The Bedrosian Identity for L<sup>p</sup> Function and the Hardy Space on Tube

Zhihong Wen; Guantie Deng; Zhihong Wen; Guantie Deng

doi:10.3934/Math.2016.1.9

AIMS Mathematics

2016, Volume 1, Issue 1: 9-23. doi: 10.3934/Math.2016.1.9

Previous Article Next Article

Research article

The Bedrosian Identity for L^p Function and the Hardy Space on Tube

Zhihong Wen ^,,
Guantie Deng

School of Mathematical Sciences, Key Laboratory of Mathematics and Complex Systems of Ministry of Education, Beijing Normal University, 100875, Beijing, China

Received: 26 March 2016 Accepted: 11 April 2016 Published: 19 April 2016

In this paper, we are devoted to establishing several necessary and sufficient conditions for $f\in L^{p}(\mathbb{R}^{n}),\ g\in L^{q}(\mathbb{R}^{n})$ with $\frac{1}{p}+\frac{1}{q}\leq 1$ to satisfy the Bedrosian identity $H(fg)=fHg$, where $H$ denotes the n-dimensional Hilbert transform. In addition, we also show that the distribution $f\in \mathcal{D}'_{L^{p}}(\mathbb{R}^{n})$ can be represented by functions in the Hardy space on tube.

Keywords:

Citation: Zhihong Wen, Guantie Deng. The Bedrosian Identity for Lp Function and the Hardy Space on Tube[J]. AIMS Mathematics, 2016, 1(1): 9-23. doi: 10.3934/Math.2016.1.9

Related Papers:

[1]	Dayang Dai, Dabuxilatu Wang . A generalized Liu-type estimator for logistic partial linear regression model with multicollinearity. AIMS Mathematics, 2023, 8(5): 11851-11874. doi: 10.3934/math.2023600
[2]	Muhammad Nauman Akram, Muhammad Amin, Ahmed Elhassanein, Muhammad Aman Ullah . A new modified ridge-type estimator for the beta regression model: simulation and application. AIMS Mathematics, 2022, 7(1): 1035-1057. doi: 10.3934/math.2022062
[3]	Sihem Semmar, Omar Fetitah, Mohammed Kadi Attouch, Salah Khardani, Ibrahim M. Almanjahie . A Bernstein polynomial approach of the robust regression. AIMS Mathematics, 2024, 9(11): 32409-32441. doi: 10.3934/math.20241554
[4]	Yanting Xiao, Wanying Dong . Robust estimation for varying-coefficient partially linear measurement error model with auxiliary instrumental variables. AIMS Mathematics, 2023, 8(8): 18373-18391. doi: 10.3934/math.2023934
[5]	Gaosheng Liu, Yang Bai . Statistical inference in functional semiparametric spatial autoregressive model. AIMS Mathematics, 2021, 6(10): 10890-10906. doi: 10.3934/math.2021633
[6]	Juxia Xiao, Ping Yu, Zhongzhan Zhang . Weighted composite asymmetric Huber estimation for partial functional linear models. AIMS Mathematics, 2022, 7(5): 7657-7684. doi: 10.3934/math.2022430
[7]	Xin Liang, Xingfa Zhang, Yuan Li, Chunliang Deng . Daily nonparametric ARCH(1) model estimation using intraday high frequency data. AIMS Mathematics, 2021, 6(4): 3455-3464. doi: 10.3934/math.2021206
[8]	Emrah Altun, Mustafa Ç. Korkmaz, M. El-Morshedy, M. S. Eliwa . The extended gamma distribution with regression model and applications. AIMS Mathematics, 2021, 6(3): 2418-2439. doi: 10.3934/math.2021147
[9]	Muhammad Amin, Saima Afzal, Muhammad Nauman Akram, Abdisalam Hassan Muse, Ahlam H. Tolba, Tahani A. Abushal . Outlier detection in gamma regression using Pearson residuals: Simulation and an application. AIMS Mathematics, 2022, 7(8): 15331-15347. doi: 10.3934/math.2022840
[10]	Zawar Hussain, Atif Akbar, Mohammed M. A. Almazah, A. Y. Al-Rezami, Fuad S. Al-Duais . Diagnostic power of some graphical methods in geometric regression model addressing cervical cancer data. AIMS Mathematics, 2024, 9(2): 4057-4075. doi: 10.3934/math.2024198

Abstract

1. Introduction

The field of mathematical analysis that deals with the study of arbitrary order integrals and derivatives is known as fractional calculus. Because of its numerous applications across a wide range of fields, this field has increased in importance and recognition over the past few years. According to researchers, this field is the most effective at identifying anomalous kinetics and has numerous uses in a variety of fields. Ordinary differential equations with fractional derivatives can be used to simulate a variety of issues, including statistical, mathematical, engineering, chemical, and biological issues. Several distinct forms of fractional integrals and derivative operators (see e.g., ^[1,2,3,4]), including Riemann-Liouville, Caputo, Riesz, Hilfer, Hadamard, Erdélyi-Kober, Saigo, Marichev-Saigo-Maeda and others, have been thoroughly investigated by researchers. From an application perspective, we suggest the readers to see the work related to the fractional differential equations presented by ^[5,6,7,8]. In ^[9], the authors studied symmetric and antisymmetric solitons in the defocused saturable nonlinearity and the PT-symmetric potential of the fractional nonlinear Schrödinger equation. In ^[10], the fractional exponential function approach is used to study a time-fractional Ablowitz-Ladik model, and bright and dark discrete soliton solutions, discrete exponential solutions, and discrete peculiar wave solutions are discovered. In ^[11], the authors presented the rich vector exact solutions for the coupled discrete conformable fractional nonlinear Schrödinger equations by taking into account the conformable fractional derivative.

