sum\limits_{n=0}^{\infty}E_{n,q}(x)\frac{t^{n}}{[n]_{q}!}$
$E_{n,q}(x)=\sum\limits_{k=0}^{n}{n\brack k}_{q}E_{k,q}x^{n-k}$ polynomials
$\left(\frac{2}{e_{q}(t)+1}\right)=\sum\limits_{n=0}^{\infty}E_{n,q}\frac{t^{n}}{[n]_{q}!}$
and numbers $E_{n,q}:=E_{n,q}(0)$ [5, 8] III. $q$-Genocchi
$\left(\frac{2t}{e_{q}(t)+1}\right)$
$\left(\frac{2t}{e_{q}(t)+1}\right)e_{q}(xt)=\sum\limits_{n=0}^{\infty}G_{n,q}(x)\frac{t^{n}}{[n]_{q}!}$
$G_{n,q}(x)=\sum\limits_{k=0}^{n}{n\brack k}_{q}G_{k,q}x^{n-k}$ polynomials
$\left(\frac{2t}{e_{q}(t)+1}\right)=\sum\limits_{n=0}^{\infty}G_{n,q}\frac{t^{n}}{[n]_{q}!}$
and numbers $G_{n,q}:=G_{n,q}(0)$ [7, 8]

Al-Salaam showed that the class of all $q$-Appell polynomials is a maximal
commutative subgroup of the group of all polynomial sets, i.e. the class of
all $q$-Appell sequences is closed under the operation of $q$-umbral
composition of polynomial sequences.

If $A_{n,q}(x)=\sum\limits_{k=0}^{n}a_{n,k;q}~{}x^{k}$ and
$B_{n,q}(x)=\sum\limits_{k=0}^{n}b_{n,k;q}~{}x^{k}$ are sequences of
$q$-polynomials, then the $q$-umbral composition of $A_{n,q}(x)$ with
$B_{n,q}(x)$ is defined to be the sequence

$(A_{n,q}\circ
B_{q})(x)=\sum\limits_{k=0}^{n}a_{n,k;q}~{}B_{k,q}(x)=\sum\limits_{0\leq k\leq
l\leq n}a_{n,k;q}b_{k,l;q}x^{l}.$ (1.13)

Under this operation the set of all $q$-Appell sequences is an abelian group
and it can be seen by considering the fact that every $q$-Appell sequence is
of the form

$A_{n,q}(x)=\left(\sum_{k=0}^{\infty}{a_{k,q}\over[k]_{q}!}D_{q}^{k}\right)x^{n}$
(1.14)

and that the umbral composition of $q$-Appell sequences corresponds to
multiplication of these formal $q$-power series in the operator $D_{q}$.

In 1982, Srivastava [14] gave several characterizations for the well known
Appell polynomials and for their basic analogues: the $q$-Appell polynomials.
In 1985, Roman proposed an approach similar to the umbral approach under the
area of nonclassical umbral calculus which is called $q$-umbral calculus [10,
9]. The $q$-Appell polynomials are studied using determinantal and umbral
approaches, see for example [6, 7]. Obtaining determinant forms for the
$q$-Appell polynomials and their members is an important aspect of such study.
The determinant forms can be helpful for computation purposes and can also be
useful in finding the solutions of general linear interpolation problems.

In this article, a family of the 2-iterated $q$-Appell polynomials is
introduced. The determinant forms and other properties of this family and also
for certain members belonging to this family are established. In Section 2,
the 2-iterated $q$-Appell polynomials are introduced by means of generating
function and series definition. Certain members belonging to the 2-iterated
$q$-Appell family and some mixed type $q$-special polynomials and related
numbers are also considered. In Section 3, the determinant forms of the
2-iterated and mixed type $q$-Appell polynomials are derived. In Section, 4,
the roots of the 2-iterated $q$-Appell and mixed type $q$-Appell polynomials
are investigated and their graphs are drawn for suitable values of indices.

##  2 2-iterated $q$-Appell polynomials

The 2-iterated $q$-Appell polynomials are introduced by means of generating
function and series definition. In order to introduce the 2I$q$AP, two
different sets of $q$-Appell polynomials $A_{n,q}^{I}(x)$ and
${A}_{n,q}^{II}(x)$ are considered.

Thus, from definitions (1.9) and (1.10), it follows that

$A_{q}^{I}(t)e_{q}(xt)=\sum\limits_{n=0}^{\infty}A_{n,q}^{I}(x)\frac{t^{n}}{[n]_{q}!},~{}0<q<1,$
(2.1)

where

$A_{q}^{I}(t):=\sum\limits_{n=0}^{\infty}A_{n,q}^{I}\frac{t^{n}}{[n]_{q}!};~{}A_{n,q}^{I}:=A_{n,q}^{I}(0);~{}A_{0,q}^{I}=1;~{}A_{q}^{I}(t)\neq
0$ (2.2)

and

$A_{q}^{II}(t)e_{q}(xt)=\sum\limits_{n=0}^{\infty}{A}_{n,q}^{II}(x)\frac{t^{n}}{[n]_{q}!},~{}0<q<1,$
(2.3)

where

$A_{q}^{II}(t):=\sum\limits_{n=0}^{\infty}{A}_{n,q}^{II}\frac{t^{n}}{[n]_{q}!};~{}{A}_{n,q}^{II}:={A}_{n,q}^{II}(0);~{}{A}_{0,q}^{II}=1;~{}A_{q}^{II}(t)\neq
0,$ (2.4)

respectively.

The generating function for the 2I$q$AP is derived by using a different
approach based on replacement techniques. For this, the following result is
proved:

###### Theorem 2.1

The following generating function for the 2-iterated $q$-Appell polynomials
holds true:

$G_{q}(x,t):=A_{q}^{I}(t)A_{q}^{II}(t)e_{q}(xt)=\sum\limits_{n=0}^{\infty}A_{n,q}^{[2]}(x)\frac{t^{n}}{[n]_{q}!},~{}0<q<1.$
(2.5)

Proof  Expanding the $q$-exponential function $e_{q}(xt)$ in the l.h.s. of
equation (2.1) and then replacing the powers of $x$, i.e.
$x^{0},~{}x^{1},~{}x^{2},\ldots,x^{n}$ by the corresponding polynomials
${A}_{0,q}^{II}(x),~{}{A}_{1,q}^{II}(x)$, $\ldots,{A}_{n,q}^{II}(x)$ in l.h.s.
and replacing $x$ by the polynomial $A_{1,q}^{II}(x)$ in the r.h.s. of the
resultant equation, it follows that

$A_{q}^{I}(t)\Big{[}1+{A}_{1,q}^{II}(x)\frac{t}{[1]_{q}!}+{A}_{2,q}^{II}(x)\frac{t^{2}}{[2]_{q}!}+\ldots+{A}_{n,q}^{II}(x)\frac{t^{n}}{[n]_{q}!}+\ldots\Big{]}=\sum\limits_{n=0}^{\infty}A_{n,q}^{I}\\{A_{1,q}^{II}(x)\\}\frac{t^{n}}{[n]_{q}!}.$
(2.6)

Summing up the series in l.h.s. and then using equation (2.3) and denoting the
resultant 2I$q$AP in the r.h.s. by $A_{n,q}^{[2]}(x)$, that is

$A_{n,q}^{[2]}(x)=A_{n,q}^{I}\\{A_{1,q}^{II}(x)\\},$ (2.7)

assertion (2.5) is proved.

Remark 2.1. The generating function (2.5) for the 2I$q$AP $A_{n,q}^{[2]}(x)$
is derived by replacing the powers of $x$ by the polynomials
${A}_{n,q}^{II}(x)~{}(n=0,1,\ldots)$ in generating function (2.1) of the
$q$-Appell polynomials $A_{n,q}^{I}(x)$. The replacements of the powers of $x$
by the polynomials ${A}_{n,q}^{I}(x)~{}(n=0,1,\ldots)$ in generating function
(2.3) of the $q$-Appell polynomials ${A}_{n,q}^{II}(x)$, yields the same
generating function.

Remark 2.2. It is important to remark that equation (2.7) is the operational
correspondence between the 2I$q$AP $A_{n,q}^{[2]}(x)$ and $q$-Appell
polynomials $A_{n,q}(x)$.

###### Theorem 2.2

The following series definition for the 2-iterated $q$-Appell polynomials
$A_{n,q}^{[2]}(x)$ holds true:

$A_{n,q}^{[2]}(x)=\sum\limits_{k=0}^{n}{n\brack
k}_{q}A_{k,q}^{I}~{}{A}_{n-k,q}^{II}(x).$ (2.8)

Proof  Using equations (2.2) and (2.3) in the l.h.s. of generating function
(2.5) and then using Cauchy-product rule in the l.h.s. of the resultant
equation, such that

$\sum\limits_{n=0}^{\infty}\sum\limits_{k=0}^{n}{n\brack
k}_{q}A_{k,q}^{I}~{}{A}_{n-k,q}^{II}(x)\frac{t^{n}}{[n]_{q}!}=\sum\limits_{n=0}^{\infty}A_{n,q}^{[2]}(x)\frac{t^{n}}{[n]_{q}!}.$
(2.9)

Equating the coefficients of same powers of $t$ in both sides of equation
(2.9), assertion (2.8) follows.

