# False Negative Distillation and Contrastive Learning for Personalized Outfit
Recommendation

Seongjae Kim, Jinseok Seol, Holim Lim, Sang-goo Lee  Seoul National
UniversityRepublic of Korea sjkim, jamie, ihl7029<EMAIL_ADDRESS>
###### Abstract.

Personalized outfit recommendation has recently been in the spotlight with the
rapid growth of the online fashion industry. However, recommending outfits has
two significant challenges that should be addressed. The first challenge is
that outfit recommendation often requires a complex and large model that
utilizes visual information, incurring huge memory and time costs. One natural
way to mitigate this problem is to compress such a cumbersome model with
knowledge distillation (KD) techniques that leverage knowledge from a
pretrained teacher model. However, it is hard to apply existing KD approaches
in recommender systems (RS) to the outfit recommendation because they require
the ranking of all possible outfits while the number of outfits grows
exponentially to the number of consisting clothing items. Therefore, we
propose a new KD framework for outfit recommendation, called False Negative
Distillation (FND), which exploits false-negative information from the teacher
model while not requiring the ranking of all candidates. The second challenge
is that the explosive number of outfit candidates amplifying the data sparsity
problem, often leading to poor outfit representation. To tackle this issue,
inspired by the recent success of contrastive learning (CL), we introduce a CL
framework for outfit representation learning with two proposed data
augmentation methods. Quantitative and qualitative experiments on outfit
recommendation datasets demonstrate the effectiveness and soundness of our
proposed methods.

Personalized Outfit Recommendation, Knowledge Distillation, False Negative
Distillation, Contrastive Learning

††doi: 10.1145/1122445.1122456

Figure 1. The user-outfit space is a representation vector space of the
trained teacher model. We define the positive boundary as an average distance
between a user and its positive outfits. We treat negative outfits inside the
positive boundary as false-negative outfits.

A figure that shows the user-outfit space. In the space, there are user,
positive outfits, negative outfits, and false-negative outfits. False-negative
outfits are outfits that lie inside the positive boundary, where the model
determines whether the outfit is preferable or not.

##  1\. Introduction

Personalized outfit recommendation is the task of determining the preference
of a user to an input outfit that consists of multiple clothing. It has
recently attracted attention with the rapid growth of the online fashion
industry, and several related studies (Lin et al., 2020; Lu et al., 2019,
2021) have been conducted. However, despite the success of existing works,
outfit recommendation has two significant challenges that should be addressed.
First, recommending outfits often requires a complex and large model that
involves the utilization of visual information (i.e., images) (Lu et al.,
2021). Such a large model incurs high latency and memory costs during the
inference phase, making it difficult to apply to real-time services (Kang et
al., 2020). The second challenge is that outfit recommendation inevitably
suffers from the data sparsity problem because the possible pool of outfit
data grows exponentially to the number of consisting clothing items (Lin et
al., 2020). The sparsity problem often leads to poor learning of outfit
representation, which hinders achieving satisfactory recommendation
performance (Xie et al., 2020).

To address the first challenge stemming from a large model, one can employ
knowledge distillation (KD) techniques that compress a model by transferring
knowledge from a large teacher model to a small student model. Accordingly,
one may try to apply existing studies (Tang and Wang, 2018; Lee et al., 2019a;
Kang et al., 2020) of KD available in recommender systems (RS) to the outfit
recommendation. However, existing methods leverage predicted ranking of all
possible outfits from the teacher model, so they are not applicable in outfit
recommendation tasks with explosively large pools. Therefore, we propose a
novel KD framework named False Negative Distillation (FND) that does not
require the ranking of all outfit candidates. Similar to most outfit
recommendation studies (Lin et al., 2020; Lu et al., 2019, 2021), FND utilizes
a ranking loss that pulls observed (positive) outfits to a user while pushing
unobserved (negative) outfits. As illustrated in Fig. 1, FND claims that
unobserved is not the same as true-negative and assumes that negative outfits
close enough to the user are false-negative outfits. We show through various
experiments that FND is effective, and the assumption is reasonable.

Figure 2. Example of contrastive learning for outfit recommendation. We
randomly alter (erase/replace) one item from an outfit to generate two
different views. In the case of replace, it substitutes an item with a similar
one using our pretrained autoencoder model. In this example, the first and the
second augmentations are erase and replace, respectively. The left outfit
erased the shoes and replaced the top. The right outfit erased the bottom and
replaced the bottom.

A figure that shows examples of outfit augmentation methods and contrastive
learning. Each outfit is augmented twice to generate two views, which should
pull each other while views from the other outfits are pushed. Augmentation
methods for outfits consist of erase and replace. Erase erases a random item
from an outfit while replace randomly replaces an item. In the figure, there
are two example outfits. One erased the shoes and replaced the top. The other
erased the bottom and replaced the bottom.

The approach for the second challenge is to deal with the problem of poor
outfit representation. As learning enhanced representation of entities is one
of the core components to achieve high performance in deep learning (Bengio et
al., 2013), numerous works (Mikolov et al., 2013; Devlin et al., 2018; Gidaris
et al., 2018; Chen et al., 2020) from diverse domains accomplished this with
self-supervised learning (SSL) techniques. There have been several studies (Ma
et al., 2020; Xin et al., 2020; Zhou et al., 2020) utilizing SSL techniques in
RS as well. Among them, more recent works (Xie et al., 2020; Yao et al., 2020;
Liu et al., 2021) exploit contrastive learning (CL), especially SimCLR (Chen
et al., 2020), which learns meaningful representations by pulling the
differently augmented view of the same data while pushing the others in the
batch. Nevertheless, leveraging CL in outfit recommendation is relatively
unexplored. Hence, as illustrated in Fig. 2, we introduce an approach to make
use of CL in outfit recommendation, along with two proposed data augmentation
methods (erase/replace).

To demonstrate the effectiveness of our proposed methods, we conduct extensive
experiments on outfit recommendation datasets. We compare our approaches with
state-of-the-art outfit recommendation methods with quantitative performance
evaluations. In addition, we study the impact of hyper-parameters and the
model size with various experiments. We use visualization to show that the
intuitive assumption of our FND illustrated in Fig. 1 is sound. We also
experiment on the cold-start scenario where users have very few outfits
interacted, and the trained model does not have any knowledge of those users.
To make appropriate recommendations to cold starters, we introduce two
practical strategies that do not require additional training of the model.

Our main contributions can be summarized as follows:

  * •

We propose a new knowledge distillation framework that can be utilized in
outfit recommendation tasks without requiring the ranking of all outfit
candidates in the system.

  * •

We propose two novel outfit data augmentation methods to leverage contrastive
learning in outfit recommendation.

  * •

We introduce two practical strategies to deal with the cold-start problem.

  * •

We demonstrate the effectiveness and soundness of our approaches with
comprehensive experiments on fashion outfit recommendation datasets.

##  2\. Related Work

###  2.1. Outfit Recommendation

Based on whether the individual preference is neglected or not, existing
outfit recommendation studies can be classified into two categories: non-
personalized (Veit et al., 2015; Vasileva et al., 2018; Tan et al., 2019;
Cucurull et al., 2019; Singhal et al., 2020; Li et al., 2017; Han et al.,
2017; Yang et al., 2020) and personalized (Hu et al., 2015; Lu et al., 2019;
Li et al., 2020; Lin et al., 2020; Lu et al., 2021) outfit recommendation. Lu
et al. (Lu et al., 2019) used pairwise scores, and they employed the weighted
hashing technique to tackle the efficiency problem. Lin et al. (Lin et al.,
2020) utilized an attention mechanism to estimate the preference score,
weighting items in an outfit differently. Lu et al. (Lu et al., 2021)
exploited Set Transformer (Lee et al., 2019b), the state-of-the-art model for
set-input problems, to capture the high-order interactions among fashion
items. They also disentangled each user into multiple anchors to accommodate
the variety of preferences. Note that methods based on graph neural networks
(Cucurull et al., 2019; Singhal et al., 2020; Li et al., 2020) or predicting
distribution over whole clothing items (Yang et al., 2020) require the test
items to be in the training set.

###  2.2. Knowledge Distillation

Knowledge distillation is a model-agnostic compression strategy for generating
efficient models. Since the early success of KD in image recognition (Hinton
et al., 2015; Romero et al., 2014), KD has been widely accepted in other
fields. In recommendation tasks, several works (Tang and Wang, 2018; Lee et
al., 2019a; Kang et al., 2020) have employed KD. They rank all items with the
teacher model and utilize the items of high rank when training the student
model. Tang et al. (Tang and Wang, 2018) considered top-$K$ items as false-
negatives and differentiated their relative importance based on their
rankings. Lee et al. (Lee et al., 2019a) trained the student to mimic the
predicted probabilities of the teacher on the sampled items of high rank. Kang
et al. (Kang et al., 2020) achieved state-of-the-art performance by
transferring both the prediction and latent knowledge of the teacher.

###  2.3. Contrastive Learning

Contrastive learning is a framework for obtaining high-quality representations
to boost the performance of downstream tasks and was first introduced in
computer vision (Chen et al., 2020). CL enhances representations by maximizing
agreement between two differently augmented views of the same data. A few
works (Xie et al., 2020; Yao et al., 2020; Liu et al., 2021) applied CL to RS,
and they showed notable success. In sequential recommendation, Xie et al. (Xie
et al., 2020) used CL by applying three augmentation methods
(crop/mask/reorder) to user interaction history. Yao et al. (Yao et al., 2020)
focused on large-scale item recommendations and employed a two-stage
augmentation consisting of masking and dropout. Liu et al. (Liu et al., 2021)
utilized CL for graph neural network based RS by randomly removing some edges.

##  3\. Approach

We recommend outfits to users based on their preference score. To compute the
preference score, we use user embeddings and vector representations of
outfits. Due to the set-like nature of fashion outfits, the representation
model requires two conditions. First, the outfit representation should be
invariant to the order of comprising fashion items. Second, the model should
be able to process input outfits of any size. To this end, we borrow the
architecture from LPAE (Lu et al., 2021) model, which uses Set Transformer
(Lee et al., 2019b) module designed to address these set-input problems.

###  3.1. Background: Computing the Preference Score to an Outfit

Figure 3. A brief architecture of computing the preference score of a user to
an outfit. First, images comprising the outfit are transformed into item
features by CNN with fully connected layers. After that, we get an outfit
representation using Set Transformer. Finally, the cosine similarity between a
user embedding and the outfit representation is the preference score of the
user to the outfit.

An outfit is composed of multiple fashion images. First, the images are
transformed into item features by CNN with fully connected layers. After that,
we can get an outfit representation using Set Transformer. Finally, the cosine
similarity between a user embedding and the outfit representation is the
preference score of the user to the outfit.

As illustrated in Fig. 3, an outfit $o_{j}$ with $n$ items is a tuple of
fashion item images: $o_{j}=(x_{j1},x_{j2},\cdots,x_{jn})\in\mathcal{I}^{n}.$
Let $f:\mathcal{I}\rightarrow\mathbb{R}^{d}$ be a Convolutional Neural Network
(CNN) with fully connected layers that encodes $x_{jk}$ into an item feature
vector $\mathbf{x}_{jk}=f(x_{jk})$, where $d$ is a feature dimension. Through
Set Transformer
$T:\mathbb{R}^{\textit{n}\times\textit{d}}\rightarrow\mathbb{R}^{\textit{d}}$,
we obtain an outfit representation $\mathbf{o}_{j}=T(\mathbf{X}_{j})$ from
item features
$\mathbf{X}_{j}=[\mathbf{x}_{j1}\,\,\mathbf{x}_{j2}\,\,\cdots\,\,\mathbf{x}_{jn}]^{\top}.$
Then, we compute the preference score with user embedding
$\mathbf{u}_{i}\in\mathbb{R}^{d}$ for each user $u_{i}$.

####  3.1.1. Set Transformer

Having benefited from the attention mechanism, Set Transformer can effectively
reflect high-order interactions among items in an outfit. As Lee et al. (Lee
et al., 2019b) have proved, Set Transformer is an order-free module that
always produces the same output regardless of the sequence order.