On the other hand, special functions like Gamma, Beta, Mittag-Leffler, et al. play a vital part in the foundation of fractional calculus. Moreover, the Mittag-Leffler function is regarded as the fundamental function in fractional calculus. The Prabhakar fractional operator containing a three-parameter version of the aforementioned function in the kernel. The M-L function has been extensively studied to construct solutions of fractional PDEs, such as dynamical characteristic of analytical fractional solitons for the space-time fractional Fokas-Lenells equation, soliton dynamics based on exact solutions of conformable fractional discrete complex cubic Ginzburg-Landau equation and Abundant fractional soliton solutions of a space-time fractional perturbed Gerdjikov-Ivanov equation by a fractional mapping method, see ^[12,13,14]. Strong generalizations of the univariate and bivariate Mittag-Leffler functions, which are known to be important in fractional calculus, are the multivariate Mittag-Leffler functions.

The well-known one-parameter Mittag-Leffler (M-L) function is defined by ^[15,16] as follows

$\begin{equation} \varepsilon_{a}(z_1) = \sum\limits_{l = 0}^{\infty}\frac{z_1^l}{\Gamma(al+1)}\,\,\,\,\,\,\,\,\,(a\in\mathbb{C};\Re(a) > 0,z_1\in\mathbb{C}), \end{equation}$ $

(1.1)

where $ \mathbb{C} $ represents the set of complex numbers and $ \Re(a) $ denotes the real part of the complex number.

The generalization of (1.1) with two parameters is defined by ^[17,18] as

$\begin{equation} \varepsilon_{a,b}(z_1) = \sum\limits_{l = 0}^{\infty}\frac{z_1^l}{\Gamma(al+b)}\,\,\,\,\,\,\,\,\,(a,b\in\mathbb{C};\Re(a) > 0,\Re(b) > 0), \end{equation}$ $

(1.2)

Later on, Agarwal ^[19], Humbert ^[20] and Humbert and Agarwal ^[21] studied the properties and applications of M-L functions. In ^[22], the generalization of (1.1) and (1.2) is defined by

$\begin{equation} \varepsilon_{a,b}^{c}(z_1) = \sum\limits_{l = 0}^{\infty}\frac{(c)_l}{\Gamma(a l+b)}\frac{z_1^l}{l!}\,\,\,\,\,\,\,\,\,(a,b,c\in\mathbb{C};\Re(a) > 0,\Re(b) > 0). \end{equation}$ $

(1.3)

In ^[23], the following generalization of the M-L function is defined by

$\begin{equation} \varepsilon_{a,b}^{c,q}(z_1) = \sum\limits_{l = 0}^{\infty}\frac{(c)_{lq}}{\Gamma(a l+b)}\frac{z_1^l}{l!}\,\,\,\,\,\,\,\,\,(a,b,c\in\mathbb{C};\Re(a) > 0,\Re(b) > 0, q > 0). \end{equation}$ $

(1.4)

In ^[24], Rahman et al. proposed the following generalized of M-L function by

$\begin{align} \varepsilon_{a,b,p}^{c,q,d}(z_1) = \sum\limits_{l = 0}^{\infty}\frac{B_p(c +lq; d-c)(d)_{lq}}{B(c, d-c)\Gamma(a l+b)}\frac{z_1^l}{l!}, \end{align}$ $

(1.5)

where $ a, b, c, d\in\mathbb{C}; \Re(c) > 0, \Re(a) > 0, \Re(b) > 0, q > 0 $ and $ B_p(x, y) = \int_0^1 t^{x-1}(1-t)^{y-1}e^{-t-\frac{p}{t}}dt $ is the extension of beta function (see ^[25]).

Gorenflo et al. ^[26] and Haubold et al. ^[27]) studied the various properties of generalized M-L function. In ^[28], a new generalization of M-L function (1.3) is presented by

$\begin{equation} \varepsilon_{a,b,p}^{c}(z_1) = \sum\limits_{l = 0}^{\infty}\frac{(c;p)_l}{\Gamma(a l+b)}\frac{z_1^l}{l!}\,\,\,\,\,\,\,\,\,(p\geq0, a,b,c\in\mathbb{C}; , \Re(a) > 0,\Re(b) > 0,), \end{equation}$ $

(1.6)

where $ (\lambda; p)_l $ is the Pochhammer symbol defined by Srivastava et al. ^[29,30] as

$\begin{equation} (\lambda;p)_{\mu} = \begin{cases} \frac{\Gamma_{p}(\lambda+\mu)}{\Gamma(\lambda)}; &(p > 0, \lambda,\mu\in\mathbb{C} )\\ \\ (\lambda)_{\mu}; & (p = 0,\ {\lambda,\mu}\in \mathbb{C}\setminus\{{0}\}.\ \ \ \ \ \ \ \ \ \ \ \ \end{cases} \end{equation}$ $

(1.7)