Remark 2.3. In view of equations (1.14) and (1.15), it is remarked that the
2-iterated $q$-Appell polynomials $A_{n,q}^{[2]}(x)$ satisfy the following
relation:

$A_{n,q}^{[2]}(x)=\left(\sum\limits_{k=0}^{\infty}\frac{{A}_{k,q}^{I}}{[k]_{q}!}D_{q}^{k}\right)A_{n,q}^{II}(x).$
(2.10)

Certain members belonging to the $q$-Appell family are given in Table 1.
Since, corresponding to each member belonging to the $q$-Appell family, there
exists a new special polynomial belonging to the 2-iterated $q$-Appell family.
Thus, by making suitable choice for the functions $A_{q}^{I}(t)$ and
$A_{q}^{II}(t)$ in equations (2.5) and (2.8), the generating function and
series definition for the corresponding member belonging to the 2-iterated
$q$-Appell family can be obtained. These resultant 2-iterated $q$-Appell
polynomials along with their notations, names, generating functions and series
definitions are given in Table 2.

Table 2. Certain members belonging to the 2-iterated $q$-Appell family

S. $A_{q}^{I}(t)=A_{q}^{II}(t)$ Notation Generating function Series definition
No. and name of the resultant 2I$q$AP I. $\left(\frac{t}{e_{q}(t)-1}\right)$
${B}_{n,q}^{[2]}(x)$:=
$\left(\frac{t}{e_{q}(t)-1}\right)^{2}e_{q}(xt)=\sum\limits_{n=0}^{\infty}{B}_{n,q}^{[2]}(x)\frac{t^{n}}{[n]_{q}!}$
${B}_{n,q}^{[2]}(x)=\sum\limits_{k=0}^{n}{n\brack
k}_{q}{B}_{k,q}{B}_{n-k,q}(x)$ 2-iterated $q$-Bernoulli polynomials (2I$q$BP)
II. $\left(\frac{2}{e_{q}(t)+1}\right)$ ${E}_{n,q}^{[2]}(x)$:=
$\left(\frac{2}{e_{q}(t)+1}\right)^{2}e_{q}(xt)=\sum\limits_{n=0}^{\infty}{E}_{n,q}^{[2]}(x)\frac{t^{n}}{[n]_{q}!}$
${E}_{n,q}^{[2]}(x)=\sum\limits_{k=0}^{n}{n\brack
k}_{q}{E}_{k,q}{E}_{n-k,q}(x)$ 2-iterated $q$-Euler polynomials (2I$q$EP) III.
$\left(\frac{2t}{e_{q}(t)+1}\right)$ ${G}_{n,q}^{[2]}(x)$:=
$\left(\frac{2t}{e_{q}(t)+1}\right)^{2}e_{q}(xt)=\sum\limits_{n=0}^{\infty}{G}_{n,q}^{[2]}(x)\frac{t^{n}}{[n]_{q}!}$
${G}_{n,q}^{[2]}(x)=\sum\limits_{k=0}^{n}{n\brack
k}_{q}{G}_{k,q}{G}_{n-k,q}(x)$ 2-iterated $q$-Genocchi polynomials (2I$q$GP)

The combinations of any two different members of the $q$-Appell family in the
2-iterated $q$-Appell family, yields a new mixed type $q$-special polynomial.
Thus, by making suitable selections for the functions $A_{q}^{I}(t)$ and
$A_{q}^{II}(t)$ in equations (2.5) and (2.8), the generating function and
series definition of the resultant mixed type $q$-special polynomials are
obtained. The possible combinations of the $q$-Bernoulli, $q$-Euler and
$q$-Genocchi polynomials (Table 1 (I-III)) are considered. The resultant mixed
type $q$-special polynomials along with their notations, names, generating
functions and series definitions are given in Table 3.

Table 3. Certain mixed type $q$-special polynomials

S. $A_{q}^{I}(t);~{}A_{q}^{II}(t)$ Notation Generating functions Series
definitions and No. name of the mixed type $q$-special polynomials I.
$\left(\frac{t}{e_{q}(t)-1}\right)$; ${}_{B}{E}_{n,q}(x):=$
$\frac{2t}{(e_{q}(t)-1)(e_{q}(t)+1)}e_{q}(xt)=\sum\limits_{n=0}^{\infty}{}_{B}{E}_{n,q}(x)\frac{t^{n}}{[n]_{q}!}$
${}_{B}{E}_{n,q}(x)=\sum\limits_{k=0}^{n}{n\brack
k}_{q}{E}_{k,q}{B}_{n-k,q}(x)$ $\left(\frac{2}{e_{q}(t)+1}\right)$
$q$-Bernoulli- Euler polynomials ($q$BEP) II.
$\left(\frac{t}{e_{q}(t)-1}\right)$; ${}_{B}{G}_{n,q}(x)$:=
$\frac{2t^{2}}{(e_{q}(t)-1)(e_{q}(t)+1)}e_{q}(xt)=\sum\limits_{n=0}^{\infty}{}_{B}{G}_{n,q}(x)\frac{t^{n}}{[n]_{q}!}$
${}_{B}{G}_{n,q}(x)=\sum\limits_{k=0}^{n}{n\brack
k}_{q}{G}_{k,q}{B}_{n-k,q}(x)$ $\left(\frac{2t}{e_{q}(t)+1}\right)$
$q$-Bernoulli- Genocchi polynomials ($q$BGP) III.
$\left(\frac{2}{e_{q}(t)+1}\right)$; ${}_{E}{G}_{n,q}(x)$:=
$\left(\frac{2t^{1/2}}{e_{q}(t)+1}\right)^{2}e_{q}(xt)=\sum\limits_{n=0}^{\infty}{}_{E}{G}_{n,q}(x)\frac{t^{n}}{[n]_{q}!}$
${}_{E}{G}_{n,q}(x)=\sum\limits_{k=0}^{n}{n\brack
k}_{q}{G}_{k,q}{E}_{n-k,q}(x)$ $\left(\frac{2t}{e_{q}(t)+1}\right)$ $q$-Euler-
Genocchi polynomials ($q$EGP)

Remark 2.4. From Remark 2.1 and Table 3, it follows that

${}_{B}{E}_{n,q}(x)\equiv{}_{E}{B}_{n,q}(x);~{}~{}{}_{B}{G}_{n,q}(x)\equiv{}_{G}{B}_{n,q}(x);~{}~{}{}_{E}{G}_{n,q}(x)\equiv{}_{G}{E}_{n,q}(x),$
(2.11)

where ${}_{E}{B}_{n,q}(x)$, ${}_{G}{B}_{n,q}(x)$ and ${}_{G}{E}_{n,q}(x)$ are
the $q$-Euler-Bernoulli polynomials ($q$EBP), $q$-Genocchi-Bernoulli
polynomials ($q$GBP) and $q$-Genocchi-Euler polynomials ($q$GEP). This is due
to the fact that the set of all $q$-Appell sequences is closed under the
operation of $q$-umbral composition of polynomial sequences and form an
abelian group.

Next, the numbers related to the 2I$q$AP $A_{n,q}^{[2]}(x)$ and also related
to certain members of this family given in Tables 2 and 3 are explored. As
mentioned in definition (1.11), the $q$-Appell numbers are the values of the
polynomials $A_{n,q}(x)$ at $x=0$ and satisfy equation (1.12).

Taking $x=0$ in equation (2.8), the following series definition for the
2-iterated $q$-Appell numbers (2I$q$AN) $A_{n,q}^{[2]}:=A_{n,q}^{[2]}(0)$ is
obtained

$A_{n,q}^{[2]}=\sum\limits_{k=0}^{n}{n\brack k}_{q}A_{k,q}^{I}A_{n-k,q}^{II},$
(2.12)

where $A_{n,q}^{I}$ and $A_{n,q}^{(2)}$ denote two different sets of
$q$-Appell numbers.

Next, the numbers related to the 2I$q$BP $B_{n,q}^{[2]}(x)$, 2I$q$EP
$E_{n,q}^{[2]}(x)$ and 2I$q$GP $G_{n,q}^{[2]}(x)$ are explored. Taking $x=0$
in series definitions of 2I$q$BP $B_{n,q}^{[2]}(x)$, 2I$q$EP
$E_{n,q}^{[2]}(x)$ and 2I$q$GP $G_{n,q}^{[2]}(x)$ given in Table 2 (I-III) and
in view of notations given in Table 1, the 2-iterated $q$-Bernoulli, $q$-Euler
and $q$-Genocchi numbers are obtained. These numbers are given in Table 4.