Attention is a map that gives the weighted sum of value vectors
$\mathbf{V}\in\mathbb{R}^{n_{\text{v}}\times d_{\text{v}}}$ with the weights
being determined by each query vector of
$\mathbf{Q}\in\mathbb{R}^{n_{\text{q}}\times d_{\text{q}}}$ and key vectors
$\mathbf{K}\in\mathbb{R}^{n_{\text{v}}\times d_{\text{q}}}$:

(1)
$\text{Attention}(\mathbf{Q},\mathbf{K},\mathbf{V})=\text{Softmax}(\frac{\mathbf{Q}\mathbf{K}^{\top}}{\sqrt{d_{\text{q}}}})\mathbf{V}.$

The multi-head attention utilizes multiple attentions through concatenation to
bear more potential relationships: for $h$ attention maps,

(2)
$\mathbf{A}_{i}=\text{Attention}(\mathbf{Q}\mathbf{W}_{i}^{\text{Q}},\mathbf{K}\mathbf{W}_{i}^{\text{K}},\mathbf{V}\mathbf{W}_{i}^{\text{V}}),$
(3)
$\text{MultiHead}(\mathbf{Q},\mathbf{K},\mathbf{V})=\text{Concat}(\mathbf{A}_{1},\mathbf{A}_{2},\cdots,\mathbf{A}_{h})\mathbf{W}^{\text{M}},$

where
$\mathbf{W}_{i}^{\text{Q}},\mathbf{W}_{i}^{\text{K}}\in\mathbb{R}^{d_{\text{q}}\times
d_{\text{q}}^{\text{M}}},\mathbf{W}_{i}^{\text{V}}\in\mathbb{R}^{d_{\text{v}}\times
d_{\text{v}}^{\text{M}}},$ and
$\mathbf{W}^{\text{M}}\in\mathbb{R}^{hd_{\text{v}}^{\text{M}}\times
d_{\text{v}}}.$ Following previous works (Lee et al., 2019b; Lu et al., 2021),
we use $d_{\text{q}}^{\text{M}}=d_{\text{q}}/h$ and
$d_{\text{v}}^{\text{M}}=d_{\text{v}}/h$.

To apply the attention mechanism to a set, Set Attention Block (SAB) uses
self-attention with residual terms: for item features $\mathbf{X}_{j}$,

(4)
$\mathbf{H}=\text{LayerNorm}(\mathbf{X}_{j}+\text{MultiHead}(\mathbf{X}_{j},\mathbf{X}_{j},\mathbf{X}_{j})),$
(5)
$\text{SAB}(\mathbf{X}_{j})=\text{LayerNorm}(\mathbf{H}+\sigma(\mathbf{H})),$

where $\sigma$ is any row-wise feed-forward layer, and
$\text{LayerNorm}(\cdot)$ is Layer Normalization (Ba et al., 2016). Multiple
SABs can be stacked to encode higher-order interactions among the items:

(6) $\mathbf{F}=\text{SAB}(\text{SAB}(\mathbf{X}_{j})).$

The final outputs of the attention blocks are then aggregated by applying
another multi-head attention on a learnable seed vector
$\mathbf{s}\in\mathbb{R}^{d}$ as follows:

(7)
$\mathbf{z}=\text{LayerNorm}(\mathbf{s}+\text{MultiHead}(\mathbf{s},\mathbf{F},\mathbf{F})),$
(8) $\mathbf{o}_{j}=\text{LayerNorm}(\mathbf{z}+\sigma(\mathbf{z})).$

The obtained $\mathbf{o}_{j}\in\mathbb{R}^{d}$ is a single compact vector
representation of an outfit $o_{j}$, holding compatibility relationships among
consisting fashion items.

####  3.1.2. Preference score prediction

Given a user $u_{i}$ and an outfit $o_{j}$, our model predicts the preference
score of the user to the outfit as follows:

(9)
$r_{ij}=\cos(\mathbf{u}_{i},\mathbf{o}_{j})=\frac{\mathbf{u}_{i}^{\top}\mathbf{o}_{j}}{\|\mathbf{u}_{i}\|\|\mathbf{o}_{j}\|}.$

###  3.2. False Negative Distillation

Large models generally show relatively higher recommendation performance
compared to their smaller counterparts. However, employing a small-sized model
is necessary to reduce latency and memory costs during the inference phase.
Therefore, we propose a novel knowledge distillation framework named False
Negative Distillation (FND) that transfers false-negative information
extracted from a well-trained large teacher model to a small student model. As
illustrated in Fig. 1, in the user-outfit space of a trained teacher model, we
assume that negative (i.e., unobserved) outfits close enough to the user are
false-negative outfits.

####  3.2.1. Teacher model

Deep learning based recommendation models adopt learning to rank framework via
deep metric learning in general. The goal is to maximize the ranking of
positive outfits given a predicted preference score. Many existing works (Tan
et al., 2019; Lin et al., 2020; Lu et al., 2019), including LPAE (Lu et al.,
2021), use triplet loss (Schroff et al., 2015) or Bayesian personalized
ranking (BPR) (Rendle et al., 2009) as an optimization objective. However,
they often suffer from poor local optima, partially because the loss function
employs only one negative outfit in each update (Sohn, 2016). To address this
problem, our model utilizes $N$-pair loss (Sohn, 2016). Aided by the
temperature-scaled cross-entropy, $N$-pair loss can take multiple negative
outfits into account per positive outfit. Let the batch $\mathcal{B}$, size of
$N$, be a set of pairs $(u_{i},o_{j})$, indicating that the user $u_{i}$
prefers the outfit $o_{j}$. Each pair in the batch has a set of negative
outfits $\\{o_{j_{k}}\\}_{k=1}^{\textit{N}}$, sampled in the training step.
Note that the negative set mainly contains randomly generated outfits and even
can include positive outfits of other users. Our objective for the teacher
model is as follows:

(10)
$\mathcal{L}_{N\text{-pair}}=-\frac{1}{N}\sum_{(u_{i},o_{j})\in\mathcal{B}}\log\frac{\exp(r_{ij}/\tau_{\text{FND}})}{\displaystyle\exp(r_{ij}/\tau_{\text{FND}})+\sum_{k=1}^{N}\exp(r_{ij_{k}}/\tau_{\text{FND}})},$

where $\tau_{\text{FND}}>0$ is a temperature hyper-parameter.

####  3.2.2. Student model

Once the teacher model is trained, we optimize our student model with the help
of additional false-negative information. The ordinary $N$-pair loss can be
interpreted as “pulling” positive outfits to a user while “pushing” the
negatives, similar to triplet loss with user-anchor. Whenever a given negative
outfit is determined as a false-negative, we wish to pull it rather than
pushing it. Concretely, we determine the false-negativeness $d_{ij_{k}}$ based
on the difference between the preference score of a negative outfit and the
average score of the positives: with $J_{i}^{+}$ a set of indices of positive
outfits for a user $u_{i}$,

(11) $\hat{r}_{i}=\frac{1}{|J_{i}^{+}|}\sum_{j^{+}\in
J_{i}^{+}}\hat{r}_{ij^{+}},$ (12)
$d_{ij_{k}}=\alpha(\hat{r}_{i}-\hat{r}_{ij_{k}}),$

where $\alpha>0$ is a distillation scaling hyper-parameter and $\hat{r}_{ij}$
denotes the predicted score from the teacher. The sign of $d_{ij_{k}}$
determines whether the given negative outfit is false-negative or not, and the
magnitude presents how much the negative should be pushed or pulled. The
student model is trained through our proposed FND loss
$\mathcal{L}_{\text{FND}}$ as follows:

(13)
$\ell_{\text{FND}}(i,j)=-\log\frac{\exp(r_{ij}/\tau_{\text{FND}})}{\displaystyle\exp(r_{ij}/\tau_{\text{FND}})+\sum_{k=1}^{N}\exp(d_{ij_{k}}r_{ij_{k}}/\tau_{\text{FND}})},$
(14)
$\mathcal{L}_{\text{FND}}=\frac{1}{N}\sum_{(u_{i},o_{j})\in\mathcal{B}}\ell_{\text{FND}}(i,j).$

Since the teacher model is frozen when training the student, $d_{ij_{k}}$
stays constant for each $i$ and $j_{k}$. Note that we use different validation
sets when training the teacher model and the student model. Otherwise, the
student model might learn information about the validation set via the
teacher, which leads to overfitting.

By examining the gradient of $\mathcal{L}_{\text{FND}}$, it can be shown that
our objective function pulls negative outfits to the user whenever they are
determined as false-negatives by $d_{ij_{k}}$. Suppose $o_{j_{k^{\prime}}}$ is
an element of the set of negative outfits $\\{o_{j_{k}}\\}_{k=1}^{N}$ for
$(u_{i},o_{j})$ in the training step. The gradient of $\ell_{\text{FND}}$
w.r.t the preference score of $u_{i}$ to $o_{j_{k^{\prime}}}$ is as follows:

(15)
$p(r_{ij_{k^{\prime}}})=\frac{\exp(d_{ij_{k^{\prime}}}r_{ij_{k^{\prime}}}/\tau_{\text{FND}})}{\exp(r_{ij}/\tau_{\text{FND}})+\sum_{k=1}^{N}\exp(d_{ij_{k}}r_{ij_{k}}/\tau_{\text{FND}})},$
(16) $\frac{\partial\ell_{\text{FND}}(i,j)}{\partial
r_{ij_{k^{\prime}}}}=\frac{d_{ij_{k^{\prime}}}}{\tau_{\text{FND}}}p(r_{ij_{k^{\prime}}}).$

The sign of the gradient is the same as $d_{ij_{k^{\prime}}}$ since
$\tau_{\text{FND}}>0$ and $p(r_{ij_{k^{\prime}}})>0$ hold. Hence, negative
outfits closer than the average of positive outfits are pulled toward the user
rather than pushed.

###  3.3. Contrastive Learning for Outfits

To obtain more enhanced outfit representations, we propose a novel approach to
leverage SimCLR (Chen et al., 2020) framework in outfit recommendation.
Specifically, we suggest two data augmentation methods for outfits.

SimCLR learns representations by maximizing the agreement between differently
augmented views of the same outfit while pushing the others in the batch.
Given the batch $\mathcal{B}=\\{(u_{i^{(n)}},o_{j^{(n)}})\\}_{n=1}^{N}$, each
outfit $o_{j^{(n)}}$ is augmented twice to create two different views
$(\overline{o}_{j^{(2n-1)}},\overline{o}_{j^{(2n)}})$, generating $2N$
augmented outfits $\\{\overline{o}_{j^{(n)}}\\}_{n=1}^{2N}$ in total. The
agreement is measured by cosine similarity between each outfit representation:
$s_{n,m}=\cos(g(\overline{\mathbf{o}}_{j^{(n)}}),g(\overline{\mathbf{o}}_{j^{(m)}}))$,
where $g(\cdot)$ is a non-linear projection layer. The objective of
contrastive learning is as follows:

(17)
$\ell_{\text{CL}}(n,m)=-\log\frac{\exp(s_{n,m}/\tau_{\text{CL}})}{\sum_{t=1}^{2N}\mathbbm{1}_{[t\neq
n]}\exp(s_{n,t}/\tau_{\text{CL}})},$ (18)
$\mathcal{L}_{\text{CL}}=\frac{1}{2N}\sum_{n=1}^{N}[\ell_{\text{CL}}(2n-1,2n)+\ell_{\text{CL}}(2n,2n-1)],$

where $\tau_{\text{CL}}>0$ is a temperature hyper-parameter.

To exploit the objective in Eq. 18, we must define appropriate data
augmentation methods which produce semantically similar outfits with an input
outfit. As illustrated in Fig. 2, we suggest two augmentation methods suitable
for outfit recommendation: erase and replace. Both augmentations randomly
alter comprising items from an outfit while preserving the semantic context.
We treat the augmentation set as a hyper-parameter and fix them at the
beginning of the training. Note that if two identical augmentations are
applied, we alter different items from the input outfit to obtain distinct
views.

####  3.3.1. Erase

Randomly erasing components from the input is a common data augmentation
method in diverse domains. In sequential recommendation, for example, Xie et
al. (Xie et al., 2020) randomly crop items from user interaction history. In
natural language processing, Wu et al. (Wu et al., 2020) erase or replace
randomly selected words in a sentence. In computer vision, DeVries et al.
(DeVries and Taylor, 2017) cut out contiguous sections of an input image,
inspired by the object occlusion problem. When it comes to outfit
recommendation, a subset of an outfit may imply or even determine the semantic
information. Motivated by this, we randomly remove one item from the outfit to
generate an augmented view.

####  3.3.2. Replace

Our proposed model computes preference scores based solely on visual
information (i.e., images), and this assumption is helpful if access to other
metadata is limited. In this situation, the semantic information of an outfit
is derived from the appearance of consisting items. Accordingly, we claim that
the visual similarity of consisting items leads to the semantic similarity of
an outfit. Based on the claim, we generate an augmented view by randomly
replacing one item from the outfit with a visually similar item from the same
category. To this end, we train a CNN autoencoder model and retrieve similar
items through their latent features.

###  3.4. Final Objective: FND-CL

The proposed losses $\mathcal{L}_{\text{FND}}$ and $\mathcal{L}_{\text{CL}}$
can be used independently; hence we can take advantage of both methods.
Therefore, our final objective is to minimize the weighted sum of both losses
as follows:

(19) $\mathcal{L}_{\text{FND-
CL}}=\mathcal{L}_{\text{FND}}+\lambda\mathcal{L}_{\text{CL}},$

where $\lambda$ is a loss weight hyper-parameter.

###  3.5. Profiling Cold Starters

In application services with recommender systems, new users can join the
service even after the model is trained and deployed. Such users, or cold
starters, have relatively few interactions in general, and the deployed model
does not have any prior knowledge of those users. In practice, fine-tuning the
model for them is a time-consuming process; thus, cold starters might starve
for the recommendation until the next iteration of deployment. Therefore, it
is necessary to have an alternative recommendation method that exploits the
already deployed model with no additional training. In personalized outfit
recommendation, only a few works (Lu et al., 2021) handled the cold-start
problem without fine-tuning the model. Here, we introduce two strategies to
compute preference scores of cold starters, analogous to memory-based
collaborative filtering.