The researchers examined the developments of these extension, (1.6) and (1.7) and studied their related features and applications. In ^[30], Srivastava et al. proposed the following generalized hypergeometric function

$\begin{equation} _sF_t[(\delta_1; p),\cdots, (\delta_s); (\zeta_1),\cdots, (\zeta_t); z_1] = \sum\limits_{l = 0}^{\infty}\frac{(\delta_{1}; p)_l\cdots (\delta_s)_l}{(\zeta_1)_l\cdots (\zeta_t)_l}\ \frac{z_1^l}{l!}\, , \end{equation}$ $

(1.8)

where $ \delta_{j}\in\mathbb{C} $ for j = 1, 2, $ \cdots $, s, $ \zeta_{k}\in\mathbb{C} $ for $ k = 1, 2, \cdots, t, $ and $ \zeta_{k}\neq $ 0, -1, -2, $ \cdots $.

The integral representation of $ (\mu; p)_\eta $ is explained by

$\begin{equation} (\mu;p)_{\eta} = \frac{1}{\Gamma(\mu)}\ \int_0^\infty\ s^{\mu+\eta-1}\ e^{-s-\frac{p}{s}} d s, \end{equation}$ $

(1.9)

where $ \Re(\rho) > 0 $ and $ \Re(\mu+\eta) > 0 $. In particular, the related confluent hypergeometric function $ _1F_1 $ and the Gauss hypergeometric function $ _2F_1 $ are given by

$\begin{equation} _2F_1[(\delta_1; p),b;\lambda; z_1] = \sum\limits_{l = 0}^{\infty}\frac{(\delta_{1}; p)_l (b)_l}{(\lambda)_l}\ \frac{z_1^l}{l!}\, , \end{equation}$ $

(1.10)

and

$\begin{equation} _1F_1[(\delta_1; p);\lambda; z_1] = \Phi[(\delta_1; p);\lambda; z_1] = \sum\limits_{l = 0}^{\infty}\frac{(\delta_{1}; p)_l}{(\lambda)_l}\ \frac{z_1^l}{l!}\, . \end{equation}$ $

(1.11)

The expansion of the generalised hypergeometric function $ _rF_s $, which was studied by ^[30], has $ r $ numerator and $ s $ denominator parameters. Researchers recently developed several extensions of special functions, together with their corresponding characteristics and applications. Using extended beta functions as its foundation, Nisar et al. ^[31], Bohner et al. ^[32] and Rahman et al. ^[33] developed an enlargement of fractional derivative operators.

The multivariate M-L function is defined by ^[34] as follows:

$\begin{align} \mathcal{E}_{(a_j), b}^{(c_j)} (z_1, z_2, \dots, z_j)& = \mathcal{E}_{(a_1, a_2,\dots,a_j), b}^{(c_1, c_2 ,\dots, c_j)} (z_1, z_2, \dots z_j) \\ = & \sum\limits_{m_1, m_2, \dots, m_j = 0}^{\infty} \frac{(c_1)_{m_1}(c_2)_{m_2} \dots (c_j)_{m_j} (z_1)^{m_1} \dots (z_j)^{m_j}}{\Gamma(a_{1}m_1+a_2 m_2+\dots a_j m_j + b)m_1! \dots m_j!}, \end{align}$ $

(1.12)

where $ z_i, a_i, b, c_i \in \mathbb{C} $; $ i = 1, 2, \ldots, j $, $ \Re{(a_i)} > 0 $, $ \Re{(b)} > 0 $ and $ \Re{(c_i)} > 0 $.

In ^{[35,36,37,38,39]}, the authors have studied various properties and applications of different type of generalized M-L functions. For real (complex) valued functions, the Lebesgue measurable space is defined by

$\begin{equation} \textsf{L}(r,s) = \left\{ \begin{array}{c} h: \|h\|_1 = \int_{r}^{s}|h(x)|dx < \infty \\ \end{array} \right\}. \end{equation}$ $

(1.13)

The left and right sides fractional integral operators of the Riemann-Liouville type are defined by ^[3,4] as follows:

$\begin{equation} (\mathfrak{I}_{r+}^{\lambda}h)(x) = \frac{1}{\Gamma(\lambda)}\int\limits_{r}^{x}\frac{h(\varrho)}{(x-\varrho)^{1-\lambda}}d\varrho,\; \; \; (x > r), \end{equation}$ $

(1.14)

and

$\begin{equation*} (\mathfrak{I}_{s-}^{\lambda}h)(x) = \frac{1}{\Gamma(\lambda)}\int\limits_{x}^{s}\frac{h(\varrho)}{(\varrho-x)^{1-\lambda}}d\varrho,\; \; \; (x < s), \end{equation*}$ $

where $ h\in \textsf{L}(r, s) $, $ \lambda\in\mathbb{C} $ and $ \Re(\lambda) > 0 $.