Table 4. 2-iterated $q$-numbers

S.No. Notation and name of the 2-iterated $q$-number Series definition I.
${B}_{n,q}^{[2]}:={B}_{n,q}^{[2]}(0)$
${B}_{n,q}^{[2]}=\sum\limits_{k=0}^{n}{n\brack k}_{q}{B}_{k,q}{B}_{n-k,q}$
2-iterated $q$-Bernoulli numbers (2I$q$BN) II.
${E}_{n,q}^{[2]}:={E}_{n,q}^{[2]}(0)$
${E}_{n,q}^{[2]}=\sum\limits_{k=0}^{n}{n\brack k}_{q}{E}_{k,q}{E}_{n-k,q}$
2-iterated $q$-Euler numbers (2I$q$EN) III.
${G}_{n,q}^{[2]}:={G}_{n,q}^{[2]}(0)$
${G}_{n,q}^{[2]}=\sum\limits_{k=0}^{n}{n\brack k}_{q}{G}_{k,q}{G}_{n-k,q}$
2-iterated $q$-Genocchi numbers (2I$q$GN)

Further, the numbers corresponding to the mixed type $q$-special polynomials
given in Table 3 are explored. Taking $x=0$ in series definitions of the
$q$BEP ${}_{B}E_{n,q}(x)$, $q$BGP ${}_{B}G_{n,q}(x)$ and $q$EGP
${}_{E}G_{n,q}(x)$ (Table 3 (I-III)) and in view of notations given in Table
1, the $q$-Bernoulli-Euler, $q$-Bernoulli-Genocchi and $q$-Euler-Genocchi
numbers are obtained. These numbers are given in Table 5.

Table 5. Mixed type $q$-numbers

S.No. Notation and name of the mixed type $q$-numbers Series definition I.
${}_{B}{E}_{n,q}:={}_{B}{E}_{n,q}(0)$
${}_{B}{E}_{n,q}=\sum\limits_{k=0}^{n}{n\brack k}_{q}{E}_{k,q}{B}_{n-k,q}$
$q$-Bernoulli-Euler numbers ($q$BEN) II. ${}_{B}{G}_{n,q}:={}_{B}{G}_{n,q}(0)$
${}_{B}{G}_{n,q}=\sum\limits_{k=0}^{n}{n\brack k}_{q}{G}_{k,q}{B}_{n-k,q}$
$q$-Bernoulli-Genocchi numbers ($q$BGN) III.
${}_{E}{G}_{n,q}:={}_{E}{G}_{n,q}(0)$
${}_{E}{G}_{n,q}=\sum\limits_{k=0}^{n}{n\brack k}_{q}{G}_{k,q}{E}_{n-k,q}$
$q$-Euler-Genocchi numbers ($q$EGN)

Note. From Remark 2.4 and Table 5, it follows that

${}_{B}{E}_{n,q}\equiv{}_{E}{B}_{n,q}:={}_{E}{B}_{n,q}(0);~{}~{}{}_{B}{G}_{n,q}\equiv{}_{G}{B}_{n,q}:={}_{G}{B}_{n,q}(0);~{}~{}{}_{E}{G}_{n,q}\equiv{}_{G}{E}_{n,q}:={}_{G}{E}_{n,q}(0),$
(2.13)

where ${}_{E}{B}_{n,q}$, ${}_{G}{B}_{n,q}$ and ${}_{G}{E}_{n,q}$ are the
$q$-Euler-Bernoulli numbers ($q$EBN), $q$-Genocchi-Bernoulli numbers ($q$GBN)
and $q$-Genocchi-Euler numbers ($q$GEN), respectively.

In the next section, the determinant forms for the 2-iterated $q$-Appell
polynomials and mixed type $q$-special polynomials are established.

##  3 Determinant approach

Keleshteri and Mahmudov [6] studied the $q$-Appell polynomials from
determinant point of view. The determinant forms of the special polynomials
are important for the computational and applied purposes. This fact provides
motivation to establish the determinant definitions of the $q$-special
polynomials introduced in previous section. In order to define the 2I$q$AP
$A_{n,q}^{[2]}(x)$ by means of determinant, the following result is proved:

###### Theorem 3.1

The following determinant form for the 2-iterated $q$-Appell polynomials
$A_{n,q}^{[2]}(x)$ of degree $n$ holds true:

$\begin{array}[]{l}A_{0,q}^{[2]}(x)=\frac{1}{\beta_{0,q}},\\\
A_{n,q}^{[2]}(x)=\frac{{(-1)}^{n}}{{(\beta_{0,q})}^{n+1}}\left|\begin{array}[]{ccccccc}1&{A}_{1,q}^{II}(x)&{A}_{2,q}^{II}(x)&\cdots&{A}_{n-1,q}^{II}(x)&{A}_{n,q}^{II}(x)\\\
\\\
{\beta}_{0,q}&{\beta}_{1,q}&{\beta}_{2,q}&\cdots&{\beta}_{n-1,q}&{\beta}_{n,q}\\\
\\\ 0&{\beta}_{0,q}&{2\brack 1}_{q}~{}{\beta}_{1,q}&\cdots&{n-1\brack
1}_{q}~{}{\beta}_{n-2,q}&{n\brack 1}_{q}~{}{\beta}_{n-1,q}\\\ \\\
0&0&{\beta}_{0,q}&\cdots&{n-1\brack 2}_{q}~{}{\beta}_{n-3,q}&{n\brack
2}_{q}~{}{\beta}_{n-2,q}\\\ .&.&.&\cdots&.&.\\\ .&.&.&\cdots&.&.\\\
0&0&0&\cdots&{\beta}_{0,q}&{n\brack
n-1}_{q}~{}{\beta}_{1,q}\end{array}\right|,\end{array}$ (3.1)

where $n=1,2,\ldots~{}~{}$ and ${A}_{n,q}^{II}(x)~{}(n=0,1,2,\ldots)$ are the
$q$-Appell polynomials of degree $n$; $\beta_{0,q}\neq 0$ and

$\begin{array}[]{l}\beta_{0,q}=\frac{1}{A_{0,q}^{I}},\\\
\beta_{n,q}=-\frac{1}{A_{0,q}^{I}}\Big{(}\sum\limits_{k=1}^{n}{n\brack
k}_{q}A_{k,q}^{I}~{}\beta_{n-k,q}\Big{)},~{}~{}n=1,2,\ldots~{}.\end{array}$
(3.2)

Proof  Let $A_{n,q}^{[2]}(x)$ be a sequence of the 2I$q$AP defined by equation
(2.5) and $A_{n,q}^{I},~{}\beta_{n,q}$, be two numerical sequences (the
coefficients of $q$-Taylor’s series expansions of functions) such that

$A_{q}^{I}(t)=A_{0,q}^{I}+\frac{t}{[1]_{q}!}A_{1,q}^{I}+\frac{t^{2}}{[2]_{q}!}A_{2,q}^{I}+\cdots+\frac{t^{n}}{[n]_{q}!}A_{n,q}^{I}+\cdots,~{}n=0,1,\ldots;~{}A_{0,q}^{I}\neq
0,$ (3.3)
$\hat{A}_{q}^{I}(t)=\beta_{0,q}+\frac{t}{[1]_{q}!}\beta_{1,q}+\frac{t^{2}}{[2]_{q}!}\beta_{2,q}+\cdots+\frac{t^{n}}{[n]_{q}!}\beta_{n,q}+\cdots,~{}n=0,1,\ldots;~{}\beta_{0,q}\neq
0,$ (3.4)

satisfying

$A_{q}^{I}(t)\hat{A}_{q}^{I}(t)=1.$ (3.5)

Then, according to the Cauchy-product rule, it follows that

$A_{q}^{I}(t)\hat{A}_{q}^{I}(t)=\sum\limits_{n=0}^{\infty}\sum\limits_{k=0}^{n}{n\brack
k}_{q}A_{k,q}^{I}~{}\beta_{n-k,q}\frac{t^{n}}{[n]_{q}!},$

which gives

$\sum\limits_{k=0}^{n}{n\brack
k}_{q}A_{k,q}^{I}~{}\beta_{n-k,q}=\begin{cases}1,~{}~{}~{}~{}~{}\textrm{if}~{}n=0,\cr
0,~{}~{}~{}~{}~{}\textrm{if}~{}n>0.\cr\end{cases}$ (3.6)

Consequently, the following holds:

$\begin{cases}\beta_{0,q}=\frac{1}{A_{0,q}^{I}},\cr\beta_{n,q}=-\frac{1}{A_{0,q}^{I}}\Big{(}\sum\limits_{k=1}^{n}{n\brack
k}_{q}A_{k,q}^{I}~{}\beta_{n-k,q}\Big{)},~{}~{}n=1,2,\ldots~{}~{}.\end{cases}$
(3.7)