Given a cold starter $u_{c}$, we define a neighborhood $\mathcal{N}_{c}$ from
the set of non-cold users $\mathcal{U}$ as follows: with a set of indices of
positive outfits $J_{c}^{+}$ for $u_{c}$,

(20) $s_{ci}=\frac{1}{|J_{c}^{+}|}\sum_{j^{+}\in J_{c}^{+}}r_{ij^{+}},$ (21)
$\mathcal{N}_{c}=\\{u_{i}\in\mathcal{U}\,|\,s_{ci}>\delta\vee i=i_{c}^{*}\\},$

where $s_{ci}$ represents the asymmetric similarity from $u_{c}$ to $u_{i}$.
$\delta$ is a similarity threshold, and $i_{c}^{*}=\text{argmax}_{i}s_{ci}$
denotes the index of the most similar user, which ensures at least one
neighbor for each $u_{c}$. To compute the preference score
$r_{cj}^{\text{cold}}$ of the cold starter $u_{c}$ to a given outfit $o_{j}$,
we aggregate the preference scores of neighbors to the outfit. Here we use two
aggregation strategies: Average and Weighted Average.

####  3.5.1. Average (avg)

A basic aggregation strategy is simply averaging the preference scores:

(22)
$r_{cj}^{\text{cold}}:=\frac{1}{|\mathcal{N}_{c}|}\sum_{u_{i}\in\mathcal{N}_{c}}r_{ij}.$

####  3.5.2. Weighted Average (w-avg)

We further utilize the similarity between the cold starter and its neighbors
as aggregation weights using cross-entropy with temperature:

(23)
$\overline{s}_{ci}=\frac{\exp(s_{ci}/\tau_{\text{w-avg}})}{\sum_{u_{i^{\prime}}\in\mathcal{N}_{c}}\exp(s_{ci^{\prime}}/\tau_{\text{w-avg}})},$
(24)
$r_{cj}^{\text{cold}}:=\sum_{u_{i}\in\mathcal{N}_{c}}\overline{s}_{ci}r_{ij},$

where $\tau_{\text{w-avg}}>0$ is a temperature hyper-parameter, and note that
$\sum_{u_{i}\in\mathcal{N}_{c}}\overline{s}_{ci}=1$ holds. We considered other
methods for deriving the aggregation weights $\overline{s}_{ci}$ from
$s_{ci}$; however, the suggested method empirically showed the best and stable
results, especially in terms of robustness to hyper-parameters.

##  4\. Experiment

###  4.1. Experimental Design

####  4.1.1. Datasets

Table 1. Dataset statistics. Dataset | # Outfits | # Items
---|---|---
Polyvore-630 | 162,945 | 199,537
Polyvore-519 | 106,806 | 178,481
Polyvore-53 | 13,707 | 26,727
Polyvore-32 | 6,565 | 18,656

We use datasets collected from the Polyvore website: Polyvore-$U$ (Lu et al.,
2019), where $U\in\\{630,519,53,32\\}$ denotes the number of users.
Polyvore-$U$ contains outfits posted by users, each consisting of three
categories: top, bottom, shoes. Outfits in Polyvore-{630, 53} have a fixed
number of items: one item for each category. Polyvore-{519, 32} include
outfits with a variable number of items (i.e., some outfits may have two
tops). We use Polyvore-{630, 519} for most of the experiments and
Polyvore-{53, 32} for cold starter tasks. Statistics of the datasets are
provided in Table 1. Following previous works (Lu et al., 2019, 2021), we
define user-posted outfits as positive outfits for each user and category-wise
random mixtures of items as negative outfits. We also discuss the results of
hard negative outfits (i.e., random samples of positive outfits of other
users) separately in Sec. 4.4. In the evaluation phase, we set the ratio
between positive and negative outfits to 1:10 for each user. We split
training, validation, and test sets to 9:2:2, and we further split the
validation set into two halves, one for the teacher model and the other for
the student model. As (Lu et al., 2019) affirmed, there are no duplicate items
between the training and the test sets for each user.

####  4.1.2. Evaluation metrics

We evaluate the ranking performance via Area Under the ROC curve (AUC) and
Normalized Discounted Cumulative Gain (NDCG), similar to previous works (Lu et
al., 2019, 2021). For each user, we rank the test outfits by the predicted
preference score of the model. We report the performance averaged over all
users.

####  4.1.3. Considered methods

We compare our methods with the following state-of-the-art non-personalized
(Vasileva et al., 2018; Tan et al., 2019; Han et al., 2017) and personalized
(Lin et al., 2020; Lu et al., 2019, 2021) outfit recommendation models. Type-
Aware (Vasileva et al., 2018) projects pairs of items onto the type-specific
subspaces. Compatibility is then measured in these subspaces and learned
through the triplet loss. SCE-Net (Tan et al., 2019) learns conditional
embeddings and their weights using an attention mechanism. Each conditional
embedding is implicitly encouraged to encode different semantic subspaces via
the triplet loss. Bi-LSTM (Han et al., 2017) considers an outfit as a sequence
of items and uses a bidirectional LSTM to learn the compatibility. The model
is trained by predicting the next and previous items in the sequence through
cross-entropy loss. OutfitNet (Lin et al., 2020) consists of two stages to
capture both general compatibility and personal taste. The objective of both
stages is to maximize the difference between positive and negative scores,
similar to BPR. FHN (Lu et al., 2019) uses pairwise scores to compute outfit
compatibility and personal preference simultaneously. We train FHN with BPR
without the binarization step, following the previous work (Lu et al., 2021).
LPAE (Lu et al., 2021) includes two models LPAE-u and LPAE-g, which mainly
handles the cold-start problem using multiple anchors for each user. Both
models utilize BPR loss, and LPAE-g has additional general anchors to model
non-personalized compatibility. For a more fair comparison, we apply
temperature scaling when using BPR or cross-entropy loss.

####  4.1.4. Implementation details

Similar to the previous work (Lu et al., 2019), we use AlexNet (Krizhevsky et
al., 2012) pretrained on ImageNet (Deng et al., 2009) as a backbone CNN. We
define two versions of AlexNet to experiment the knowledge distillation. One
is AlexNet-large, which is the original AlexNet. The other is AlexNet-small, a
downsized version of AlexNet that all fully-connected layers are removed and
with a global average pooling at the end (Lin et al., 2013). Only the teacher
model uses AlexNet-large, and the others use AlexNet-small. We set the feature
dimension $d=128$ for all methods. For simplicity, we set $\delta$ to zero,
the median of the possible range of cosine similarity. We use
$\tau_{\text{FND}}=\tau_{\text{CL}}=0.1$, $\tau_{\text{w-avg}}=0.2$,
$\lambda=0.2$, and the number of heads $h=8$. We set $\alpha$ to 1.25 for
Polyvore-630 and 1.5 for Polyvore-519. When it comes to CL, the pair of
augmentations are (erase, replace) for Polyvore-630 and (erase, erase) for
Polyvore-519. SGD with momentum (Qian, 1999) is used to train all methods, and
the batch size is set to 32. For each method, we report the test performance
with their optimal hyper-parameters searched via the validation set unless
otherwise specified.

###  4.2. Performance Comparison

Table 2. Performance comparison of different methods on Polyvore datasets. | Polyvore-630 | Polyvore-519
---|---|---
Method | AUC | NDCG | AUC | NDCG
Type-Aware (Vasileva et al., 2018) | 75.87 | 57.06 | 78.03 | 60.33
SCE-Net (Tan et al., 2019) | 77.76 | 59.83 | 78.86 | 61.37
Bi-LSTM (Han et al., 2017) | 78.30 | 62.30 | 80.25 | 65.54
OutfitNet (Lin et al., 2020) | 85.03 | 72.87 | 85.04 | 71.96
FHN (Lu et al., 2019) | 87.15 | 76.33 | 87.89 | 77.07
LPAE-u (Lu et al., 2021) | 87.82 | 77.49 | 89.19 | 79.23
LPAE-g (Lu et al., 2021) | 87.05 | 76.33 | 89.59 | 81.05
FND | 89.91 | 82.18 | 91.86 | 84.74
FND-CL | 90.28 | 82.47 | 92.30 | 85.51

As shown in Table 2, our proposed FND outperforms baseline methods under all
datasets and metrics. Furthermore, the performance of FND-CL shows the
effectiveness of the outfit CL framework.

Recall that LPAE-u adopts multiple anchors for representing each user, and
LPAE-g further leverages non-personalized compatibility. Comparison with LPAE
models shows that FND can effectively achieve improved performance without
auxiliary parameters and structures.

###  4.3. Performance on Cold Starters

Table 3. Comparison of different methods on cold starters. For our FND and FND-CL, w/o and w/ “(w)” represent the avg and w-avg strategies, respectively. We use AUC as an evaluation metric. | Polyvore-53 | Polyvore-32
---|---|---
# Outfits | 1 | 5 | 1 | 5
Type-Aware (Vasileva et al., 2018) | 72.79 | 72.79 | 77.44 | 77.44
SCE-Net (Tan et al., 2019) | 76.55 | 76.55 | 78.97 | 78.97
Bi-LSTM (Han et al., 2017) | 78.61 | 78.61 | 81.69 | 81.69
LPAE-u (Lu et al., 2021) | 77.58 | 77.92 | 80.72 | 80.97
LPAE-g (Lu et al., 2021) | 77.65 | 77.35 | 82.62 | 82.53
FND | 78.68 | 79.42 | 83.84 | 84.10
FND (w) | 79.72 | 80.98 | 84.53 | 85.31
FND-CL | 79.59 | 80.09 | 85.26 | 85.43
FND-CL (w) | 80.33 | 81.56 | 85.51 | 86.41

We evaluate the ranking performance of recommending to the cold starters.
Concretely, we test a scenario where a model is trained in Polyvore-{630,
519}, and cold starters in Polyvore-{53, 32} desire recommendations. Following
the previous work (Lu et al., 2021), we experiment on the circumstances that
each cold starter has only 1 or only 5 interacted outfits. We compare our
methods with non-personalized and LPAE methods, which do not require
additional training of the model. For LPAE methods, we use the anchor-search
(Lu et al., 2021), which is known to be the most effective strategy in the
cold-start case. In FND and FND-CL, we evaluate both avg and w-avg strategies.
We conduct experiments 10 times and report the average results in Table 3. The
results show that our approaches consistently outperform baseline methods even
though the primary purpose of LPAE methods is to deal with the cold-start
problem. The w-avg strategy is more effective than the avg strategy in both
FND and FND-CL, implying the importance of considering neighbors differently
based on similarity rather than treating them equally.

###  4.4. Performance on Hard Negative Outfits

Table 4. Comparison of different methods on hard negative outfits. | Polyvore-630-H | Polyvore-519-H
---|---|---
Method | AUC | NDCG | AUC | NDCG
OutfitNet (Lin et al., 2020) | 79.19 | 60.40 | 79.83 | 60.77
FHN (Lu et al., 2019) | 79.67 | 60.61 | 80.15 | 60.20
LPAE-u (Lu et al., 2021) | 81.69 | 65.26 | 82.17 | 65.79
LPAE-g (Lu et al., 2021) | 80.88 | 63.99 | 80.53 | 62.35
FND | 83.55 | 68.74 | 83.90 | 69.11
FND-CL | 84.24 | 69.99 | 84.76 | 70.50

We test a more challenging case where negative outfits in evaluation are
composed of positive outfits of other users (i.e., hard negatives). We only
compare with personalized methods since non-personalized methods cannot
distinguish users. Following the previous work (Lu et al., 2019), we set half
of the negative outfits to hard negative outfits when training the model. We
report the results in Table 4. The results show that LPAE-g, which
additionally considers non-personalized compatibility, performs poorly in this
task compared to LPAE-u. We can see that our FND and FND-CL outperform the
baseline methods. Note that the effectiveness of the CL framework is more
apparent on hard negative outfits than the results in Table 2. We believe that
the functionality of CL to distinguish outfits enables the model to capture
more meaningful outfit representations, especially in the hard negative
setting.

###  4.5. Performance with Different $\alpha$

Figure 4. Comparison of different $\alpha$ on Polyvore datasets.

We report the AUC of FND according to alpha on Polyvore datasets. In addition,
the performance of N-pair and BPR is also shown in the same figure. For each
dataset, FND outperforms other baselines when appropriate alpha is chosen.

We evaluate the performance of FND under various $\alpha$ (see Eq. 12). The
results are reported in Fig. 4, and we also show the performance of $N$-pair
and BPR in the same figure. As mentioned in Sec. 3.2.1, $N$-pair overcomes the
partial shortcoming of BPR by considering multiple negative outfits in each
update and thus clearly outperforms BPR. Different datasets tend to have
different optimal $\alpha$, but given adequate value, FND can surpass the
performance of a strong $N$-pair.