The left and right sides Riemann-Liouville fractional derivatives for the function $ h(x)\in \textsf{L}(r, s) $, $ \lambda\in\mathbb{C} $, $ \Re(\lambda) > 0 $ and $ n = [\Re(\lambda)]+1 $ are defined in ^[3,4] by

$\begin{equation} (\mathfrak{D}_{r+}^{\lambda}h)(x) = (\frac{d}{dx})^n(\mathfrak{I}_{r+}^{n-\lambda}h)(x) \end{equation}$ $

(1.15)

and

$\begin{equation*} (\mathfrak{D}_{s-}^{\lambda}h)(x) = (-\frac{d}{dx})^n(\mathfrak{I}_{s-}^{n-\lambda}h)(x), \end{equation*}$ $

respectively. The generalized differential operator $ \mathfrak{D}_{r+}^{\lambda, v} $ of order $ 0 < \lambda < 1 $ and type $ 0 < v < 1 $ with respect to $ x $ can be found in ^[2,4] as

$\begin{equation} (\mathfrak{D}_{r+}^{\lambda,v}h) = \left( \begin{array}{c} \mathfrak{I}_{r+}^{v(1-\lambda)}\frac{d}{dx}(\mathfrak{I}_{r+}^{(1-v)(1-\lambda)}h) \\ \end{array} \right)(x). \end{equation}$ $

(1.16)

In particular, if $ v = 0, $ then (1.16) will lead to the operator $ \mathfrak{D}_{r+}^{\lambda} $ defined in (1.15).

We also take into account the aforementioned well-known results.

Theorem 1.1. In ^[40], the following result for the fractional integral is presented by

$\begin{equation} \mathfrak{I}_{r+}^{\lambda}(\varrho-r)^{\eta-1} = \frac{\Gamma(\eta)}{\Gamma(\lambda+\eta)}(x-r)^{\lambda+\eta-1}, \end{equation}$ $

(1.17)

where $ \lambda $, $ \eta\in\mathbb{C} $, $ \Re(\lambda) > 0 $, $ \Re(\eta) > 0 $,

Theorem 1.2. ^[41] Suppose that the function $ h(z) $ has a power series expansion $ h(z) = \sum\limits_{k = 0}^{\infty}k_nz^k $ and it is analytic in the disc $ |z| < R $, then we have the following result

$\begin{equation*} \label{a11} \mathfrak{D}_{z}^{\lambda}\{z^{\eta-1}h(z)\} = \frac{\Gamma(\eta)}{\Gamma(\lambda+\eta)}\sum\limits_{n = 0}^{\infty}\frac{a_n(\eta)_n}{(\lambda+\eta)_n}z^n. \end{equation*}$ $

Lemma 1.1. (Srivastava and Tomovski ^[42]) Suppose that $ x > r $, $ \lambda\in(0, 1) $, $ v\in[0, 1] $, $ \Re(\eta) > 0 $ and $ \Re(\lambda) > 0 $, then we have

$\begin{equation} \mathfrak{D}_{r+}^{\lambda}[(\varrho-r)^{\eta-1}](x) = \frac{\Gamma(\eta)}{\Gamma(\eta-\lambda)}(x-r)^{\eta-\lambda-1}. \end{equation}$ $

(1.18)

The generalized multivariate M-L function (1.12) is then defined in terms of the modified Pochhammer symbol (1.7) and its different features as well as the accompanying integral operators are examined. This is driven by the aforementioned modifications of special functions.

Motivated by the above results and literature, the paper has the following structure: First, we describe and investigate a novel generalization of the multivariate M-L function using a generalized Pochhammer symbol. Secondly, we offer a few differential and fractional integral formulas for the explored multivariate M-L function. By using the new form of the multivariate M-L function, a new generalization of the fractional integral operator is introduced, and some fundamental characteristics of the operator are discussed.

2. The generalized multivariate of M-L function

We are in a position to present the generalized multivariate M-L function by utilizing the extended Pochhammer symbol in (1.7) as follows:

$\begin{equation} \varepsilon_{(a_j),b;p}^{(c_j)}(z_1,z_2,\cdots,z_j) = \sum\limits_{l_1,\cdots,l_j = 0}^{\infty}\frac{(c_1;p)_{l_1}(c_2)_{l_2}\cdots(c_j)_{l_j}}{\Gamma(a_1 l_1+a_2l_2+\cdots+c_jl_j+b)}\frac{z_1^{l_1} z_2^{l_2}\cdots z_j^{l_j}}{l_1!\cdots l_j!}, \end{equation}$ $

(2.1)

where $ a_i, b, c_i\in\mathbb{C}; \Re(a_i) > 0, \Re(b) > 0, , p\geq0 $ for $ i = 1, 2, \cdots, j $. The special case for $ a_1 = 1 $ and $ l_2 = \cdots = l_j = 0 $ in (2.1) can be reduced to extended confluent hypergeometric function (1.11) as follows:

$\begin{equation} \varepsilon_{1,b;p}^{c_1}(z_1) = \frac{1}{\Gamma(b)} {_1F_1}[(c_1; p);b; z_1] = \frac{1}{\Gamma(b)} \Phi[(c_1; p);b; z_1]. \end{equation}$ $

(2.2)

In coming results, we demonstrate some fundamental characteristics and integral representations of the generalized multivariate M-L function.

Theorem 2.1. For the multivariate M-L function defined in (2.1), the following relation holds true:

$\begin{equation} \varepsilon_{(a_j),b;p}^{(c_j)}(z_1,z_2,\cdots,z_j) = b \varepsilon_{(a_j),b+1;p}^{(c_j)}(z_1,z_2,\cdots,z_j) \end{equation}$ $

(2.3)

$\begin{equation*} +[a_1z_1\frac{d}{dz_1}+\cdots+ a_jz_j\frac{d}{dz_j}] \varepsilon_{(a_j),b+1;p}^{(c_j)}(z_1,\cdots,z_j), \end{equation*}$ $

where $ a_i, b, c_i\in\mathbb{C}; \Re(a_i) > 0, \Re(b) > 0, , p\geq0 $ for $ i = 1, 2, \cdots, j $.