Multiplication of equation (2.5) by $\hat{A}_{q}^{I}(t)$ gives

$A_{q}^{I}(t)\hat{A}_{q}^{I}(t){A}_{q}^{II}(t)e_{q}(xt)=\hat{A}_{q}^{I}(t)\sum\limits_{n=0}^{\infty}A_{n,q}^{[2]}(x)\frac{t^{n}}{n!},$

which in view of equations (3.4), (3.5) and (2.3) gives

$\sum\limits_{n=0}^{\infty}{A}_{n,q}^{II}(x)\frac{t^{n}}{[n]_{q}!}=\sum\limits_{n=0}^{\infty}A_{n,q}^{[2]}(x)\frac{t^{n}}{[n]_{q}!}\sum\limits_{k=0}^{\infty}\beta_{k,q}\frac{t^{k}}{[k]_{q}!}.$
(3.8)

Again, multiplication of the series on the r.h.s. of equation (3.8) according
to Cauchy-product rule leads to the following system of infinite equations in
the unknowns $A_{n,q}^{[2]}(x)~{}(n=0,1,\ldots$):

$\begin{cases}A_{0,q}^{[2]}(x)\beta_{0,q}=1,\cr\\\
A_{0,q}^{[2]}(x)\beta_{1,q}+A_{1,q}^{[2]}(x)\beta_{0,q}={A}_{1,q}^{II}(x),\cr\\\
A_{0,q}^{[2]}(x)\beta_{2,q}+{2\brack
1}_{q}A_{1,q}^{[2]}(x)\beta_{1,q}+A_{2,q}^{[2]}(x)\beta_{0,q}={A}_{2,q}^{II}(x),\cr\vdots\\\
A_{0,q}^{[2]}(x)\beta_{n-1,q}+{n-1\brack
1}_{q}A_{1,q}^{[2]}(x)\beta_{n-2,q}+\cdots+A_{n,q}^{[2]}(x)\beta_{0,q}={A}_{n-1,q}^{II}(x),\cr\\\
A_{0,q}^{[2]}(x)\beta_{n,q}+{n\brack
1}_{q}A_{1,q}^{[2]}(x)\beta_{n-1,q}+\cdots+A_{n,q}^{[2]}(x)\beta_{0,q}={A}_{n,q}^{II}(x),\cr\vdots\end{cases}$
(3.9)

First equation of system (3.9), proves the first part of assertion (3.1).
Also, the special form of system (3.9) (lower triangular) allows to work out
the unknowns $A_{n,q}^{[2]}(x)$. Operating with the first $n+1$ equations
simply by applying the Cramer’s rule, it follows that

$\begin{array}[]{l}A_{n,q}^{[2]}(x)=\frac{\left|\begin{array}[]{cccccc}\beta_{0,q}&0&0&\cdots&0&1\\\
\\\ \beta_{1,q}&\beta_{0,q}&0&\cdots&0&{A}_{1,q}^{II}(x)\\\ \\\
\beta_{2,q}&{2\brack
1}_{q}~{}\beta_{1,q}&\beta_{0,q}&\cdots&0&{A}_{2,q}^{II}(x)\\\ \\\
.&.&.&\cdots&.&.\\\ \beta_{n-1,q}&{n-1\brack 1}_{q}~{}\beta_{n-2,q}&{n-1\brack
2}_{q}~{}\beta_{n-3,q}&\cdots&\beta_{0,q}&{A}_{n-1,q}^{II}(x)\\\ \\\
\beta_{n,q}&{n\brack 1}_{q}~{}\beta_{n-1,q}&{n\brack
2}_{q}~{}\beta_{n-2,q}&\cdots&{n\brack
n-1}_{q}~{}\beta_{1,q}&{A}_{n,q}^{II}(x)\\\
\end{array}\right|}{\left|\begin{array}[]{cccccc}\beta_{0,q}&0&0&\cdots&0&0\\\
\\\ \beta_{1,q}&\beta_{0,q}&0&\cdots&0&0\\\ \\\ \beta_{2,q}&{2\brack
1}_{q}~{}\beta_{1,q}&\beta_{0,q}&\cdots&0&0\\\ .&.&.&\cdots&.&.\\\
\beta_{n-1,q}&{n-1\brack 1}_{q}~{}\beta_{n-2,q}&{n-1\brack
2}_{q}~{}\beta_{n-3,q}&\cdots&\beta_{0,q}&0\\\ \\\ \beta_{n,q}&{n\brack
1}_{q}~{}\beta_{n-1,q}&{n\brack 2}_{q}~{}\beta_{n-2,q}&\cdots&{n\brack
n-1}_{q}~{}\beta_{1,q}&\beta_{0,q}\\\ \end{array}\right|}\end{array},$ (3.10)

where, $~{}n=1,2,\ldots~{}$.

Now, taking the transpose of the determinant in the numerator and expanding
the determinant in the denominator, so that

$\begin{array}[]{l}A_{n,q}^{[2]}(x)=\frac{1}{(\beta_{0,q})^{n+1}}\left|\begin{array}[]{cccccc}\beta_{0,q}&\beta_{1,q}&\beta_{2,q}&\cdots&\beta_{n-1,q}&\beta_{n,q}\\\
\\\ 0&\beta_{0,q}&{2\brack 1}_{q}~{}\beta_{1,q}&\cdots&{n-1\brack
1}_{q}~{}\beta_{n-2,q}&{n\brack 1}_{q}~{}\beta_{n-1,q}\\\ \\\
0&0&\beta_{0,q}&\cdots&{n-1\brack 2}_{q}~{}\beta_{n-3,q}&{n\brack
2}_{q}~{}\beta_{n-2,q}\\\ \\\ .&.&.&\cdots&.&.\\\
0&0&0&\cdots&\beta_{0,q}&{n\brack n-1}_{q}~{}\beta_{1,q}\\\ \\\
1&{A}_{1,q}^{II}(x)&{A}_{2,q}^{II}(x)&\cdots&{A}_{n-1,q}^{II}(x)&{A}_{n,q}^{II}(x)\end{array}\right|,\end{array}$
(3.11)

which after $n$ circular row exchanges, that is after moving the $i^{th}$ row
to the $(i+1)^{th}$ position for $i=1,2,\ldots,n-1$, yields second part of
assertion (3.1).

Remark 3.1. Since the polynomials $B_{n,q}(x)$, $E_{n,q}(x)$ and $G_{n,q}(x)$
are particular members of the $q$-Appell family $A_{n,q}(x)$. Therefore, the
determinant forms of $B_{n,q}(x)$, $E_{n,q}(x)$ and $G_{n,q}(x)$ can be
obtained by giving suitable values to the coefficients $\beta_{0,q}$ and
${\beta}_{i,q}~{}(i=1,2,\cdots,n)$ in the definition of $A_{n,q}(x)$.

Taking $\beta_{0,q}=1$, ${\beta}_{i,q}=\frac{1}{[i+1]_{q}}~{}(i=1,2,\cdots,n)$
in the determinant definition of the $q$-Appell polynomials $A_{n,q}(x)$ [6,
p.12(19)], the following determinant definition of the $q$-Bernoulli
polynomials $B_{n,q}(x)$ is obtained:

Definition 3.1. The $q$-Bernoulli polynomials $B_{n,q}(x)$ of degree $n$ are
defined by

$\begin{array}[]{l}B_{0,q}(x)=1,\\\
B_{n,q}(x)=(-1)^{n}\left|\begin{array}[]{ccccccc}1&x&x^{2}&\cdots&x^{n-1}&x^{n}\\\
\\\
1&\frac{1}{[2]_{q}}&\frac{1}{[3]_{q}}&\cdots&\frac{1}{[n]_{q}}&\frac{1}{[n+1]_{q}}\\\
\\\ 0&1&{2\brack 1}_{q}~{}\frac{1}{[2]_{q}}&\cdots&{n-1\brack
1}_{q}~{}\frac{1}{[n-1]_{q}}&{n\brack 1}_{q}~{}\frac{1}{[n]_{q}}\\\ \\\
0&0&1&\cdots&{n-1\brack 2}_{q}~{}\frac{1}{[n-2]_{q}}&{n\brack
2}_{q}~{}\frac{1}{[n-1]_{q}}\\\ .&.&.&\cdots&.&.\\\ .&.&.&\cdots&.&.\\\
0&0&0&\cdots&1&{n\brack
n-1}_{q}~{}\frac{1}{[2]_{q}}\end{array}\right|,~{}~{}n=1,2,\cdots~{}.\end{array}$
(3.12)

The particular case of definition (3.12), for $n=4$ is considered in [15,
p.250].