###  4.6. Performance with Different Augmentations

Figure 5. Comparison of different augmentation methods on Polyvore datasets.
Augmentation pair XY represents that the first and the second augmentations
are X and Y. We also include the identity function as an augmentation method.
I/E/R indicates identity/erase/replace augmentations.

We report the AUC of FND-CL according to augmentations on Polyvore datasets.
In addition, the performance of FND is also shown in the same figure.

To test the performance of FND-CL for all possible augmentation methods, we
put the identity function into the set of augmentations. The results are shown
in Fig. 5, and we also report the performance of FND in the same figure. Note
that we did not experiment on the (identity, identity) augmentation pair
because two augmented views should be different. Regardless of which
augmentation pair we use, FND-CL outperforms FND in almost all cases, and the
optimal augmentation pair is different for each dataset. Concretely, replace
and erase augmentation methods tend to be more effective at Polyvore-630 and
Polyvore-519, respectively. Therefore, we can see that both erase and replace
are meaningful augmentation methods and that the CL framework is effective in
outfit recommendation.

###  4.7. Performance with Different Model Sizes

Figure 6. Comparison of different model sizes on Polyvore datasets.

We report the AUC of N-pair, FND, and FND-CL according to model sizes on
Polyvore datasets. The performance of the teacher model is also shown in the
same figure. We also report the number of parameters for each model size in
the same figure.

We study the impact of model size on performance, especially in the case of
the student model. We consider student models with three different sizes
(i.e., XS, S, M) and the teacher model. XS uses AlexNet-small, and S and M use
a downsized version of AlexNet that output dimensions of all fully-connected
layers are reduced to 1/4 and 1/2, respectively. Fig. 6 shows the results. It
is clear that the larger the size, the better the performance. Moreover, the
fact that FND-CL outperforms FND and FND outperforms $N$-pair is consistent
regardless of the size of the model, supporting that our approaches are
meaningful.

The performance gap between FND and $N$-pair appears to shrink with the
increasing size of the student model. Such a tendency implies that the
effectiveness of FND depends on the performance gap between the teacher and
the student model, which implicitly emphasizes the importance of utilizing the
superior teacher model. On the other hand, we can see that the effectiveness
of the CL framework is hardly affected by the size of the model, as expected.

Detailed information on inference efficiency is measured for each model of
different sizes and reported in Table 5. We conduct experiments using FND for
the student model and $N$-pair for the teacher model. Since FND affects only
the training step, FND and $N$-pair share the same inference efficiency. Note
that if CL is added, only the number of parameters increases by about 0.03M.
In all inference tests, we use PyTorch with CUDA from Tesla P100 SXM2 GPU and
Xeon E5-2690 v4 CPU. From the results, we can see that as the size of the
model increases, all metrics that indicate inefficiency (i.e., Time, Memory, #
Params) also increase.

###  4.8. Performance with Different Batch Sizes

Table 5. Model compactness and inference efficiency. “Time” denotes model inference time for making recommendations to every user in each dataset, and we report the mean and standard deviation of 10 runs. “Memory” represents GPU memory usage. “Ratio” indicates the relative parameter size of the student model compared to the teacher model. Dataset | Model | Time | Memory | # Params | Ratio
---|---|---|---|---|---
Polyvore -630 | Teacher | 80.5s$\pm$1.7 | 2.89GB | 65.99M | -
XS | 74.1s$\pm$1.7 | 2.64GB | 2.85M | 4.3%
S | 75.2s$\pm$2.9 | 2.68GB | 13.88M | 21.0%
M | 76.3s$\pm$3.3 | 2.75GB | 28.10M | 42.6%
Polyvore -519 | Teacher | 65.4s$\pm$2.7 | 3.05GB | 65.97M | -
XS | 58.3s$\pm$3.0 | 2.80GB | 2.83M | 4.3%
S | 61.6s$\pm$2.7 | 2.84GB | 13.86M | 21.0%
M | 64.4s$\pm$1.9 | 2.91GB | 28.09M | 42.6%

Figure 7. Comparison of different batch sizes on Polyvore datasets.

We report the AUC of FND-CL according to batch sizes on Polyvore datasets.

To study the impact of the batch size, we test the performance of FND-CL with
different batch sizes. Note that we use a linear scaling of the learning rate
when training with different batch sizes (Krizhevsky, 2014; Goyal et al.,
2017). Fig. 7 shows the results of the experiment. The performance tends to be
improved as the batch size increases and appears to converge when the batch
size exceeds a certain threshold. We believe that the number of negative
samples (i.e., outfits and augmented views) proportional to the batch size is
the primary factor of this tendency. Note that except for the number of
negative samples, we do not use any other factors significantly influenced by
the batch size, such as batch normalization (Ioffe and Szegedy, 2015).
Therefore, we can see from the results in Fig. 7 as well as Fig. 4 that it is
important to exploit a sufficient number of negative samples per each update
when using the ranking loss.

###  4.9. Visualization of the User-Outfit Space

We visualize the user-outfit space of the teacher model to support the
intuition of FND (see Fig. 1). The visualization uses $t$-SNE (Van der Maaten
and Hinton, 2008) and shows three users and their positive and negative
outfits from the training set. We focus on the training phase since the
approach of FND is to distill knowledge from the teacher model when training
the student model. The results are shown in Fig. 8. Recall that negative
outfits are randomly generated, and thus a positive outfit from the test set
can appear as a negative sample by pure chance in the training step. With the
help of the teacher model, the student model can treat such samples as false-
negatives, denoted as a dash-bordered rectangle in the figure. Moreover, other
negative outfits close to a user share a similar style with positive outfits,
showing the possibility of being false-negatives. Hence, we can conclude that
the approach of FND that utilizes false-negative information from the teacher
model is reasonable.

Figure 8. The $t$-SNE visualization result of the user-outfit space. The “X”
symbol denotes a user embedding vector distinguished by different colors. Each
rectangle is an outfit representation vector. A rectangle with a black border
indicates a negative outfit. For rectangles with a colored border, they
represent the positive outfits of the user corresponding to each color. Among
colored rectangles, a dashed border shows a false-negative outfit found in the
test set.

We use t-SNE to visualize the user-outfit space of the teacher model on a 2D
map. We show users, positive outfits, negative outfits, and false-negative
outfits in the figure.

##  5\. Conclusion

In this paper, we study how to leverage knowledge distillation (KD) and
contrastive learning (CL) framework for personalized outfit recommendation. We
propose a new KD framework named False Negative Distillation (FND) that does
not require the ranking of all possible outfits. We also propose two novel
data augmentation methods to make use of the CL framework in outfit
recommendation. Quantitative experiments show that our FND and CL achieve
notable success in outfit recommendation tasks. In detail, FND outperforms the
state-of-the-art methods under fair conditions and achieves improved
performance than without using FND in the same model. The outfit CL framework
also contributes to the recommendation performance by allowing the model to
obtain a more meaningful outfit representation. We support the soundness of
our FND by visualizing the user-outfit space of the teacher model. One
interesting future work is to apply a contrastive learning framework in a
supervised manner by treating each user as a class.

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# On the Distributions of Product and Quotient of two Independent
$\hat{I}$-function variates

Vilma D’Souza1, Shantha Kumari Kurumujji${}^{1^{*}}$, Arjun K. Rathie2 1
Department of Mathematics, A J Institute of Engineering and Technology,
Mangaluru-575006(Affiliated to Visvesvaraya Technological University (VTU),
Belagavi), Karnataka, INDIA<EMAIL_ADDRESS><EMAIL_ADDRESS>2
Department of Mathematics, Vedant College of Engineering & Technology
(Rajasthan Technical University), Village: Tulsi, Post : Jakhamund, Dist.
Bundi, Rajasthan State, India<EMAIL_ADDRESS>
###### Abstract.

The study of probability distributions for random variables and their
algebraic combinations has been a central focus driving the advancement of
probability and statistics. Since the 1920s, the challenge of calculating the
probability distributions of sums, differences, products, and quotients of
independent random variables have drawn the attention of numerous
statisticians and mathematicians who studied the algebraic properties and
relationships of random variables. Statistical distributions are highly
helpful in data science and machine learning, as they provide a range of
possible values for the variables, aiding in the development of a deeper
understanding of the underlying problem. In this paper, we have presented a
new probability distribution based on the $\hat{I}$-function. Also, we have
discussed the applications of the $\hat{I}$ function, particularly in deriving
the distributions of product and the quotient involving two independent
$\hat{I}$ function variates. Additionally, it has been shown that both the
product and quotient of two independent $\hat{I}$-function variates also
follow the $\hat{I}$-function distribution. Furthermore, the new distribution,
known as the $\hat{I}$-function distribution, includes several well-known
classical distributions such as the gamma, beta, exponential, normal
H-function, and G-function distributions, among others, as special cases.
Therefore, the $\hat{I}$-function distribution can be considered a
characterization or generalization of the above-mentioned distributions.

###### Key words and phrases:

I function, Distributions, Mellin Transform, Random variable

###### 1991 Mathematics Subject Classification:

33C60, 44A20, 60E05

$*$ Corresponding Author

##  1\. Introduction

The study of special functions and their distributions is a fundamental area
of mathematical analysis, with numerous applications in statistics, physics,
and engineering. Among these special functions, the H-function has been
extensively researched for its rich algebraic and analytical properties. The
H-function is interconnected with several other important mathematical
functions. It is related to the generalized hypergeometric function
${}_{p}F_{q}$, the Meijer G-function, the generalized Wright function
${}_{p}\psi_{q}$, and the Meiger’s G-function[11]. These relationships are
thoroughly documented in the works of Mathai et al. [12, 13], providing a
comprehensive framework for understanding the H-function and its various
applications. Subsequent research into the H-function distribution was carried
out by Cook [5], who examined its algebraic properties, and later by
Bodenschatz [4], who further explored its statistical characteristics. Their
studies provided a more comprehensive understanding of the mathematical
structure and potential applications of the H-function.

Inayat-Hussain [7] introduced a significant generalization of the H-function,
known as the $\bar{H}$-function. Further generalization beyond the
$\bar{H}$-function has been provided by Rathie[15], who extended the framework
to include a broader class of special functions, including the
$\bar{H}$-function, generalized Riemann zeta function, the polylogarithmic
function of complex order, and the exact partition function of the Gaussian
free energy model in statistical mechanics, which are not special cases of the
H-function. The extension of the I-function to two variables and its
properties were studied by shantha et al.[10].

Recent advancements have focused on novel methods for generating Fox’s
H-function distributed random variables, essential for modeling fading in
wireless communication. These methods employ standard normal and Gamma random
variables, validated via Monte Carlo simulations to closely approximate
analytical H-function probability density functions (PDFs) across various
parameters[16]. This is a topic of considerable interest due to its
applications in various domains, including fading channels within wireless
communication fields. The applications of I-function to wireless communication
were also studied by Ansari et al.[1, 2]. Motivated by this, our paper focuses
on obtaining the distributions of the product and quotient of two independent
random variables having densities in terms of the $\hat{I}$ \- functions
(which is a particular case of the Rathie’s ${I}$ \- function). For another
form of distribution, readers are suggested to refer to [8].

In 1997, Rathie introduced the I-function[15] as a generalization of the
H-function defined by Fox[6] and studied by Braaksma[3]. It is represented by
the following Mellin-Barnes type contour integral:

$\displaystyle\qquad\qquad\mathrm{I_{p,q}^{m,n}}(z)$
$\displaystyle\equiv\mathrm{I_{p,q}^{m,n}}\left[\begin{array}[]{c}z\end{array}\left|\begin{array}[]{l}\left(a_{1},e_{1},A_{1}),\ldots,(a_{p},e_{p},A_{p}\right)\\\
\left(b_{1},f_{1},B_{1}),\ldots,(b_{q},f_{q},B_{q}\right)\end{array}\right.\right]$
(4) $\displaystyle=\frac{1}{2\pi i}\int_{\mathcal{L}}\theta(s)z^{s}ds$ (5)

where

$\qquad\qquad\theta(s)=\frac{\prod_{j=1}^{m}\Gamma^{B_{j}}\left(b_{j}-f_{j}s\right)\prod_{j=1}^{n}\Gamma^{A_{j}}\left(1-a_{j}+e_{j}s\right)}{\prod_{j=m+1}^{q}\Gamma^{B_{j}}\left(1-b_{j}+f_{j}s\right)\prod_{j=n+1}^{p}\Gamma^{A_{j}}\left(a_{j}-e_{j}s\right)}$
(6)

Also

  1. (i)

$i$ is the imaginary unit, $i=\sqrt{-1}$;

  2. (ii)

$z$ is a non zero complex variable, signifying the argument of the I-function;
;

  3. (iii)

$m,n,p,q$ are integers satisfying the constraints, $0\leq m\leq q$, $0\leq
n\leq p$;

  4. (iv)

The contour $\mathcal{L}$ is a carefully chosen path in the complex plane for
the integration;

  5. (v)

an empty product is to be interpreted as unity;

  6. (vi)

The parameters $e_{j}$ and $f_{j}$, with $j$ ranging from $1$ to $p$ and $q$
respectively, represent positive numbers;

  7. (vii)

The parameters $A_{j}$ and $B_{j}$, with $j$ ranging from $1$ to $p$ and $q$
respectively, also denote positive numbers;

  8. (viii)

The parameters $a_{j}$ and $b_{j}$, with $j$ ranging from $1$ to $p$ and $q$,
respectively, are complex numbers, subject to the condition that no
singularity of $\Gamma^{B_{j}}\left(b_{j}-f_{j}s\right)$, for $j=1,\ldots,m$,
coincides with any singularity of $\Gamma^{A_{j}}\left(1-a_{j}+e_{j}s\right)$,
for $j=1,\ldots,n$. It is important to note that these singularities are not
necessarily poles.