Proof. From (2.1), we have

$\begin{align*} &b \varepsilon_{(a_j),b+1,p}^{(c_j)}(z_1,\cdots,z_j)+[a_1z_1\frac{d}{dz_1}+\cdots+ a_jz_j\frac{d}{dz_j}]\varepsilon_{(a_j),b+1;p}^{(c_j)}(z_1,\cdots,z_j)\\ = &b \sum\limits_{l_1,\cdots,l_j = 0}^{\infty}\frac{ (c_1,p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j+b+1)}\frac{z_1^{l_1}\cdots z_j^{l_j}}{l_1!\cdots l_j!}\\+&[a_1z_1\frac{d}{dz_1}+\cdots+ a_jz_j\frac{d}{dz_j}]\sum\limits_{l_1,\cdots,l_j = 0}^{\infty}\frac{ (c_1,p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j+b+1)}\frac{z_1^{l_1}\cdots z_j^{l_j}}{l_1!\cdots l_j!}\\ = &b \sum\limits_{l_1,\cdots,l_j = 0}^{\infty}\frac{ (c_1,p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j+b+1)}\frac{z_1^{l_1}\cdots z_j^{l_j}}{l_1!\cdots l_j!}\\+&[a_1z_1 \frac{d}{dz_1}+\cdots +a_jz_j \frac{d}{dz_j}]\sum\limits_{l_1,\cdots,l_j = 0}^{\infty}\frac{ (c_1,p)_{l_1}(c_2)_{l_2}\cdots(c_j)_{l_j}}{\Gamma(a_1 l_1+\cdots+a_jl_j+b+1)}\frac{z_1^{l_1}\cdots z_jl_j}{l_1!\cdots l_j!}\\ = &b \sum\limits_{l_1,\cdots,l_j = 0}^{\infty}\frac{ (c_1,p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j+b+1)}\frac{z_1^{l_1}\cdots z_j^{l_j}}{l_1!\cdots l_j!}\\+&\sum\limits_{l_1,\cdots,l_j = 0}^{\infty}\frac{(c_1,p)_{l_1}(c_2)_{l_2}\cdots(c_j)_{l_j} }{\Gamma(a_1 l_1+\cdots+a_jl_j+b+1)}\frac{z_1^{l_1}\cdots z_j^{l_j}}{l_1!\cdots l_j!}(a_1l_1+\cdots+a_jl_j)\\ = & \sum\limits_{l_1,\cdots,l_j = 0}^{\infty}\frac{ (c_1,p)_{l_1}(c_2)_{l_2}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j+b+1)}\frac{z_1^{l_1}\cdots z_j^{l_j}}{l_1!\cdots l_j!}(a_1l_1+\cdots+a_jl_j+b)\ \ (\mbox{using} \ \Gamma(z_1+1) = z_1 \Gamma(z_1))\\ = &\sum\limits_{l = 0}^{\infty}\frac{(c_1,p)_{l_1}(c_2)_{l_2}\cdots(c_j)_{l_j} }{ \Gamma(a_1l_1+\cdots+a_jl_j+b)}\frac{z_1^{l_1}\cdots z_j^{l_j}}{l_1!\cdots l_j!}\\ = &\varepsilon_{(a_j),b,p}^{(c_j)}(z_1,z_2,\cdots,z_j), \end{align*}$ $

which is the desired result (2.3).

Theorem 2.2. For the generalized multivariate M-L function defined in (1.12), the following relations hold true:

$\begin{equation} \big(\frac{d}{dz_1}\big)^m\cdots(\frac{d}{dz_j}\big)^m \varepsilon_{(a_j),b;p}^{(c_j)}(z_1,z_2,\cdots,z_j) = (c_1)_m\cdots(c_j)_m \varepsilon_{(a_j),b+(a_j)m;p}^{(c_j)+m}(z_1,\cdots,z_j), \end{equation}$ $

(2.4)

and

$\begin{equation} \big(\frac{d}{dz_1}\big)^m [z_1^{b-1} \varepsilon_{(a_j),b;p}^{(c_j)}(\varpi_1 z_1^{a_1},\cdots,\varpi_j z_1^{a_j}))] = z_1^{b-m-1} \varepsilon_{(a_j),b-m;p}^{(c_j)}(\varpi_1 z_1^{a_1},\cdots,\varpi_j z_1^{a_j}), \end{equation}$ $

(2.5)

where $ a_i, b, c_i\in\mathbb{C}; \Re(a_i) > 0, \Re(b) > 0, , p\geq0 $ for $ i = 1, 2, \cdots, j $, and $ \Re(b-m) > 0 $ with $ m\in\mathbb{N}. $

Proof. Differentiating (1.12) $ m $ times with respect to $ z_1, z_2, \cdots, z_j $ respectively, we get