Next, taking $\beta_{0,q}=1$, ${\beta}_{i,q}=\frac{1}{2}~{}(i=1,2,\cdots,n)$
in the determinant definition of the $q$-Appell polynomials $A_{n,q}(x)$ [6,
p.12(19)], the following determinant definition of the $q$-Euler polynomials
$E_{n,q}(x)$ is obtained:

Definition 3.2.The $q$-Euler polynomials $E_{n,q}(x)$ of degree $n$ are
defined by

$\begin{array}[]{l}E_{0,q}(x)=1,\\\
E_{n,q}(x)=(-1)^{n}\left|\begin{array}[]{ccccccc}1&x&x^{2}&\cdots&x^{n-1}&x^{n}\\\
\\\ 1&\frac{1}{2}&\frac{1}{2}&\cdots&\frac{1}{2}&\frac{1}{2}\\\ \\\
0&1&\frac{1}{2}{2\brack 1}_{q}&\cdots&\frac{1}{2}{n-1\brack
1}_{q}&\frac{1}{2}{n\brack 1}_{q}\\\ \\\ 0&0&1&\cdots&\frac{1}{2}{n-1\brack
2}_{q}&\frac{1}{2}{n\brack 2}_{q}\\\ .&.&.&\cdots&.&.\\\ .&.&.&\cdots&.&.\\\
0&0&0&\cdots&1&\frac{1}{2}{n\brack n-1}_{q}\\\
\end{array}\right|,~{}~{}n=1,2,\cdots~{}.\end{array}$ (3.13)

Further, taking $\beta_{0,q}=1$,
${\beta}_{i,q}=\frac{1}{2[i+1]_{q}}~{}(i=1,2,\cdots,n)$ in the determinant
definition of the $q$-Appell polynomials $A_{n,q}(x)$ [6, p.12(19)], the
following determinant definition of the $q$-Genocchi polynomials $G_{n,q}(x)$
is obtained:

Definition 3.3. The $q$-Genocchi polynomials $G_{n,q}(x)$ of degree $n$ are
defined by

$\begin{array}[]{l}G_{0,q}(x)=1,\\\
G_{n,q}(x)=(-1)^{n}\left|\begin{array}[]{ccccccc}1&x&x^{2}&\cdots&x^{n-1}&x^{n}\\\
\\\
1&\frac{1}{2[2]_{q}}&\frac{1}{2[3]_{q}}&\cdots&\frac{1}{2[n]_{q}}&\frac{1}{2[n+1]_{q}}\\\
\\\ 0&1&{2\brack 1}_{q}~{}\frac{1}{2[2]_{q}}&\cdots&{n-1\brack
1}_{q}~{}\frac{1}{2[n-1]_{q}}&{n\brack 1}_{q}~{}\frac{1}{2[n]_{q}}\\\ \\\
0&0&1&\cdots&{n-1\brack 2}_{q}~{}\frac{1}{2[n-2]_{q}}&{n\brack
2}_{q}~{}\frac{1}{2[n-1]_{q}}\\\ .&.&.&\cdots&.&.\\\ .&.&.&\cdots&.&.\\\
0&0&0&\cdots&1&{n\brack
n-1}_{q}~{}\frac{1}{2[2]_{q}}\end{array}\right|,~{}~{}n=1,2,\cdots~{}.\end{array}$
(3.14)

Remark 3.2. Taking suitable values of the coefficients $\beta_{0,q}$ and
${\beta}_{i,q}~{}(i=1,2,\cdots,n)$ in the determinant definition of the
2I$q$AP family, the determinant definitions for the 2-iterated $q$-members
belonging to this family and certain mixed type $q$-special polynomials can be
obtained.

First, the determinant definitions for the members of the 2I$q$AP given in
Table 2 are obtained. Taking $\beta_{0,q}=1$,
${\beta}_{i,q}=\frac{1}{[i+1]_{q}}~{}(i=1,2,\cdots,n)$ and
$A_{n,q}^{II}(x)=B_{n,q}(x)$ in equation (3.1), the following determinant
definition of the 2I$q$BP $B_{n,q}^{[2]}(x)$ is obtained:

Definition 3.4. The 2-iterated $q$-Bernoulli polynomials $B_{n,q}^{[2]}(x)$ of
degree $n$ are defined by

$\begin{array}[]{l}B_{0,q}^{[2]}(x)=1,\\\
B_{n,q}^{[2]}(x)=(-1)^{n}\left|\begin{array}[]{ccccccc}1&B_{1,q}(x)&B_{2,q}(x)&\cdots&B_{n-1,q}(x)&B_{n,q}(x)\\\
\\\
1&\frac{1}{[2]_{q}}&\frac{1}{[3]_{q}}&\cdots&\frac{1}{[n]_{q}}&\frac{1}{[n+1]_{q}}\\\
\\\ 0&1&{2\brack 1}_{q}~{}\frac{1}{[2]_{q}}&\cdots&{n-1\brack
1}_{q}~{}\frac{1}{[n-1]_{q}}&{n\brack 1}_{q}~{}\frac{1}{[n]_{q}}\\\ \\\
0&0&1&\cdots&{n-1\brack 2}_{q}~{}\frac{1}{[n-2]_{q}}&{n\brack
2}_{q}~{}\frac{1}{[n-1]_{q}}\\\ .&.&.&\cdots&.&.\\\ .&.&.&\cdots&.&.\\\
0&0&0&\cdots&1&{n\brack
n-1}_{q}~{}\frac{1}{[2]_{q}}\end{array}\right|,~{}~{}n=1,2,\cdots~{},\end{array}$
(3.15)

where $B_{n,q}(x)~{}(n=0,1,2,\ldots)$ are the $q$-Bernoulli polynomials of
degree $n$.

Taking $\beta_{0,q}=1$, ${\beta}_{i,q}=\frac{1}{2}~{}(i=1,2,\cdots,n)$ and
$A_{n,q}^{II}(x)=E_{n,q}(x)$ in equation (3.1), the following determinant
definition of the 2I$q$EP $E_{n,q}^{[2]}(x)$ is obtained:

Definition 3.5. The 2-iterated $q$-Euler polynomials $E_{n,q}^{[2]}(x)$ of
degree $n$ are defined by

$\begin{array}[]{l}E_{0,q}^{[2]}(x)=1,\\\
E_{n,q}^{[2]}(x)={(-1)}^{n}\left|\begin{array}[]{ccccccc}1&E_{1,q}(x)&E_{2,q}(x)&\cdots&E_{n-1,q}(x)&E_{n,q}(x)\\\
\\\ 1&\frac{1}{2}&\frac{1}{2}&\cdots&\frac{1}{2}&\frac{1}{2}\\\ \\\
0&1&\frac{1}{2}{2\brack 1}_{q}&\cdots&\frac{1}{2}{n-1\brack
1}_{q}&\frac{1}{2}{n\brack 1}_{q}\\\ \\\ 0&0&1&\cdots&\frac{1}{2}{n-1\brack
2}_{q}&\frac{1}{2}{n\brack 2}_{q}\\\ .&.&.&\cdots&.&.\\\ .&.&.&\cdots&.&.\\\
0&0&0&\cdots&1&\frac{1}{2}{n\brack n-1}_{q}\\\
\end{array}\right|,~{}n=1,2,\cdots,\end{array}$ (3.16)

where $E_{n,q}(x)~{}(n=0,1,2,\ldots)$ are the $q$-Euler polynomials of degree
$n$.

Taking $\beta_{0,q}=1$,
${\beta}_{i,q}=\frac{1}{2[i+1]_{q}}~{}(i=1,2,\cdots,n)$ and
$A_{n,q}^{II}(x)=G_{n,q}(x)$ in equation (3.1), the following determinant
definition of the 2I$q$GP $G_{n,q}^{[2]}(x)$ is obtained:

Definition 3.6. The 2-iterated $q$-Genocchi polynomials $G_{n,q}^{[2]}(x)$ of
degree $n$ are defined by

$\begin{array}[]{l}G_{0,q}^{[2]}(x)=1,\\\
G_{n,q}^{[2]}(x)=(-1)^{n}\left|\begin{array}[]{ccccccc}1&G_{1,q}(x)&G_{2,q}(x)&\cdots&G_{n-1,q}(x)&G_{n,q}(x)\\\
\\\
1&\frac{1}{2[2]_{q}}&\frac{1}{2[3]_{q}}&\cdots&\frac{1}{2[n]_{q}}&\frac{1}{2[n+1]_{q}}\\\
\\\ 0&1&{2\brack 1}_{q}~{}\frac{1}{2[2]_{q}}&\cdots&{n-1\brack
1}_{q}~{}\frac{1}{2[n-1]_{q}}&{n\brack 1}_{q}~{}\frac{1}{2[n]_{q}}\\\ \\\
0&0&1&\cdots&{n-1\brack 2}_{q}~{}\frac{1}{2[n-2]_{q}}&{n\brack
2}_{q}~{}\frac{1}{2[n-1]_{q}}\\\ .&.&.&\cdots&.&.\\\ .&.&.&\cdots&.&.\\\
0&0&0&\cdots&1&{n\brack
n-1}_{q}~{}\frac{1}{2[2]_{q}}\end{array}\right|,~{}~{}n=1,2,\cdots~{},\end{array}$
(3.17)

where $G_{n,q}(x)~{}(n=0,1,2,\ldots)$ are the $q$-Genocchi polynomials of
degree $n$.