  9. (ix)

Additionally, the contour $\mathcal{L}$ is described to go from
$\sigma-i\infty$ to $\sigma+i\infty$ ($\sigma$ real), with the requirement
that the singularities of $\Gamma^{B_{j}}\left(b_{j}-f_{j}s\right)$ for
$j=1,\ldots,m$ lie to the right of $\mathcal{L}$, and the singularities of
$\Gamma^{A_{j}}\left(1-a_{j}+e_{j}s\right)$ for $j=1,\ldots,n$ lie to the left
of $\mathcal{L}$.

In a compact notation, (4) is represented as,

$\mathrm{I_{\>p,\;q}^{\>m,\;n}}\left[\begin{array}[]{c}z\end{array}\left|\begin{array}[]{l}{}_{1}(a_{j},e_{j},A_{j})_{p}\\\
{}_{1}(b_{j},f_{j},B_{j})_{q}\end{array}\right.\right]$

The function defined by (4) is convergent if

$\Delta>0,\quad|arg(z)|<\frac{1}{2}\Delta\pi,$ (7)

where

$\Delta=\sum_{j=1}^{m}B_{j}f_{j}-\sum_{j=m+1}^{q}B_{j}f_{j}+\sum_{j=1}^{n}A_{j}e_{j}-\sum_{j=n+1}^{p}A_{j}e_{j}.$
(8)

When $A_{1}=A_{2}=\cdots=A_{p}=1=B_{1}=B_{2}=\cdots=B_{q}$, (4) reduces to the
H-function introduced by Fox[6] and studied by Braaksma[3].

When the exponents $A_{j}=1$ for $j=1,2,\cdots,n$ and $B_{j}=1$ for
$j=1,2,\cdots,m$, the $I$-function defined by (4) reduces to
$\hat{I}$-function, which is represented by the following Mellin Barnes type
contour integral:

$\displaystyle\hat{I}(z)$
$\displaystyle=\hat{I}_{\;p,\;q}^{\;m,\;n}\left[z\left|\begin{array}[]{c}_{1}(a_{j},e_{j},1)_{n},\;_{n+1}(a_{j},e_{j},A_{j})_{p}\\\
_{1}(b_{j},f_{j},1)_{m},\;_{m+1}(b_{j},f_{j},B_{j})_{q}\end{array}\right]\right.$
(11) $\displaystyle=\frac{1}{2\pi i}\int_{\mathcal{L}}\theta(s)z^{s}ds$

where

$\theta(s)=\frac{\prod_{j=1}^{m}\Gamma\left(b_{j}-f_{j}s\right)\prod_{j=1}^{n}\Gamma\left(1-a_{j}+e_{j}s\right)}{\prod_{j=m+1}^{q}\Gamma^{B_{j}}\left(1-b_{j}+f_{j}s\right)\prod_{j=n+1}^{p}\Gamma^{A_{j}}\left(a_{j}-e_{j}s\right)}$
(13)

and the function defined by (13) is convergent if

$\Delta_{1}>0,\quad|arg(z)|<\frac{1}{2}\Delta_{1}\pi$ (14)

where

$\Delta_{1}=\sum_{j=1}^{m}f_{j}-\sum_{j=m+1}^{q}B_{j}f_{j}+\sum_{j=1}^{n}e_{j}-\sum_{j=n+1}^{p}A_{j}e_{j}.$
(15)

Also, from Rathie [15], it can be proven that
$\hat{I}(z)\sim z^{c}$ for small $z$, where $\;c=\displaystyle\min_{1\leq
j\leq m}\left(Re\left[\frac{b_{j}}{f_{j}}\right]\right).$
and
$\hat{I}(z)\sim z^{d}$ for large $z$, where $\;d=\displaystyle\max_{1\leq
j\leq n}\left(Re\left[\frac{a_{j}-1}{e_{j}}\right]\right).$

By the definition of $\hat{I}$-function, the following properties are
immediate.

$\displaystyle\hat{I}_{\;p,\;q}^{\;m,\;n}$
$\displaystyle\left[z^{\sigma}\left|\begin{array}[]{c}_{1}(a_{j},e_{j},1)_{n},\;_{n+1}(a_{j},e_{j},A_{j})_{p}\\\
_{1}(b_{j},f_{j},1)_{m},\;_{m+1}(b_{j},f_{j},B_{j})_{q}\end{array}\right]\right.$
(18)
$\displaystyle=\frac{1}{\sigma}\hat{I}_{\;p,\;q}^{\;m,\;n}\left[z\left|\begin{array}[]{c}_{1}(a_{j},\frac{e_{j}}{\sigma},1)_{n},\;_{n+1}(a_{j},\frac{e_{j}}{\sigma},A_{j})_{p}\\\
_{1}(b_{j},\frac{f_{j}}{\sigma},1)_{m},\;_{m+1}(b_{j},\frac{f_{j}}{\sigma},B_{j})_{q}\end{array}\right]\right.,~{}~{}~{}\sigma>0$
(21) $\displaystyle z^{\sigma}\;\hat{I}_{\;p,\;q}^{\;m,\;n}$
$\displaystyle\left[z\left|\begin{array}[]{c}_{1}(a_{j},e_{j},1)_{n},\;_{n+1}(a_{j},e_{j},A_{j})_{p}\\\
_{1}(b_{j},f_{j},1)_{m},\;_{m+1}(b_{j},f_{j},B_{j})_{q}\end{array}\right]\right.$
(24)
$\displaystyle=\hat{I}_{\;p,\;q}^{\;m,\;n}\left[z\left|\begin{array}[]{c}_{1}(a_{j}+\sigma
e_{j},e_{j},1)_{n},\;_{n+1}(a_{j}+\sigma e_{j},e_{j},A_{j})_{p}\\\
_{1}(b_{j}+\sigma f_{j},f_{j},1)_{m},\;_{m+1}(b_{j}+\sigma
f_{j},f_{j},B_{j})_{q}\end{array}\right]\right.$ (27)

In addition to this, the Mellin transform of a continuous positive random
variable $X$ with $\operatorname{pdf}f_{X}(x)$ is defined by

$\displaystyle\mathcal{M}_{f_{X}}(s)=E\left[X^{s-1}\right]=\int_{0}^{\infty}x^{s-1}f_{X}(x)dx$
(28)

There exists an inversion integral

$\displaystyle f_{X}(x)=\frac{1}{2\pi
i}\int_{\mathcal{L}}x^{-s}\mathcal{M}_{f_{X}}(s)ds.$ (29)

where $\mathcal{L}$ is a suitable contour.

##  2\. Mellin Transform of $\hat{I}$-function

In this section, we shall establish the Mellin transform of
$\hat{I}$-function.
By the definition of $\hat{I}$-function (11), we have

$\displaystyle\hat{I}(z\;x^{\sigma})$
$\displaystyle=\hat{I}_{\;p,\;q}^{\;m,\;n}\left[z\;x^{\sigma}\left|\begin{array}[]{c}_{1}(a_{j},e_{j},1)_{n},\;_{n+1}(a_{j},e_{j},A_{j})_{p}\\\
_{1}(b_{j},f_{j},1)_{m},\;_{m+1}(b_{j},f_{j},B_{j})_{q}\end{array}\right]\right.$
(32) $\displaystyle=\frac{1}{2\pi
i}\int_{\mathcal{L}}\theta(s)(z\;x^{\sigma})^{-s}ds$ (33)

Replacing $\sigma s$ by $t$ and $ds$ by $\frac{dt}{\sigma}$, we get

$\displaystyle\hat{I}(z\;x^{\sigma})=\frac{1}{2\pi
i}\int_{\mathcal{L}}x^{-t}\;\theta\left(\frac{t}{\sigma}\right)\;z^{\frac{-t}{\sigma}}\;\frac{dt}{\sigma}$
(34)

By the definition of Inverse Mellin Transform (29), we have

$\displaystyle\mathcal{M}\\{\hat{I}(z\;x^{\sigma})\\}=\frac{\theta(\frac{s}{\sigma})}{\sigma\;z^{\frac{s}{\sigma}}}$
(35)

In other way, the result (35) can be written in the following form:

$\displaystyle\int_{0}^{\infty}x^{s-1}\;$
$\displaystyle\hat{I}_{\;p,\;q}^{\;m,\;n}\left[z\;x^{\sigma}\left|\begin{array}[]{c}_{1}(a_{j},e_{j},1)_{n},\;_{n+1}(a_{j},e_{j},A_{j})_{p}\\\
_{1}(b_{j},f_{j},1)_{m},\;_{m+1}(b_{j},f_{j},B_{j})_{q}\end{array}\right]\right.dx$
(38)
$\displaystyle=\frac{1}{\sigma\;z^{\frac{s}{\sigma}}}\;\frac{\prod_{j=1}^{m}\Gamma\left(b_{j}-f_{j}\frac{s}{\sigma}\right)\prod_{j=1}^{n}\Gamma\left(1-a_{j}+e_{j}\frac{s}{\sigma}\right)}{\prod_{j=m+1}^{q}\Gamma^{B_{j}}\left(1-b_{j}+f_{j}\frac{s}{\sigma}\right)\prod_{j=n+1}^{p}\Gamma^{A_{j}}\left(a_{j}-e_{j}\frac{s}{\sigma}\right)}$
(39)

where

$\displaystyle Re\left[s+\sigma\min_{1\leq j\leq
m}\left(\frac{b_{j}}{f_{j}}\right)\right]>0,$ $\displaystyle
Re\left[s+\sigma\min_{1\leq j\leq
n}\left(\frac{a_{j}-1}{e_{j}}\right)\right]>0,$
$\displaystyle\Delta_{1}>0,~{}~{}~{}~{}\;|argz|<\frac{\Delta_{1}\pi}{2},$

where $\Delta_{1}$ is the same as defined by (15).

##  3\. Mellin Transform of the product of two $\hat{I}$-functions

In this section, we shall establish the Mellin transform of the product of two
$\hat{I}$-functions.

For this, let

$\displaystyle\hat{I}(z_{1}x^{\sigma})$
$\displaystyle=\hat{I}_{\;p_{1},\;q_{1}}^{\;m_{1},\;n_{1}}\left[z_{1}x^{\sigma}\left|\begin{array}[]{c}_{1}(a_{j},e_{j},1)_{n_{1}},\;_{n_{1}+1}(a_{j},e_{j},A_{j})_{p_{1}}\\\
_{1}(b_{j},f_{j},1)_{m_{1}},\;_{m_{1}+1}(b_{j},f_{j},B_{j})_{q_{1}}\end{array}\right]\right.$
(42) $\displaystyle=\frac{1}{2\pi
i}\int_{\mathcal{L}}\theta_{1}(s_{1})z_{1}^{-s_{1}}ds_{1}$ (43)

and

$\displaystyle\hat{I}(z_{2}x^{\mu})$
$\displaystyle=\hat{I}_{\;p_{2},\;q_{2}}^{\;m_{2},\;n_{2}}\left[z_{2}x^{\mu}\left|\begin{array}[]{c}_{1}(a_{j}^{\prime},e_{j}^{\prime},1)_{n_{2}},\;_{n_{2}+1}(a_{j}^{\prime},e_{j}^{\prime},A_{j}^{\prime})_{p_{2}}\\\
_{1}(b_{j}^{\prime},f_{j}^{\prime},1)_{m_{2}},\;_{m_{1}+1}(b_{j}^{\prime},f_{j}^{\prime},B_{j}^{\prime})_{q_{2}}\end{array}\right]\right.$
(46) $\displaystyle=\frac{1}{2\pi
i}\int_{\mathcal{L}}\theta_{2}(s_{2})z_{2}^{-s_{2}}ds_{2}$ (47)

where

$\theta_{1}(s_{1})=\frac{\prod_{j=1}^{n_{1}}\Gamma\left(1-a_{j}-e_{j}s_{1}\right)\prod_{j=1}^{m_{1}}\Gamma\left(b_{j}+f_{j}s_{1}\right)}{\prod_{j=n_{1}+1}^{p_{1}}\Gamma^{A_{j}}\left(a_{j}+e_{j}s_{1}\right)\prod_{j=m_{1}+1}^{q_{1}}\Gamma^{B_{j}}\left(1-b_{j}-f_{j}s_{1}\right)}$
(48)

and

$\displaystyle\theta_{2}(s_{2})=\frac{\prod_{j=1}^{n_{2}}\Gamma\left(1-a_{j}^{\prime}-e_{j}^{\prime}s_{2}\right)\prod_{j=1}^{m_{2}}\Gamma\left(b_{j}^{\prime}+f_{j}^{\prime}s_{2}\right)}{\prod_{j=n_{2}+1}^{p_{2}}\Gamma^{A_{j}^{\prime}}\left(a_{j}^{\prime}+e_{j}^{\prime}s_{2}\right)\prod_{j=m_{2}+1}^{q_{2}}\Gamma^{B_{j}^{\prime}}\left(1-b_{j}^{\prime}-f_{j}^{\prime}s_{2}\right)}$
(49)