$\begin{align*} & (\frac{d}{dz_1}\big)^m\cdots(\frac{d}{dz_j}\big)^m \varepsilon_{(a_j),b;p}^{(c_j)}(z_1,\cdots,z_j) = (\frac{d}{dz_1}\big)^m\cdots(\frac{d}{dz_j}\big)^m \sum\limits_{l_1 = l_2 = \cdots = l_j = 0}^{\infty}\frac{(c_1;p)_{l_1}(c_2)_{l_2}\cdots(c_j)_{l_j}}{\Gamma(a_1 l_1+\cdots+a_jl_j+b)}\frac{z_1^{l_1}\cdots z_j^{l_j}}{l_1!\cdots l_j!}\\ = &\sum\limits_{l_1 = \cdots = l_j = m}^{\infty}\frac{(c_1;p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j+b)}\frac{l_1!\cdots l_j!\ z_1^{l_1-m}\cdots z_j^{l_j-m}}{(l_1-m)!\cdots(l_j-m)!\ l1!\cdots l_j!}\\ & = \sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1;p)_{l_1+m}\cdots(c_j)_{l_j+m}}{\Gamma(a_1(l_1+m)+\cdots a_j(l_j+m)+b)}\frac{z_1^{l_1}\cdots z_j^{l_j}}{l_1!\cdots l_j!}\ \ (\mbox{Replacing}\ l_i \ \mbox{by}\ l_i+m) \\ & = \sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1)_m\cdots(c_j)_m\ (c_1+m;p)_{l_1}\cdots(c_j+m)_{l_j}}{\Gamma(a_1l_1+\cdots a_jl_j+b+(a_1+\cdots+a_j)m)}\frac{z_1^{l_1}\cdots z_j^{l_j}}{l_1!\cdots l_j!}. \end{align*}$ $

Now using $ (\lambda; \sigma)_{\mu+p} = (\lambda)_{\mu} (\lambda+\mu; \sigma)_p $ and $ (\lambda)_{\mu+p} = (\lambda)_{\mu} (\lambda+\mu)_p $, we get

$\begin{align*} & (\frac{d}{dz_1}\big)^m\cdots(\frac{d}{dz_j}\big)^m \varepsilon_{(a_j),b;p}^{(c_j)}(z_1,\cdots,c_j)\\ = &(c_1)_m \cdots(c_j)_m \sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1+m;p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots a_jl_j+b+(a_1+\cdots +a_j)m)}\frac{z_1^{l}\cdots z_j^{l_j}}{l_1!\cdots l_j!}\\ & = (c_1)_m\cdots (c_j)_{m} \ \varepsilon_{(a_j),b+(a_j)m;p}^{(c_j)+m}(z_1,z_2,\cdots, z_j), \end{align*}$ $

which is the desired result (2.4). Similarly, to prove (2.5), we have

$\begin{align*} &\big(\frac{d}{dz_1}\big)^m [z_1^{b-1} \varepsilon_{(a_j),b;p}^{(c_j)}(\varpi_1 z_1^{a_1},\cdots \varpi_j z_j^{a_j})]\\ = &\big(\frac{d}{dz_1}\big)^m z_1^{b-1}\sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1;p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j+b)}\frac{(\varpi_1 z_1^{a_1})^{l_1}\cdots(\varpi_jz_1^{a_j})^{l_j}}{l_1!\cdots l_j!}\\ = &\big(\frac{d}{dz_1}\big)^m \sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1;p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+ a_jl_j+b)} \frac{z_1^{b-1+a_1l_1+\cdots+a_jl_j}}{l_1!\cdots l_j!} \varpi_1^{l_1}\cdots\varpi_j^{l_j}\\ = &\sum\limits_{l_1 = \cdots l_j = 0}^{\infty}\frac{(c_1;p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j+b)} \frac{\varpi_1^{l_1}\cdots\varpi_j^{l_j}}{l_1!\cdots l_j!} \frac{(a_1l_1+\cdots+a_jl_j+b-1)!}{(a_1l_1+\cdots+a_jl_j+b-m-1)!}\ z_1^{a_1l_1+\cdots+a_jl_j+b-m-1}. \end{align*}$ $

Differentiating $ m $ times and using the relation $ l(l-1)! = l! $, we get

$\begin{align*} &\big(\frac{d}{dz_1}\big)^m [z_1^{b-1} \varepsilon_{(a_j),b;p}^{(c_j)}(\varpi_1 z_1^{a_1},\cdots \varpi_j z_j^{a_j})]\\ = &\sum\limits_{l_1 = \cdots l_j = 0}^{\infty}\frac{(c_1;p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j+b)} \frac{\Gamma(a_1l_1+\cdots+a_jl_j+b)}{\Gamma(a_1l_1+\cdots+a_jl_j+b-m)}\, \varpi_1^{l_1}\cdots\varpi_j^{l_j} \, \frac{z_1^{a_1l_1+\cdots+a_jl_j+b-1-m}}{l_1!\cdots l_j!} \\ = & z_1^{b-m-1}\sum\limits_{l_1 = \cdots l_j = 0}^{\infty}\frac{(c_1;p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j+b-m)}\frac{(\varpi_1 z_1^{a_1})^{l_1}\cdots(\varpi_1 z_1^{a_j})^{l_j}}{l_1!\cdots l_j!}\\ = &z_1^{b-m-1}\varepsilon_{(a_j),b-m;p}^{(c_j)}(\varpi_1 z_1^{a_1},\cdots,\varpi_j z_1^{a_j}). \end{align*}$ $

The proof is completed.