Next, the determinant definitions for the mixed type $q$-special polynomials
given in Table 3 are obtained.

Taking $\beta_{0,q}=1$, ${\beta}_{i,q}=\frac{1}{2}~{}(i=1,2,\cdots,n)$ and
$A_{n,q}^{II}(x)=B_{n,q}(x)$ in equation (3.1), the following determinant
definition of the $q$BEP ${}_{B}E_{n,q}(x)$ is obtained:

Definition 3.7. The $q$-Bernoulli-Euler polynomials ${}_{B}E_{n,q}(x)$ of
degree $n$ are defined by

$\begin{array}[]{l}{}_{B}E_{0,q}(x)=1,\\\
{}_{B}E_{n,q}(x)=(-1)^{n}\left|\begin{array}[]{ccccccc}1&B_{1,q}(x)&B_{2,q}(x)&\cdots&B_{n-1,q}(x)&B_{n,q}(x)\\\
\\\ 1&\frac{1}{2}&\frac{1}{2}&\cdots&\frac{1}{2}&\frac{1}{2}\\\ \\\
0&1&\frac{1}{2}{2\brack 1}_{q}&\cdots&\frac{1}{2}{n-1\brack
1}_{q}&\frac{1}{2}{n\brack 1}_{q}\\\ \\\ 0&0&1&\cdots&\frac{1}{2}{n-1\brack
2}_{q}&\frac{1}{2}{n\brack 2}_{q}\\\ .&.&.&\cdots&.&.\\\ .&.&.&\cdots&.&.\\\
0&0&0&\cdots&1&\frac{1}{2}{n\brack n-1}_{q}\\\
\end{array}\right|,~{}n=1,2,\cdots~{},\end{array}$ (3.18)

where $B_{n,q}(x)~{}(n=0,1,2,\ldots)$ are the $q$-Bernoulli polynomials of
degree $n$.

Taking $\beta_{0,q}=1$,
${\beta}_{i,q}=\frac{1}{2[i+1]_{q}}~{}(i=1,2,\cdots,n)$ and
$A_{n,q}^{II}(x)=B_{n,q}(x)$ in equation (3.1), the following determinant
definition of the $q$BGP ${}_{B}G_{n,q}(x)$ is obtained:

Definition 3.8. The $q$-Bernoulli-Genocchi polynomials ${}_{B}G_{n,q}(x)$ of
degree $n$ are defined by

$\begin{array}[]{l}{}_{B}G_{0,q}(x)=1,\\\
{}_{B}G_{n,q}(x)=(-1)^{n}\left|\begin{array}[]{ccccccc}1&B_{1,q}(x)&B_{2,q}(x)&\cdots&B_{n-1,q}(x)&B_{n,q}(x)\\\
\\\
1&\frac{1}{2[2]_{q}}&\frac{1}{2[3]_{q}}&\cdots&\frac{1}{2[n]_{q}}&\frac{1}{2[n+1]_{q}}\\\
\\\ 0&1&{2\brack 1}_{q}~{}\frac{1}{2[2]_{q}}&\cdots&{n-1\brack
1}_{q}~{}\frac{1}{2[n-1]_{q}}&{n\brack 1}_{q}~{}\frac{1}{2[n]_{q}}\\\ \\\
0&0&1&\cdots&{n-1\brack 2}_{q}~{}\frac{1}{2[n-2]_{q}}&{n\brack
2}_{q}~{}\frac{1}{2[n-1]_{q}}\\\ .&.&.&\cdots&.&.\\\ .&.&.&\cdots&.&.\\\
0&0&0&\cdots&1&{n\brack
n-1}_{q}~{}\frac{1}{2[2]_{q}}\end{array}\right|,~{}n=1,2,\cdots~{},\end{array}$
(3.19)

where $B_{n,q}(x)~{}(n=0,1,2,\ldots)$ are the $q$-Bernoulli polynomials of
degree $n$.

Taking $\beta_{0,q}=1$,
${\beta}_{i,q}=\frac{1}{2[i+1]_{q}}~{}(i=1,2,\cdots,n)$ and
$A_{n,q}^{II}(x)=E_{n,q}(x)$ in equation (3.1), the following determinant
definition of the $q$EGP ${}_{E}G_{n,q}(x)$ is obtained:

Definition 3.9. The $q$-Euler-Genocchi polynomials ${}_{E}G_{n,q}(x)$ of
degree $n$ are defined by

$\begin{array}[]{l}{}_{E}G_{0,q}(x)=1,\\\
{}_{E}G_{n,q}(x)=(-1)^{n}\left|\begin{array}[]{ccccccc}1&E_{1,q}(x)&E_{2,q}(x)&\cdots&E_{n-1,q}(x)&E_{n,q}(x)\\\
\\\
1&\frac{1}{2[2]_{q}}&\frac{1}{2[3]_{q}}&\cdots&\frac{1}{2[n]_{q}}&\frac{1}{2[n+1]_{q}}\\\
\\\ 0&1&{2\brack 1}_{q}~{}\frac{1}{2[2]_{q}}&\cdots&{n-1\brack
1}_{q}~{}\frac{1}{2[n-1]_{q}}&{n\brack 1}_{q}~{}\frac{1}{2[n]_{q}}\\\ \\\
0&0&1&\cdots&{n-1\brack 2}_{q}~{}\frac{1}{2[n-2]_{q}}&{n\brack
2}_{q}~{}\frac{1}{2[n-1]_{q}}\\\ .&.&.&\cdots&.&.\\\ .&.&.&\cdots&.&.\\\
0&0&0&\cdots&1&{n\brack
n-1}_{q}~{}\frac{1}{2[2]_{q}}\end{array}\right|,~{}~{}n=1,2,\cdots~{},\end{array}$
(3.20)

where $E_{n,q}(x)~{}(n=0,1,2,\ldots)$ are the $q$-Euler polynomials of degree
$n$.

Remark 3.3. Taking $x=0$ in determinant definitions (3.12)-(3.14) of
$B_{n,q}(x)$, $E_{n,q}(x)$ and $G_{n,q}(x)$ and on expanding the determinants
with respect to first row and using suitable notations from Table 1 (I-III),
the determinant definitions of the $B_{n,q}$, $E_{n,q}$ and $G_{n,q}$ can be
obtained.

Remark 3.4. Taking $x=0$ in determinant definitions (3.15)-(3.20) of
$B_{n,q}^{[2]}(x)$, $E_{n,q}^{[2]}(x)$ and $G_{n,q}^{[2]}(x)$,
${}_{B}E_{n,q}(x)$, ${}_{B}G_{n,q}(x)$ and ${}_{E}G_{n,q}(x)$ and then using
suitable notations from Tables 4 and 5 (I-III), the determinant definitions of
the $B_{n,q}^{[2]}$, $E_{n,q}^{[2]}$, $G_{n,q}^{[2]}$, ${}_{B}E_{n,q}(x)$,
${}_{B}G_{n,q}$ and ${}_{E}G_{n,q}$ can be obtained.

To give the applications of the operational correspondence introduced in
Section 2, the following identities for the $q$-Appell polynomials are
considered [6].

$\begin{array}[]{llll}A_{n,q}(x)&=\frac{1}{\beta_{0,q}}\Big{(}x^{n}-\sum\limits_{k=0}^{n-1}~{}{n\brack
k}_{q}~{}\beta_{n-k,q}~{}A_{k,q}(x)\Big{)},~{}~{}~{}n=1,2,\cdots~{}~{}~{},\\\
x^{n}&=\sum\limits_{k=0}^{n}~{}{n\brack
k}_{q}\beta_{n-k,q}~{}A_{k,q}(x),~{}~{}~{}n=1,2,\cdots~{}~{}~{}.\\\
\end{array}$ (3.21)

Replacing the powers of $x$, i.e. $x^{1}$ and $x^{n}$ by the corresponding
polynomials ${A}_{1,q}(x)$ and ${A}_{n,q}(x)$ in above equations and then
using operational representation (2.7) in the resultant equations, the
following identities for the 2I$q$AP $A_{n,q}^{[2]}(x)$ are obtained:

$\begin{array}[]{lll}A_{n,q}^{[2]}(x)&=\frac{1}{\beta_{0,q}}\Big{(}{A}_{n,q}(x)-\sum\limits_{k=0}^{n-1}~{}{n\brack
k}_{q}~{}\beta_{n-k,q}~{}A_{k,q}^{[2]}(x)\Big{)},~{}~{}~{}n=1,2,\cdots~{}~{}~{},\\\
{A}_{n,q}(x)&=\sum\limits_{k=0}^{n}~{}{n\brack
k}_{q}\beta_{n-k,q}~{}A_{k,q}^{[2]}(x),~{}~{}~{}n=1,2,\cdots~{}~{}~{}.\\\
\end{array}$ (3.22)

The above examples illustrate that the operational correspondence established
in this article can be applied to derive the results for the newly introduced
$q$-special polynomials given in Tables 2 and 3 from the results of the
corresponding member belonging to the $q$-Appell family.