Then by the definition of Mellin transform,

$\displaystyle M_{\hat{I}[z_{1}x^{\sigma}]\;\hat{I}[z_{2}x^{\mu}]}(s)$
$\displaystyle=\int_{0}^{\infty}x^{s-1}\hat{I}[z_{1}x^{\sigma}]\;\hat{I}[z_{2}x^{\mu}]\;dx$
$\displaystyle=\int_{0}^{\infty}\left\\{{x^{s-1}}{\hat{I}_{\>p_{1},\;q_{1}}^{\>m_{1},\;n_{1}}}\left[\begin{array}[]{c}z_{1}\;x^{\sigma}\end{array}\left|\begin{array}[]{c}_{1}(a_{j},e_{j},1)_{n_{1}},\;_{n_{1}+1}(a_{j},e_{j},A_{j})_{p_{1}}\\\
_{1}(b_{j},f_{j},1)_{m_{1}},\;_{m_{1}+1}(b_{j},f_{j},B_{j})_{q_{1}}\end{array}\right.\right]\right.$
(53)
$\displaystyle\qquad\left.\times{\hat{I}_{\>p_{2},\;q_{2}}^{\>m_{2},\;n_{2}}}\left[\begin{array}[]{c}z_{2}\;x^{\mu}\end{array}\left|\begin{array}[]{c}_{1}(a_{j}^{\prime},e_{j}^{\prime},1)_{n_{2}},\;_{n_{2}+1}(a_{j}^{\prime},e_{j}^{\prime},A_{j}^{\prime})_{p_{2}}\\\
_{1}(b_{j}^{\prime},f_{j}^{\prime},1)_{m_{2}},\;_{m_{2}+1}(b_{j}^{\prime},f_{j}^{\prime},B_{j}^{\prime})_{q_{2}}\end{array}\right.\right]\right\\}dx$
(57)

Expressing one of the $\hat{I}$-functions by its definition, we have

$\displaystyle
M_{\hat{I}[z_{1}x^{\sigma}]\;\hat{I}[z_{2}x^{\mu}]}(s)=\int_{0}^{\infty}x^{s-1}\hat{I}[z_{1}x^{\sigma}]\;\frac{1}{2\pi
i}\int_{\mathcal{L}}\theta_{2}(s_{2})\;(z_{2}x^{\mu})^{-s_{2}}\ ds_{2}\;dx$

By interchanging the order of integration, we have

$\displaystyle M_{\hat{I}[z_{1}x^{\sigma}]\;\hat{I}[z_{2}x^{\mu}]}(s)$
$\displaystyle=\int_{\mathcal{L}}\theta_{2}(s_{2})\;z_{2}^{-s_{2}}\int_{0}^{\infty}x^{s-\mu
s_{2}-1}\hat{I}[z_{1}x^{\sigma}]\;dx\;ds_{2}$

Using (28),

$\displaystyle
M_{\hat{I}[z_{1}x^{\sigma}]\;\hat{I}[z_{2}x^{\mu}]}(s)=\int_{\mathcal{L}}\theta_{2}(s_{2})\;z_{2}^{-s_{2}}\;M_{\hat{I}[z_{1}x^{\sigma}]}(s-\mu
s_{2})\;ds_{2}$

Using (35),

$\displaystyle
M_{\hat{I}[z_{1}x^{\sigma}]\;\hat{I}[z_{2}x^{\mu}]}(s)=\int_{\mathcal{L}}\theta_{2}(s_{2})\;z_{2}^{-s_{2}}\;\frac{1}{\sigma
z_{1}^{\frac{s}{\sigma}-\frac{\mu
s_{2}}{\sigma}}}\;\theta_{1}\left(\frac{s-\mu s_{2}}{\sigma}\right)\;ds_{2}$

Using (11),we have finally

$\displaystyle M_{\hat{I}[z_{1}x^{\sigma}]\;\hat{I}[z_{2}x^{\mu}]}(s)$
$\displaystyle=\frac{1}{\sigma{z_{1}}^{\frac{s}{\sigma}}}\;{I_{\>q_{1}+p_{2},\;p_{1}+q_{2}}^{\>n_{1}+m_{2},\;m_{1}+n_{2}}}\left[\begin{array}[]{c}\frac{z_{2}}{z_{1}^{\frac{-\mu}{\sigma}}}\end{array}\left|\begin{array}[]{c}_{1}(a_{j}^{\prime},e_{j}^{\prime},1)_{n_{2}},~{}~{}{}_{1}(1-b_{j}-\frac{s}{\sigma}f_{j},\frac{\mu}{\sigma}f_{j},1)_{m_{1}},\\\
_{1}(b_{j}^{\prime},f_{j}^{\prime},1)_{m_{2}},\;_{1}(1-a_{j}-\frac{s}{\sigma}e_{j},\frac{\mu}{\sigma}e_{j},1)_{n_{1}}\end{array}\right.\right.$
(61)
$\displaystyle\qquad\qquad\qquad\left.\left.\begin{array}[]{c}_{m_{1}+1}(1-b_{j}-\frac{s}{\sigma}f_{j},\frac{\mu}{\sigma}f_{j},B_{j})_{q_{1}},\;_{n_{2}+1}(a_{j}^{\prime},e_{j}^{\prime},A_{j}^{\prime})_{p_{2}}\\\
_{n_{1}+1}(1-a_{j}-\frac{s}{\sigma}e_{j},\frac{\mu}{\sigma}e_{j},A_{j})_{p_{1}},\;_{{m_{2}}+1}(b_{j}^{\prime},f_{j}^{\prime},B_{j}^{\prime})_{q_{2}}\end{array}\right.\right]$
(64)

In either way, the result (3) can be written in the following form:

$\displaystyle\int_{0}^{\infty}$ $\displaystyle
x^{s-1}\;\hat{I}_{\;p_{1},\;q_{1}}^{\;m_{1},\;n_{1}}\left[z_{1}x^{\sigma}\left|\begin{array}[]{c}_{1}(a_{j},e_{j},1)_{n_{1}},\;_{n_{1}+1}(a_{j},e_{j},A_{j})_{p_{1}}\\\
_{1}(b_{j},f_{j},1)_{m_{1}},\;_{m_{1}+1}(b_{j},f_{j},B_{j})_{q_{1}}\end{array}\right]\right.$
(67)
$\displaystyle\times\hat{I}_{\;p_{2},\;q_{2}}^{\;m_{2},\;n_{2}}\left[z_{2}x^{\mu}\left|\begin{array}[]{c}_{1}(a_{j}^{\prime},e_{j}^{\prime},1)_{n_{2}},\;_{n_{2}+1}(a_{j}^{\prime},e_{j}^{\prime},A_{j}^{\prime})_{p_{2}}\\\
_{1}(b_{j}^{\prime},f_{j}^{\prime},1)_{m_{2}},\;_{m_{1}+1}(b_{j}^{\prime},f_{j}^{\prime},B_{j}^{\prime})_{q_{2}}\end{array}\right]\right.\;dx$
(70)
$\displaystyle=\frac{1}{\sigma{z_{1}}^{\frac{s}{\sigma}}}\;{I_{\>q_{1}+p_{2},\;p_{1}+q_{2}}^{\>n_{1}+m_{2},\;m_{1}+n_{2}}}\left[\begin{array}[]{c}z_{2}z_{1}^{\frac{\mu}{\sigma}}\end{array}\left|\begin{array}[]{c}_{1}(a_{j}^{\prime},e_{j}^{\prime},1)_{n_{2}},\;_{1}(1-b_{j}-\frac{s}{\sigma}f_{j},\frac{\mu}{\sigma}f_{j},1)_{m_{1}},\\\
_{1}(b_{j}^{\prime},f_{j}^{\prime},1)_{m_{2}},\;_{1}(1-a_{j}-\frac{s}{\sigma}e_{j},\frac{\mu}{\sigma}e_{j},1)_{n_{1}}\end{array}\right.\right.$
(74)
$\displaystyle\qquad\qquad\qquad\left.\left.\begin{array}[]{c}_{m_{1}+1}(1-b_{j}-\frac{s}{\sigma}f_{j},\frac{\mu}{\sigma}f_{j},B_{j})_{q_{1}},\;_{n_{2}+1}(a_{j}^{\prime},e_{j}^{\prime},A_{j}^{\prime})_{p_{2}}\\\
_{n_{1}+1}(1-a_{j}-\frac{s}{\sigma}e_{j},\frac{\mu}{\sigma}e_{j},A_{j})_{p_{1}},\;_{{m_{2}}+1}(b_{j}^{\prime},f_{j}^{\prime},B_{j}^{\prime})_{q_{2}}\end{array}\right.\right]$
(77)

where

$\displaystyle Re\left[s+\sigma\min_{1\leq j\leq
m_{1}}\left(\frac{b_{j}}{f_{j}}\right)+\mu\min_{1\leq j\leq
m_{2}}\left(\frac{b_{j}^{\prime}}{f_{j}^{\prime}}\right)\right]>0,$
$\displaystyle Re\left[s+\sigma\min_{1\leq j\leq
n_{1}}\left(\frac{a_{j}-1}{e_{j}}\right)+\mu\min_{1\leq j\leq
n_{2}}\left(\frac{a_{j}^{\prime}-1}{e_{j}^{\prime}}\right)\right]>0,$
$\displaystyle\Delta_{11}>0,\;|argz_{1}|<\frac{\Delta_{11}\pi}{2},\;\Delta_{12}>0,\;|argz_{2}|<\frac{\Delta_{12}\pi}{2},$

where $\Delta_{11}$ and $\Delta_{12}$ are defined by

$\Delta_{11}=\sum_{j=1}^{m_{1}}f_{j}-\sum_{j=m_{1}+1}^{q_{1}}B_{j}f_{j}+\sum_{j=1}^{n_{1}}e_{j}-\sum_{j=n_{1}+1}^{p_{1}}A_{j}e_{j}$

and

$\Delta_{12}=\sum_{j=1}^{m_{2}}f_{j}^{\prime}-\sum_{j=m_{2}+1}^{q_{2}}B_{j}^{\prime}f_{j}^{\prime}+\sum_{j=1}^{n_{2}}e_{j}^{\prime}-\sum_{j=n_{2}+1}^{p_{2}}A_{j}^{\prime}e_{j}^{\prime}$

##  4\. Distributions of product and quotient of two independent
$\hat{I}$-function variates

In this section, we shall obtain the distribution of product and quotient of
two independent $\hat{I}$-function variates asserted in the following theorem.

######  Theorem 4.1.

If $X_{1}$ and $X_{2}$ be the independent $\hat{I}$-function variates with
p.d.f. given respectively by

$\displaystyle
f_{1}(X_{1})=\sigma\;z_{1}^{\frac{s_{1}}{\sigma}}\;\phi_{1}(s_{1})\;x^{s_{1}-1}\hat{I}_{\;p_{1},\;q_{1}}^{\;m_{1},\;n_{1}}\left[z_{1}\;x^{\sigma}\left|\begin{array}[]{c}_{1}(a_{j},e_{j},1)_{n_{1}},\;_{n_{1}+1}(a_{j},e_{j},A_{j})_{p_{1}}\\\
_{1}(b_{j},f_{j},1)_{m_{1}},\;_{m_{1}+1}(b_{j},f_{j},B_{j})_{q_{1}}\end{array}\right]\right.$
(80)

and

$\displaystyle
f_{2}(X_{2})=\mu\;z_{2}^{\frac{s_{2}}{\mu}}\;\phi_{2}(s_{2})\;x^{s_{2}-1}\hat{I}_{\;p_{2},\;q_{2}}^{\;m_{2},\;n_{2}}\left[z_{2}\;x^{\mu}\left|\begin{array}[]{c}_{1}(a_{j}^{\prime},e_{j}^{\prime},1)_{n_{2}},\;_{n_{2}+1}(a_{j}^{\prime},e_{j}^{\prime},A_{j}^{\prime})_{p_{2}}\\\
_{1}(b_{j}^{\prime},f_{j}^{\prime},1)_{m_{2}},\;_{m_{2}+1}(b_{j}^{\prime},f_{j}^{\prime},B_{j}^{\prime})_{q_{2}}\end{array}\right]\right.$
(83)

where $0<\sigma<\infty$, $0<\mu<\infty$, $z_{1}$, $z_{2}$, $s_{1}$, $s_{2}>0$,
with