Corollary 2.1. The generalized multivariate M-L function has the following integral representations:

$\begin{equation*} \int_{0}^{z_1}t^{b-1}\varepsilon_{(a_j),b;p}^{(c_j)}(\varpi_1 t^{a_1},\cdots,\varpi_j t^{a_j}) d t = z_1^{b}\varepsilon_{(a_j),b+1;p}^{(c_j)}(\varpi_1 z_1^{a_1},\cdots,\varpi_jz_1^{a_j}), \end{equation*}$ $

where $ a_i, b, c_i, \varpi_i\in\mathbb{C}; \Re(a_i) > 0, \Re(b) > 0, p\geq0 $ for $ i = 1, 2, \cdots, j $.

3. Fractional integral and differential formulas of generalized M-L function

In this section, we present some fractional integration and differentiation formulas of generalized M-L function given in (2.1).

Theorem 3.1. Suppose $ x > r\, (r\in\mathbb{R_+} = [0, \infty)) $, $ a_i $, $ b $, $ c_i $, $ \varpi\in\mathbb{C} $, $ \Re(a_i) > 0 $ and $ \Re(c_i) > 0 $, $ \Re(b) > 0 $ and $ \Re(\lambda) > 0 $, then the following relations hold true:

$\begin{align} &\mathfrak{I}_{r+}^{\lambda}\left[(\varrho-r)^{b-1}\varepsilon_{(a_j),b;p}^{(c_j)}(\varpi_1(\varrho-r)^{a_1},\cdots,\varpi_j(\varrho-r)^{a_j})\right](x) \\ = &(x-r)^{\lambda+b-1} \varepsilon_{(a_i), b+\lambda;p}^{(c_i)}(\varpi_1(x-r)^{a_1},\cdots,\varpi_j(x-r)^{a_j}), \end{align}$ $

(3.1)

$\begin{align} &\mathfrak{D}_{r+}^{\lambda}\left[(\varrho-r)^{b-1}\varepsilon_{(a_i),b;p}^{( c_i)}(\varpi_1(\varrho-r)^{a_1},\cdots,\varpi_j(\varrho-r)^{a_j})\right](x) \\ = &(x-r)^{b-\lambda-1} \varepsilon_{(a_i),b-\lambda;p}^{(c_i)}(\varpi_1(x-r)^{a_1},\cdots,\varpi_j(x-r)^{a_j}) \end{align}$ $

(3.2)

and

$\begin{align} &\mathfrak{D}_{r+}^{\lambda,v}\left[(\varrho-r)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(x-r)^{a_1},\cdots,\varpi_j(x-r)^{a_j})\right](x) \\ = &(x-r)^{b-\lambda-1} \varepsilon_{(a_i),b-\lambda;p}^{(c_i)}(\varpi_1(x-r)^{a_1},\cdots,\varpi_j(x-r)^{a_j}). \end{align}$ $

(3.3)

Proof. Consider

$\begin{align*} &\mathfrak{I}_{r+}^{\lambda}\left[(\varrho-r)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(x-r)^{a_1},\cdots,\varpi_j(x-r)^{a_j})\right](x)\\ = &\frac{1}{\Gamma(\lambda)}\int_{r}^{x}\frac{(x-r)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(\varrho-r)^{a_1},\cdots,\varpi_1(\varrho-r)^{a_j})} {(x-\varrho)^{1-\lambda}}d\varrho\\ = &\frac{1}{\Gamma(\lambda)}\sum\limits_{n = 0}^{\infty}\frac{(c_1;p,v)_{l_1}\cdots(c_j)_{l_n}\varpi^{l_1}\cdots\varpi^{l_j}}{\Gamma(a_1 l_1+\cdots+a_jl_j+b)l_1!\cdots l_j!}\int_{r}^{x}(\varrho-r)^{b+a_1 l_1+\cdots+a_jl_j-1}(x-\varrho)^{\lambda-1}d\varrho\\ = &\sum\limits_{n = 0}^{\infty}\frac{(c_1;p,v)_{l_1}\cdots(c_j)_{l_n}\varpi^{l_1}\cdots\varpi^{l_j}}{\Gamma(a_1 l_1+\cdots+a_jl_j+b)l_1!\cdots l_j!} \left( \mathfrak{I}_{r+}^{\lambda}[(\varrho-r)^{b+a_1 l_1+\cdots+a_jl_j-1}] \right). \end{align*}$ $

By the use of (1.17), we have

$\begin{align*} &\mathfrak{I}_{r+}^{\lambda}\left[(\varrho-r)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(x-r)^{a_1},\cdots,\varpi_j(x-r)^{a_j})\right](x)\\ = &\sum\limits_{n = 0}^{\infty}\frac{(c_1;p,v)_{l_1}\cdots(c_j)_{l_n}\varpi^{l_1}\cdots\varpi^{l_j}}{\Gamma(a_1 l_1+\cdots+a_jl_j+b)l_1!\cdots l_j!} (x-r)^{b+\lambda+a_1l_1+\cdots+a_jl_j-1}. \frac{\Gamma(a_1l_1+\cdots+a_jl_j+b)}{\Gamma(a_1l_1+\cdots+a_jl_j+b+\lambda)}\\ = &(x-r)^{b+\lambda-1}\sum\limits_{n = 0}^{\infty}\frac{(c_1;p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j+b+\lambda)} \frac{[\varpi_1^{l_1}(x-r)^{a_1l_1}\cdots\varpi_j^{l_j}(x-r)^{a_jl_j}]}{l_1!\cdots l_j!}\\ = &(x-r)^{b+\lambda-1}\varepsilon_{(a_i),b+\lambda;p}^{(c_i)}(\varpi_1(x-r)^{a_1},\cdots,\varpi_j(x-r)^{a_j}), \end{align*}$ $

which gives the proof of (3.1).