In the next section, the shapes of the 2I$q$BP $B_{n,q}^{[2]}(x)$, 2I$q$EP
$E_{n,q}^{[2]}(x)$, 2I$q$GP $G_{n,q}^{[2]}(x)$, $q$BEP ${}_{B}E_{n,q}(x)$,
$q$BGP ${}_{B}G_{n,q}(x)$ and $q$EGP ${}_{E}G_{n,q}(x)$ are displayed. The
zeros of these polynomials are also computed by using Matlab.

##  4 Graphical representation and roots

There has been increasing interest in solving mathematical problems with the
aid of computers. Numerical investigation of the roots of certain
$q$-polynomials are considered in [12]. Also, the shapes of a new class of
$q$-Bernoulli polynomials are explored in [11]. Here, the plots of the 2I$q$BP
$B_{n,q}^{[2]}(x)$, 2I$q$EP $E_{n,q}^{[2]}(x)$, 2I$q$GP $G_{n,q}^{[2]}(x)$,
$q$BEP ${}_{B}E_{n,q}(x)$, $q$BGP ${}_{B}G_{n,q}(x)$ and $q$EGP
${}_{E}G_{n,q}(x)$ are drawn for $n=1,2,3,4$ and $q=\frac{1}{2}$ $(0<q<1)$.

This shows the four plots combined into one for each of these polynomials. For
this, the values of the first five $B_{n,q}$, $E_{n,q}$ and $G_{n,q}$ are
required. The values of first five $B_{n,q}$, $E_{n,q}$ and $G_{n,q}$ [5, 7]
are listed in Table 6.

Table 6. Values of first five $B_{n,q}$, $E_{n,q}$ and $G_{n,q}$

$n$ 0 1 2 3 4 $B_{n,q}$ 1 $-(1+q)^{-1}$ $q^{2}([3]_{q}!)^{-1}$
$(1-q)q^{3}([2]_{q})^{-1}([4]_{q})^{-1}$
$q^{4}(1-q^{2}-2q^{3}-q^{4}+q^{6})([2]_{q}^{2}[3]_{q}[5]_{q})^{-1}$ $E_{n,q}$
1 $-\frac{1}{2}$ $\frac{1}{4}(-1+q)$ $\frac{1}{8}(-1+2q+2q^{2}-q^{3})$
$\frac{1}{16}(q-1)[3]_{q}!(q^{2}-4q+1)$ $G_{n,q}$ 1 $\frac{q}{1+q}$
$\frac{-(q^{3}+3q^{2}+4q+3)}{(1+q)(1+q+q^{2})}$
$\frac{1}{(1+q)^{2}}(2q^{3}+q^{2})$
$\frac{1}{1+q^{2}}(\frac{2q^{3}+q^{2}}{(1+q)^{2}}+\frac{q^{3}+3q^{2}+4q+3}{(1+q)(1+q+q^{2})})-\frac{q}{q+1}-\frac{1}{[5]_{q}}-1$

The expressions of the first five $B_{n,q}(x)$, $E_{n,q}(x)$ and $G_{n,q}(x)$
are obtained by making use of the values of the first five $B_{n,q}$,
$E_{n,q}$ and $G_{n,q}$ in the series definitions given in Table 1 (I-III).
The expressions of first five $B_{n,q}(x)$, $E_{n,q}(x)$ and $G_{n,q}(x)$ are
listed in Table 7.

Table 7. Expressions of first five $B_{n,q}(x)$, $E_{n,q}(x)$ and $G_{n,q}(x)$

$n$ 0 1 2 3 4 $B_{n,q}(x)$ $1$ $x-\frac{1}{1+q}$
$x^{2}-\frac{[2]_{q}}{1+q}x+\frac{q^{2}}{[3]_{q}[2]_{q}}$
$x^{3}-\frac{[3]_{q}x^{2}}{1+q}+\frac{q^{2}x}{[2]_{q}}+\frac{(1-q)q^{3}}{[2]_{q}[4]_{q}}$
$x^{4}-\frac{[4]_{q}}{1+q}x^{3}+\frac{[4]_{q}q^{2}}{[2]_{q}^{2}}x^{2}+\frac{(1-q)q^{3}}{[2]_{q}}x$
$+q^{4}(1-q^{2}-2q^{3}-q^{4}+q^{6})([2]_{q}^{2}[3]_{q}[5]_{q})^{-1}$
$E_{n,q}(x)$ $1$ $x-\frac{1}{2}$ $x^{2}-\frac{[2]_{q}}{2}x+\frac{1}{4}(-1+q)$
$x^{3}-\frac{[3]_{q}}{2}x^{2}+\frac{[3]_{q}}{4}(-1+q)x$
$x^{4}-\frac{[4]_{q}}{2}x^{3}+\frac{[4]_{q}[3]_{q}(q-1)}{4[2]_{q}}x^{2}+$
$+\frac{1}{8}(-1+2q+2q^{2}-q^{3})$
$\frac{[4]_{q}(-1+2q+2q^{2}-q^{3})}{8}x+\frac{(q-1)[3]_{q}!(q^{2}-4q+1)}{16}$
$G_{n,q}(x)$ $1$ $x+\frac{q}{1+q}$
$x^{2}+qx-\frac{(q^{3}+3q^{2}+4q+3)}{(1+q)(1+q+q^{2})}$
$x^{3}+\frac{[3]_{q}q}{(1+q)}x^{2}-\frac{(q^{3}+3q^{2}+4q+3)}{(1+q)}x$
$x^{4}+\frac{[4]_{q}q}{q+1}x^{3}-\frac{[4]-q[3]_{q}(q^{3}+3q^{2}+4q+3)}{[2]_{q}(1+q)(1+q+q^{2})}x^{2}$
$+\frac{(2q^{3}+q^{2})}{(1+q)}$
$\frac{[4]_{q}(2q^{3}+q^{4})}{(1+q)^{2}}x+\frac{1}{1+q^{2}}(\frac{2q^{3}+q^{2}}{(1+q)^{2}}+\frac{q^{3}+3q^{2}+4q+3}{(1+q)(1+q+q^{2})})$
$-\frac{q}{q+1}-\frac{1}{[5]_{q}}-1$

By making appropriate substitutions from Tables 6 and 7 in the series
definitions given in Tables 2 and 3 (I-III), the expressions of the first five
2I$q$BP $B_{n,q}^{[2]}(x)$, 2I$q$EP $E_{n,q}^{[2]}(x)$, 2I$q$GP
$G_{n,q}^{[2]}(x)$, $q$BEP ${}_{B}E_{n,q}(x)$, $q$BGP ${}_{B}G_{n,q}(x)$ and
$q$EGP ${}_{E}G_{n,q}(x)$ for $q=\frac{1}{2}$ are obtained. These expressions
are given in Table 8.

Table 8. Expressions of first five $B_{n,1/2}^{[2]}(x)$, $E_{n,1/2}^{[2]}(x)$,
$G_{n,1/2}^{[2]}(x)$, ${}_{B}E_{n,1/2}(x)$, ${}_{B}G_{n,1/2}(x)$ and
${}_{E}G_{n,1/2}(x)$