$\displaystyle\phi_{1}(s_{1})=\frac{\prod_{j=n_{1}+1}^{p_{1}}\Gamma^{A_{j}}\left(a_{j}+\frac{s_{1}}{\sigma}e_{j}\right)\prod_{j=m_{1}+1}^{q_{1}}\Gamma^{B_{j}}\left(1-b_{j}-\frac{s_{1}}{\sigma}f_{j}\right)}{\prod_{j=1}^{n_{1}}\Gamma\left(1-a_{j}-\frac{s_{1}}{\sigma}e_{j}\right)\prod_{j=1}^{m_{1}}\Gamma\left(b_{j}+\frac{s_{1}}{\sigma}f_{j}\right)}$
(84)

and

$\displaystyle\phi_{2}(s_{2})=\frac{\prod_{j=n_{2}+1}^{p_{2}}\Gamma^{A_{j}^{\prime}}\left(a_{j}^{\prime}+\frac{s_{2}}{\mu}e_{j}^{\prime}\right)\prod_{j=m_{2}+1}^{q_{2}}\Gamma^{B_{j}^{\prime}}\left(1-b_{j}^{\prime}-\frac{s_{2}}{\mu}f_{j}^{\prime}\right)}{\prod_{j=1}^{n_{2}}\Gamma\left(1-a_{j}^{\prime}-\frac{s_{2}}{\mu}e_{j}^{\prime}\right)\prod_{j=1}^{m_{2}}\Gamma\left(b_{j}^{\prime}+\frac{s_{2}}{\mu}f_{j}^{\prime}\right)}$
(85)

then the p.d.f. of the variate $Y=X_{1}\;X_{2}$ is given by

$\displaystyle g(Y)=$
$\displaystyle\frac{\sigma\;\mu}{y}\;\phi_{1}(s_{1})\;\phi_{2}(s_{2})$
$\displaystyle\times\hat{I}_{\;p_{1}+p_{2},\;q_{1}+q_{2}}^{\;m_{1}+m_{2},\;n_{1}+n_{2}}\left[z_{1}^{\mu}\;z_{2}^{\sigma}\;y^{\mu\sigma}\left|\begin{array}[]{c}_{1}(a_{j}+\frac{s_{1}}{\sigma}e_{j},\mu
e_{j},1)_{n_{1}},\;_{1}(a_{j}^{\prime}+\frac{s_{2}}{\mu}e_{j}^{\prime},\sigma
e_{j}^{\prime},1)_{n_{2}},\\\ _{1}(b_{j}+\frac{s_{1}}{\sigma}f_{j},\mu
f_{j},1)_{m_{1}},\;_{1}(b_{j}^{\prime}+\frac{s_{2}}{\mu}f_{j}^{\prime},\sigma
f_{j}^{\prime},1)_{m_{2}},\end{array}\right.\right.$ (88)
$\displaystyle\left.\left.\begin{array}[]{c}_{n_{1}+1}(a_{j}+\frac{s_{1}}{\sigma}e_{j},\mu
e_{j},A_{j})_{p_{1}},\;_{n_{2}+1}(a_{j}^{\prime}+\frac{s_{2}}{\mu}e_{j}^{\prime}1,\sigma
e_{j}^{\prime},A_{j}^{\prime})_{p_{2}}\\\
_{m_{1}+1}(b_{j}+\frac{s_{1}}{\sigma}f_{j},\mu
f_{j},B_{j})_{q_{1}},\;_{m_{2}+1}(b_{j}^{\prime}+\frac{s_{2}}{\mu}f_{j}^{\prime},\sigma
f_{j}^{\prime},B_{j}^{\prime})_{q_{2}}\end{array}\right.\right]$ (91)

###### Proof.

It is not difficult to see that

$\displaystyle E(X_{1}^{s-1})$
$\displaystyle=\int_{0}^{\infty}x_{1}^{s-1}\;f_{1}(X_{1})\;dx_{1}$
$\displaystyle=z_{1}^{\frac{1-s}{\sigma}}\;\phi_{1}(s_{1})\;\theta_{1}\left(\frac{s_{1}-1+s}{\sigma}\right)$
(92)

and

$\displaystyle E(X_{2}^{s-1})$
$\displaystyle=\int_{0}^{\infty}x_{2}^{s-1}\;f_{2}(X_{2})\;dx_{2}$
$\displaystyle=z_{2}^{\frac{1-s}{\mu}}\;\phi_{2}(s_{2})\;\theta_{2}\left(\frac{s_{2}-1+s}{\mu}\right)$
(93)

Now, since $X_{1},X_{2}$ are independent random variables, so

$\displaystyle
E(Y^{s-1})=E[(X_{1}\;X_{2})^{s-1}]=E(X_{1}^{s-1})\;E(X_{2}^{s-1})$ (94)

and

$\displaystyle g(Y)=\frac{1}{2\pi i}\int_{\mathcal{L}}E(Y^{s-1})\;y^{-s}\;ds$
(95)

Using (4) and (4) in (95), we have

$\displaystyle g(Y)$ $\displaystyle=\frac{1}{2\pi
i}\int_{\mathcal{L}}z_{1}^{\frac{1-s}{\sigma}}\;\phi_{1}(s_{1})\;\theta_{1}\left(\frac{s_{1}-1+s}{\sigma}\right)\;z_{2}^{\frac{1-s}{\mu}}\;\phi_{2}(s_{2})\;\theta_{2}\left(\frac{s_{2}-1+s}{\mu}\right)\;ds$
$\displaystyle=z_{1}^{\frac{1}{\sigma}}\;z_{2}^{\frac{1}{\mu}}\;\phi_{1}(s_{1})\;\phi_{2}(s_{2})$
$\displaystyle~{}~{}~{}~{}~{}\times\frac{1}{2\pi
i}\int_{\mathcal{L}}\theta_{1}\left(\frac{s_{1}-1+s}{\sigma}\right)\;\theta_{2}\left(\frac{s_{2}-1+s}{\mu}\right)\;\left(y\;z_{1}^{\frac{1}{\sigma}}\;z_{2}^{\frac{1}{\mu}}\right)^{-s}ds$
(96)

By using the definition of $\hat{I}$-function (11), we get

$\displaystyle g(Y)$
$\displaystyle=z_{1}^{\frac{1}{\sigma}}\;z_{2}^{\frac{1}{\mu}}\;\phi_{1}(s_{1})\;\phi_{2}(s_{2})$
$\displaystyle\times\hat{I}_{\;p_{1}+p_{2},\;q_{1}+q_{2}}^{\;m_{1}+m_{2},\;n_{1}+n_{2}}\left[y\;z_{1}^{\frac{1}{\sigma}}\;z_{2}^{\frac{1}{\mu}}\left|\begin{array}[]{c}_{1}(a_{j}+\frac{s_{1}-1}{\sigma}e_{j},\frac{e_{j}}{\sigma},1)_{n_{1}},\;_{1}(a_{j}^{\prime}+\frac{s_{2}-1}{\mu}e_{j}^{\prime},\frac{e_{j}^{\prime}}{\mu},1)_{n_{2}},\\\
_{1}(b_{j}+\frac{s_{1}-1}{\sigma}f_{j},\frac{f_{j}}{\sigma},1)_{m_{1}},\;1(b_{j}^{\prime}+\frac{s_{2}-1}{\mu}f_{j}^{\prime},\frac{f_{j}^{\prime}}{\mu},1)_{m_{2}},\end{array}\right.\right.$
(99)
$\displaystyle\qquad\qquad\qquad\left.\left.\begin{array}[]{c}_{n_{1}+1}(a_{j}+\frac{s_{1}-1}{\sigma}e_{j},\frac{e_{j}}{\sigma},A_{j})_{p_{1}},\;_{n_{2}+1}(a_{j}^{\prime}+\frac{s_{2}-1}{\mu}e_{j}^{\prime},\frac{e_{j}^{\prime}}{\mu},A_{j}^{\prime})_{p_{2}}\\\
\;_{m_{1}+1}(b_{j}+\frac{s_{1}-1}{\sigma}f_{j},\frac{f_{j}}{\sigma},B_{j})_{q_{1}},\;_{m_{2}+1}(b_{j}^{\prime}+\frac{s_{2}-1}{\mu}f_{j}^{\prime},\frac{f_{j}^{\prime}}{\mu},B_{j}^{\prime})_{q_{2}}\end{array}\right.\right]$
(102)

Upon using (18) and (24), we finally arrive at (4.1).
This completes the proof of Theorem 1. ∎

######  Theorem 4.2.

If $X_{1}$ and $X_{2}$ be two independent $\hat{I}$-function variates with
p.d.f. given respectively by the equations (80) and (83), where
$0<\sigma<\infty,0<\mu<\infty,z_{1},z_{2},s_{1},s_{2}>0$, and
$\phi_{1}(s_{1})$ and $\phi_{2}(s_{2})$ are also the same given by equations
(84) and (85) respectively, then the p.d.f. of the variate
$Y=\frac{X_{1}}{X_{2}}$ is given by

$\displaystyle g(Y)=$
$\displaystyle\frac{\sigma\;\mu}{y}\;\phi_{1}(s_{1})\;\phi_{2}(s_{2})$
$\displaystyle\times\hat{I}_{\;p_{1}+q_{2},\;q_{1}+p_{2}}^{\;m_{1}+n_{2},\;n_{1}+m_{2}}\left[z_{1}^{\mu}\;z_{2}^{-\sigma}\;y^{\mu\sigma}\left|\begin{array}[]{c}_{1}(a_{j}+\frac{s_{1}}{\sigma}e_{j},\mu
e_{j},1)_{n_{1}},\\\ _{1}(b_{j}+\frac{s_{1}}{\sigma}f_{j},\mu
f_{j},1)_{m_{1}},\end{array}\right.\right.$ (105)
$\displaystyle\qquad\qquad\qquad\qquad\left.\left.\begin{array}[]{c}_{1}(1-b_{j}^{\prime}-\frac{s_{2}}{\mu}f_{j}^{\prime},\sigma
f_{j}^{\prime},1)_{m_{2}},\;_{n_{1}+1}(a_{j}+\frac{s_{1}}{\sigma}e_{j},\mu
e_{j},A_{j})_{p_{1}},\\\
_{1}(1-a_{j}^{\prime}-\frac{s_{2}}{\mu}e_{j}^{\prime},\sigma
e_{j}^{\prime},1)_{n_{2}},\;_{m_{1}+1}(b_{j}+\frac{s_{1}}{\sigma}f_{j},\mu
f_{j},B_{j})_{q_{1}},\end{array}\right.\right.$ (108)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\left.\left.\begin{array}[]{c}_{m_{2}+1}(1-b_{j}^{\prime}-\frac{s_{2}}{\mu}f_{j}^{\prime},\sigma
f_{j}^{\prime}1,B_{j}^{\prime})_{q_{2}}\\\
_{n_{2}+1}(1-a_{j}^{\prime}-\frac{s_{2}}{\mu}e_{j}^{\prime},\sigma
e_{j}^{\prime},A_{j}^{\prime})_{p_{2}}\end{array}\right.\right]$ (111)

###### Proof.

We have

$\displaystyle
E(Y^{s-1})=E\left[\left(\frac{X_{1}}{X_{2}}\right)^{s-1}\right]$
$\displaystyle=\frac{E(X_{1}^{s-1})}{E(X_{2}^{s-1})}=E(X_{1}^{s-1}X_{2}^{1-s})$

Since $X_{1}$ and $X_{2}$ are independent random variables,

$\displaystyle E(Y^{s-1})=E(X_{1}^{s-1})E(X_{2}^{1-s})$

Using (4) and (4) in (95), we have

$\displaystyle g(Y)$ $\displaystyle=\frac{1}{2\pi
i}\int_{\mathcal{L}}z_{1}^{\frac{1-s}{\sigma}}\;\phi_{1}(s_{1})\;\theta_{1}\left(\frac{s_{1}-1+s}{\sigma}\right)\;z_{2}^{\frac{s-1}{\mu}}\;\phi_{2}(s_{2})\;\theta_{2}\left(\frac{s_{2}+1-s}{\mu}\right)\;ds$
$\displaystyle=z_{1}^{\frac{1}{\sigma}}\;z_{2}^{-\frac{1}{\mu}}\;\phi_{1}(s_{1})\;\phi_{2}(s_{2})$
$\displaystyle~{}~{}~{}~{}~{}\times\frac{1}{2\pi
i}\int_{\mathcal{L}}\theta_{1}\left(\frac{s_{1}-1+s}{\sigma}\right)\;\theta_{2}\left(\frac{s_{2}+1-s}{\mu}\right)\;\left(y\;z_{1}^{\frac{1}{\sigma}}\;z_{2}^{-\frac{1}{\mu}}\right)^{-s}ds$
(112)

By using the definition of $\hat{I}$-function (11), we get

$\displaystyle g(Y)$
$\displaystyle=z_{1}^{\frac{1}{\sigma}}\;z_{2}^{-\frac{1}{\mu}}\;\phi_{1}(s_{1})\;\phi_{2}(s_{2})$
$\displaystyle\times\hat{I}_{\;p_{1}+q_{2},\;q_{1}+p_{2}}^{\;m_{1}+n_{2},\;n_{1}+m_{2}}\left[y\;z_{1}^{\frac{1}{\sigma}}\;z_{2}^{-\frac{1}{\mu}}\left|\begin{array}[]{c}_{1}(a_{j}+\frac{s_{1}-1}{\sigma}e_{j},\frac{e_{j}}{\sigma},1)_{n_{1}},\\\
_{1}(b_{j}+\frac{s_{1}-1}{\sigma}f_{j},\frac{f_{j}}{\sigma},1)_{m_{1}},\end{array}\right.\right.$
(115)
$\displaystyle\qquad\qquad\left.\left.\begin{array}[]{c}_{1}(1-b_{j}^{\prime}-\frac{s_{2}+1}{\mu}f_{j}^{\prime},\frac{f_{j}^{\prime}}{\mu},1)_{m_{2}},\;_{n_{1}+1}(a_{j}+\frac{s_{1}-1}{\sigma}e_{j},\frac{e_{j}}{\sigma},A_{j})_{p_{1}},\\\
_{1}(1-a_{j}^{\prime}-\frac{s_{2}+1}{\mu}e_{j}^{\prime},\frac{e_{j}^{\prime}}{\mu},1)_{n_{2}},\;_{m_{1}+1}(b_{j}+\frac{s_{1}-1}{\sigma}f_{j},\frac{f_{j}}{\sigma},B_{j})_{q_{1}},\end{array}\right.\right.$
(118)
$\displaystyle\qquad\qquad\qquad\qquad\left.\left.\begin{array}[]{c}_{m_{2}+1}(1-b_{j}^{\prime}-\frac{s_{2}+1}{\mu}f_{j}^{\prime},\frac{f_{j}^{\prime}}{\mu},B_{j}^{\prime})_{q_{2}}\\\
_{n_{2}+1}(1-a_{j}^{\prime}-\frac{s_{2}+1}{\mu}e_{j}^{\prime},\frac{e_{j}^{\prime}}{\mu},A_{j}^{\prime})_{p_{2}}\end{array}\right.\right]$
(121)

Upon using (18) and (24), we finally arrive at (4.2).
This completes the proof of Theorem 2. ∎

##  5\. Corollaries

### Corollary 1

If $A_{j}=1$ for $j=n_{1}+1,\cdots,p_{1}$, $B_{j}=1$ for
$j=m_{1}+1,\cdots,q_{1}$, $A_{j}^{\prime}=1$ for $j=n_{2}+1,\cdots,p_{2}$, and
$B_{j}^{\prime}=1$ for $j=m_{2}+1,\cdots,q_{2}$, then the $\hat{I}$-function
in Thereom 1 reduces to H-function, and we get the corresponding result for
the H-function as follows.
If $X_{1}$ and $X_{2}$ are independent $H$-function variates with p.d.f. given
respectively by

$\displaystyle
f_{1}(X_{1})=\sigma\;z_{1}^{\frac{s_{1}}{\sigma}}\;\phi_{1}(s_{1})\;x^{s_{1}-1}H_{\;p_{1},\;q_{1}}^{\;m_{1},\;n_{1}}\left[z_{1}\;x^{\sigma}\left|\begin{array}[]{c}_{1}(a_{j},e_{j})_{p_{1}}\\\
_{1}(b_{j},f_{j})_{q_{1}}\end{array}\right]\right.,x>0$ (124)

and

$\displaystyle
f_{2}(X_{2})=\mu\;z_{2}^{\frac{s_{2}}{\mu}}\;\phi_{2}(s_{2})\;x^{s_{2}-1}H_{\;p_{2},\;q_{2}}^{\;m_{2},\;n_{2}}\left[z_{2}\;x^{\mu}\left|\begin{array}[]{c}_{1}(a_{j}^{\prime},e_{j}^{\prime})_{p_{2}}\\\
_{1}(b_{j}^{\prime},f_{j}^{\prime})_{q_{2}}\end{array}\right]\right.,x>0$
(127)

where $0<\sigma<\infty,0<\mu<\infty,z_{1},z_{2},s_{1},s_{2}>0$,
also

$\displaystyle\phi_{1}(s_{1})=\frac{\prod_{j=n_{1}+1}^{p_{1}}\Gamma\left(a_{j}+\frac{s_{1}}{\sigma}e_{j}\right)\prod_{j=m_{1}+1}^{q_{1}}\Gamma\left(1-b_{j}-\frac{s_{1}}{\sigma}f_{j}\right)}{\prod_{j=1}^{n_{1}}\Gamma\left(1-a_{j}-\frac{s_{1}}{\sigma}e_{j}\right)\prod_{j=1}^{m_{1}}\Gamma\left(b_{j}+\frac{s_{1}}{\sigma}f_{j}\right)}$
(128)

and

$\displaystyle\phi_{2}(s_{2})=\frac{\prod_{j=n_{2}+1}^{p_{2}}\Gamma\left(a_{j}^{\prime}+\frac{s_{2}}{\mu}e_{j}^{\prime}\right)\prod_{j=m_{2}+1}^{q_{2}}\Gamma\left(1-b_{j}^{\prime}-\frac{s_{2}}{\mu}f_{j}^{\prime}\right)}{\prod_{j=1}^{n_{2}}\Gamma\left(1-a_{j}^{\prime}-\frac{s_{2}}{\mu}e_{j}^{\prime}\right)\prod_{j=1}^{m_{2}}\Gamma\left(b_{j}^{\prime}+\frac{s_{2}}{\mu}f_{j}^{\prime}\right)}$
(129)

then the p.d.f. of the variate $Y=X_{1}\;X_{2}$ is given by

$\displaystyle g(Y)$
$\displaystyle=\frac{\sigma\;\mu}{y}\;\phi_{1}(s_{1})\;\phi_{2}(s_{2})$
$\displaystyle\times
H_{\;p_{1}+p_{2},\;q_{1}+q_{2}}^{\;m_{1}+m_{2},\;n_{1}+n_{2}}\left[z_{1}^{\mu}\;z_{2}^{\sigma}\;y^{\mu\sigma}\left|\begin{array}[]{c}_{1}(a_{j}+\frac{s_{1}}{\sigma}e_{j},\mu
e_{j})_{n_{1}},\\\ _{1}(b_{j}+\frac{s_{1}}{\sigma}f_{j},\mu
f_{j})_{m_{1}},\end{array}\right.\right.$ (132)
$\displaystyle\left.\left.\begin{array}[]{c}_{1}(a_{j}^{\prime}+\frac{s_{2}}{\mu}e_{j}^{\prime},\sigma
e_{j}^{\prime})_{n_{2}},\;_{n_{1}+1}(a_{j}+\frac{s_{1}}{\sigma}e_{j},\mu
e_{j})_{p_{1}},\;_{n_{2}+1}(a_{j}^{\prime}+\frac{s_{2}}{\mu}e_{j}^{\prime},\sigma
e_{j}^{\prime})_{p_{2}}\\\
_{1}(b_{j}^{\prime}+\frac{s_{2}}{\mu}f_{j}^{\prime},\sigma
f_{j}^{\prime})_{m_{2}},\;_{m_{1}+1}(b_{j}+\frac{s_{1}}{\sigma}f_{j},\mu
f_{j})_{q_{1}},\;_{m_{2}+1}(b_{j}^{\prime}+\frac{s_{2}}{\mu}f_{j}^{\prime},\sigma
f_{j}^{\prime})_{q_{2}}\end{array}\right.\right]$ (135)

The p.d.f. of the variate $Y=\frac{X_{1}}{X_{2}}$ is given by

$\displaystyle h(Y)$
$\displaystyle=\frac{\sigma\;\mu}{y}\;\phi_{1}(s_{1})\;\phi_{2}(s_{2})$
$\displaystyle\times
H_{\;p_{1}+q_{2},\;q_{1}+p_{2}}^{\;m_{1}+n_{2},\;n_{1}+m_{2}}\left[z_{1}^{\mu}\;z_{2}^{-\sigma}\;y^{\mu\sigma}\left|\begin{array}[]{c}_{1}(a_{j}+\frac{s_{1}}{\sigma}e_{j},\mu
e_{j})_{n_{1}},\\\ _{1}(b_{j}+\frac{s_{1}}{\sigma}f_{j},\mu
f_{j})_{m_{1}},\end{array}\right.\right.$ (138)
$\displaystyle\left.\left.\begin{array}[]{c}_{1}(1-b_{j}^{\prime}-\frac{s_{2}}{\mu}f_{j}^{\prime},\sigma
f_{j}^{\prime})_{q_{2}},\;_{n_{1}+1}(a_{j}+\frac{s_{1}}{\sigma}e_{j},\mu
e_{j})_{p_{1}}\\\ _{1}(1-a_{j}^{\prime}-\frac{s_{2}}{\mu}e_{j}^{\prime},\sigma
e_{j}^{\prime})_{p_{2}},\;_{m_{1}+1}(b_{j}+\frac{s_{1}}{\sigma}f_{j},\mu
f_{j})_{q_{1}}\end{array}\right.\right]$ (141)

### Corollary 2

If we take
$\sigma=\mu=r,z_{1}=z_{2}=z,m_{1}=m_{2}=m,n_{1}=n_{2}=n,p_{1}=p_{2}=p,q_{1}=q_{2}=q$
in Theorem 2, then we get the following result after little simplification.
If $X_{1}$ and $X_{2}$ are independent $\hat{I}$-function variates with p.d.f.
given respectively by

$\displaystyle
f_{1}(X_{1})=r\;z^{\frac{s_{1}}{r}}\;\phi_{1}(s_{1})\;x^{s_{1}-1}\hat{I}_{\;p,\;q}^{\;m,\;n}\left[z\;x^{r}\left|\begin{array}[]{c}_{1}(a_{j},e_{j},1)_{n},\;_{n+1}(a_{j},e_{j},A_{j})_{p}\\\
_{1}(b_{j},f_{j},1)_{m},\;_{m+1}(b_{j},f_{j},B_{j})_{q}\end{array}\right]\right.$
(144)

and

$\displaystyle
f_{2}(X_{2})=r\;z^{\frac{s_{2}}{r}}\;\phi_{2}(s_{2})\;x^{s_{2}-1}\hat{I}_{\;p,\;q}^{\;m,\;n}\left[z\;x^{r}\left|\begin{array}[]{c}_{1}(a_{j},e_{j},1)_{n},\;_{n+1}(a_{j},e_{j},A_{j})_{p}\\\
_{1}(b_{j},f_{j},1)_{m},\;_{m+1}(b_{j},f_{j},B_{j})_{q}\end{array}\right]\right.$
(147)

where $0<r<\infty,z,s_{1},s_{2}>0$,
The p.d.f. of the variate $Y=\frac{X_{1}}{X_{2}}$ is given by

$\displaystyle g(Y)=$ $\displaystyle
r\;y^{-s_{2}-1}\;\phi(s_{1})\;\phi(s_{2})$
$\displaystyle\times\hat{I}_{\;p+q,\;q+p}^{\;m+n,\;n+m}\left[y^{r}\left|\begin{array}[]{c}_{1}(1-b_{j},f_{j},1)_{m},\;_{m+1}(1-b_{j},f_{j},B_{j})_{q},\\\
_{1}(b_{j}+\frac{s_{1}+s_{2}}{r}f_{j},f_{j},1)_{m},\;_{m+1}(b_{j}+\frac{s_{1}+s_{2}}{r}f_{j},f_{j},B_{j})_{q},\end{array}\right.\right.$
(150)
$\displaystyle\left.\left.\begin{array}[]{c}_{1}(a_{j}+\frac{s_{1}+s_{2}}{r}e_{j},e_{j},1)_{n},\;_{n+1}(a_{j}+\frac{s_{1}+s_{2}}{r}e_{j},e_{j},A_{j})_{p}\\\
_{1}(1-a_{j},e_{j},1)_{n},\;_{n+1}(1-a_{j},e_{j},A_{j})_{p}\end{array}\right.\right]$
(153)

Further, in (5), if we take $A_{j}=1$ for $j=n+1,\cdots,p$ and $B_{j}=1$ for
$j=m+1,\cdots,q$, then we get a known theorem due to Mathai and Saxena [14].
Similarly, other special cases can be considered.

## Concluding Remarks

In this paper, in the beginning, we have obtained the Mellin transforms of the
$\hat{I}$ \- function and Mellin transforms of the product of two $\hat{I}$ \-
functions. Thereafter, a general probability distribution involving
$\hat{I}$-function has been introduced. Finally, we have established the
distributions of the product and quotient of two independent random variables
having densities in terms of the $\hat{I}$ \- functions (which is a particular
case of the Rathie’s ${I}$ \- function). Since $\hat{I}$\- function is the
most general function of one variable which includes not only the well known
functions such as Inayat Hussain’s $\bar{H}$-function[7], Fox’s H-function[6,
12] and Meijer’s G-function[12], but also a large number of elementary
functions, so on specializing the parameters therein, we can obtain
distributions of product and quotient of two independent random variables
having different densities. It is hoped that the results presented in this
paper could be of potential use in the area of statistics, applied
mathematics, mathematical physics and wireless communication.

We conclude this paper by remarking that all the results established in the
paper contain proper convergence conditions.

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