$n$ 0 1 2 3 4 $B_{n,1/2}^{[2]}(x)$ $1$ $x-\frac{4}{3}$ $x^{2}-2x+\frac{6}{7}$
$x^{3}-\frac{7}{3}x^{2}+\frac{3}{2}x-\frac{8}{45}$
$x^{4}-\frac{5}{2}x^{3}+\frac{45}{24}x^{2}-\frac{8}{24}x-\frac{221}{7812}$
$E_{n,1/2}^{[2]}(x)$ $1$ $x-1$ $x^{2}-\frac{3}{2}x-\frac{1}{16}$
$x^{3}-\frac{7}{4}x^{2}+\frac{7}{32}x+\frac{5}{16}$
$x^{4}-\frac{15}{8}x^{3}+\frac{35}{128}x^{2}+\frac{125}{1024}x+\frac{71}{1024}$
$G_{n,1/2}^{[2]}(x)$ $1$ $x+\frac{2}{3}$ $x^{2}+x-\frac{181}{42}$
$x^{3}+\frac{7}{6}x^{2}-\frac{181}{24}x-\frac{37}{18}$
$x^{4}+\frac{5}{4}x^{3}-\frac{905}{96}x^{2}-\frac{585}{144}x+\frac{358499}{31248}$
${}_{B}E_{n,1/2}(x)$ $1$ $x-\frac{7}{6}$ $x^{2}-\frac{3}{2}x+\frac{51}{168}$
$x^{3}-\frac{49}{24}x^{2}+\frac{79}{96}x+\frac{379}{2880}$
$x^{4}-\frac{35}{16}x^{3}-\frac{445}{384}x^{2}-\frac{461}{1536}x-\frac{402305}{9999360}$
${}_{B}G_{n,1/2}(x)$ $1$ $x-\frac{1}{3}$ $x^{2}-\frac{1}{2}x-\frac{52}{21}$
$x^{3}-\frac{7}{12}x^{2}-\frac{5}{12}x+\frac{21}{70}$
$x^{4}-\frac{15}{24}x^{3}-\frac{5460}{1008}x^{2}+\frac{5502}{1008}x-\frac{22203615}{27888840}$
${}_{E}G_{n,1/2}(x)$ $1$ $x-\frac{1}{6}$ $x^{2}-\frac{1}{4}x-\frac{439}{168}$
$x^{3}-\frac{7}{24}x^{2}-\frac{439}{96}x+\frac{1187}{576}$
$x^{4}-\frac{5}{16}x^{3}-\frac{595}{384}x^{2}+\frac{3553}{32256}x+\frac{1226437}{19999872}$

With the help of Matlab and by using the expressions for the first five
$B_{n,1/2}^{[2]}(x)$, $E_{n,1/2}^{[2]}(x)$, $G_{n,1/2}^{[2]}(x)$,
${}_{B}E_{n,1/2}(x)$, ${}_{B}G_{n,1/2}(x)$ and ${}_{E}G_{n,1/2}(x)$ from Table
8 for $n=1,2,3,4$, the following graphs are drawn:

Figure 4.1 Figure 4.2

Figure 4.3 Figure 4.4

Figure 4.5 Figure 4.6

Next, we find the real and complex zeros of the polynomials given in Table 8.
The computation of these zeros by hand is too complicated, therefore, we use
Matlab to investigate these zeros. The investigation in this direction will
lead to a new approach employing numerical methods in the field of these
$q$-special polynomials to appear in mathematics and physics.

The real zeros of $B_{n,1/2}^{[2]}(x)$, $E_{n,1/2}^{[2]}(x)$,
$G_{n,1/2}^{[2]}(x)$, ${}_{B}E_{n,1/2}(x)$, ${}_{B}G_{n,1/2}(x)$ and
${}_{E}G_{n,1/2}(x)$ are computed by using Matlab. These zeros are given in
Table 9.

Table 9. Real zeros of $B_{n,1/2}^{[2]}(x)$, $E_{n,1/2}^{[2]}(x)$,
$G_{n,1/2}^{[2]}(x)$, ${}_{B}E_{n,1/2}(x)$, ${}_{B}G_{n,1/2}(x)$ and
${}_{E}G_{n,1/2}(x)$

$\textrm{Degree}~{}n$ $B_{n,1/2}^{[2]}(x)$ $E_{n,1/2}^{[2]}(x)$
$G_{n,1/2}^{[2]}(x)$ ${}_{B}E_{n,1/2}(x)$ ${}_{B}G_{n,1/2}(x)$
${}_{E}G_{n,1/2}(x)$ $1$ $1.3333$ $1.0000$ $-0.6667$ $1.1667$ $0.3333$
$0.1667$ $2$ $0.6220$, $1.3780$ $-0.0406$, $1.5406$ $-2.6353$, $1.6353$
$0.2411$, $1.2589$ $-1.3433$, $1.8433$ $-1.4963$, $1.7463$ $3$ $0.1522$,
$0.9446$, $-3.2768$, $-0.2642$, $-3.2768$, $-0.2642$, $-0.1213$, $0.7910$,
$-0.6795$ $-2.2055$, $0.4583$, $1.2365$ $2.3743$ $2.3743$ $1.3719$ $2.0388$
$4$ $-0.0617$, $0.3823$ $0.5479$, $1.6488$ $-3.3847$, $-1.3512$, $-0.2476$,
$2.6664$ $-2.4871$, $0.1776$, $-1.1179$, $-0.1703$, $1.0514$, $2.4705$
$0.8751$, $2.0594$ $0.2360$, $1.3647$

Again, with the help of Matlab, we find the complex zeros of these
polynomials. These complex zeros are given in Table 10.

Table 10. Complex zeros of $B_{n,1/2}^{[2]}(x)$, $E_{n,1/2}^{[2]}(x)$,
$G_{n,1/2}^{[2]}(x)$, ${}_{B}E_{n,1/2}(x)$, ${}_{B}G_{n,1/2}(x)$ and
${}_{E}G_{n,1/2}(x)$

$\textrm{Degree}~{}n$ $B_{n,1/2}^{[2]}(x)$ $E_{n,1/2}^{[2]}(x)$
$G_{n,1/2}^{[2]}(x)$ ${}_{B}E_{n,1/2}(x)$ ${}_{B}G_{n,1/2}(x)$
${}_{E}G_{n,1/2}(x)$ $1$ $-$ $-$ $-$ $-$ $-$ $-$ $2$ $-$ $-$ $-$ $-$ $-$ $-$
$3$ $-$ $-$ $-$ $-$ -$0.6314+0.2068i$, $-$ $0.6314-0.2068i$ $4$
$1.0897+0.1112i$, $-0.1609+0.225i$, $-$ $-0.1157+0.2181i$, $-$ $-$
$1.0897-0.1112i$ $-0.1609-0.225i$ $-0.1157-0.2181i$

Further, we plot the zeros of the polynomials $B_{n,1/2}^{[2]}(x)$,
$E_{n,1/2}^{[2]}(x)$, $G_{n,1/2}^{[2]}(x)$, ${}_{B}E_{n,1/2}(x)$,
${}_{B}G_{n,1/2}(x)$ and ${}_{E}G_{n,1/2}(x)$ for $n=1,2,3,4$.

Figure 4.7 Figure 4.8

Figure 4.9 Figure 4.10

Figure 4.11 Figure 4.12

Remark 4.1. From Tables 9 and 10, the following general relation is observed.
The number of real zeros lying on the real plane $\textrm{Im}~{}(x)=0$, i.e.,

$\textrm{Real zeros of}~{}A_{n,q}^{[2]}(x)=n-\textrm{Complex zeros
of}~{}A_{n,q}^{[2]}(x),$

where $n$ is the degree of polynomial.

In order to make the above discussion more clear, we draw the combined graphs
of shapes and zeros of the polynomials $B_{n,1/2}^{[2]}(x)$,
$E_{n,1/2}^{[2]}(x)$, $G_{n,1/2}^{[2]}(x)$, ${}_{B}E_{n,1/2}(x)$,
${}_{B}G_{n,1/2}(x)$ and ${}_{E}G_{n,1/2}(x)$ for $n=4$.

Figure 4.13 Figure 4.14

Figure 4.15 Figure 4.16

Figure 4.17 Figure 4.18

It is to be noted that in Figures 4.13, 4.14 and 4.16 out of total two complex
zeros only one, with positive imaginary part is visible, due to the absence of
negative imaginary axis in these graphs.

The numerical results for approximate solutions of real zeros of
$B_{n,1/2}^{[2]}(x)$, $E_{n,1/2}^{[2]}(x)$, $G_{n,1/2}^{[2]}(x)$,
${}_{B}E_{n,1/2}(x)$, ${}_{B}G_{n,1/2}(x)$ and ${}_{E}G_{n,1/2}(x)$ for
$(n=1,2,3,4)$ are displayed in Table 9.

Also, we note that the real zeros of these polynomials displayed in Table 9
are giving the numerical results for the approximate solutions of
$B_{n,1/2}^{[2]}(x)=0$, $E_{n,1/2}^{[2]}(x)=0$, $G_{n,1/2}^{[2]}(x)=0$,
${}_{B}E_{n,1/2}(x)=0$, ${}_{B}G_{n,1/2}(x)=0$ and ${}_{E}G_{n,1/2}(x)=0$ for
$n=1,2,3,4$.

Note. It is important to note that the 2-iterated $q$-Appell polynomials
introduced in this article are actually the $q$-Appell polynomials, since
their generating function is of the type $A^{\star}_{q}(t)e_{q}(xt)$, with a
suitable choice for $A^{\star}_{q}(t)$. The results established in this
article for the 2-iterated and mixed type $q$-special polynomials can be used
to solve the existing as well as new emerging problems of certain branches of
mathematics, physics and engineering.

Acknowledgement

This work has been done under Senior Research Fellowship (Award letter No.
F1-17.1/2012-13, MANF-MUS-UTT-9243) awarded to the second author by the
University Grants Commission, Government of India, New Delhi.

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