Next, we have

$\begin{align*} &\mathfrak{D}_{r+}^{\lambda}\left[(\varrho-r)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(\varrho-r)^{a_1},\cdots,\varpi_j(\varrho-r)^{a_j})\right]\\ = &(\frac{d}{dx})^n \left\{ \mathfrak{I}_{r+}^{n-\lambda}[(\varrho-r)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(\varrho-r)^{a_1},\cdots,\varpi_j(\varrho-r)^{a_j})] \right\}, \end{align*}$ $

which on using (3.1) takes the following form:

$\begin{align*} &\mathfrak{D}_{r+}^{\lambda}\left[(\varrho-r)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(\varrho-r)^{a_1},\cdots,\varpi_j(\varrho-r)^{a_j})\right] \\ = &(\frac{d}{dx})^n \left\{ (x-r)^{b-\lambda+n-1}\varepsilon_{(a_i),b-\lambda+n;p}^{(c_i)}(\varpi_1(x-r)^{a_1},\cdots,\varpi_j(x-r)^{a_j}) \right\}. \end{align*}$ $

Applying (2.5), we get

$\begin{align*} &\mathfrak{D}_{r+}^{\lambda}\left[(\varrho-r)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(x-r)^{a_1},\cdots,\varpi_j(x-r)^{a_j})\right](x) \\ = &\left\{ (x-r)^{\eta-\lambda-1}\varepsilon_{(a_i),b-\lambda;p}^{(c_i)}(\varpi_1(x-r)^{a_1},\cdots, \varpi_j(x-r)^{a_j}) \right\}, \end{align*}$ $

which gives the proof of (3.2).

To obtain (3.3), we have

$\begin{align*} &\left( \mathfrak{D}_{r+}^{\lambda,v}\left[(\varrho-r)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(\varrho-r)^{a_1},\cdots,\varpi_j(\varrho-r)^{a_j})\right] \right)(x)\\ = &\left( \mathfrak{D}_{r+}^{\lambda,v}\left[\sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1;p,v)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j +b)}\frac{\varpi^{l_1}\cdots\varpi^{l_j}}{l_1!\cdots l_j!}(\varrho-r)^{a_1 l_1+\cdots+a_jl_j+b-1}\right] \right)(x)\\ = &\sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1;p,v)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j +b)}\frac{\varpi^{l_1}\cdots\varpi^{l_j}}{l_1!\cdots l_j!}\\ \times&\left( \mathfrak{D}_{r+}^{\lambda,v}[(\varrho-r)^{a_1 l_1+\cdots+a_jl_j+b-1}] \right)(x). \end{align*}$ $

By applying (1.18), we get

$\begin{align*} &\left( \mathfrak{D}_{r+}^{\lambda,v}[(\varrho-r)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(\varrho-r)^{a_1},\cdots,\varpi_j(\varrho-r)^{a_j})] \right)(x)\\ = &\sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1;p,v)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j +b)}\frac{\varpi^{l_1}\cdots\varpi^{l_j}}{l_1!\cdots l_j!}\\ \times&\frac{\Gamma(a_1l_1+\cdots+ a_jl_j+b)}{\Gamma(a_1l_1+\cdots+a_jl_j +b-\lambda)}(x-r)^{a_1l_1+ \cdots+a_jl_j+b-\lambda-1}\\ = &(x-r)^{b-\lambda-1}\sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1;p,v)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j +b-\lambda)}\frac{\varpi^{l_1}(x-r)^{a_1}\cdots\varpi^{l_j}(x-r)^{a_j}}{l_1!\cdots l_j!}\\ = &(x-r)^{b-\lambda-1}\varepsilon_{(a_i),b-\lambda;p}^{(c_i)}(\varpi^{l_1}(x-r)^{a_1},\cdots,\varpi^{l_j}(x-r)^{a_j}), \end{align*}$ $

which completes the required proof.

Remark 3.1. Applying Theorem 3.1 for $ p = 0 $, then we obtain the result presented in ^[34].

4. Generalization of fractional integral and its properties

In this section, we define a fractional integral involving newly defined multivariate M-L function and discuss its properties.

Definition 4.1. Let $ b, a_i, c_i, \varpi_i\in\mathbb{C} $, $ \Re(c_i) > 0 $, $ \Re(a_i) > 0 $ and $ \Re(b) > 0 $ and $ h\in L(r, s) $. Then the generalized left and right sided fractional integrals are defined by

$\begin{align} &\left(\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}h\right)(x) = \int_{r}^{x}(x-\varrho)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)} (\varpi_1(x-\varrho)^{a_1},\cdots,\varpi_j(x-\varrho)^{a_j})h(\varrho)d\varrho, (x > r) \end{align}$ $

(4.1)

and

$\begin{align} &\left(\mathfrak{R}_{s-;(a_i),b;p}^{(\varpi_i);(c_i)}h\right)(x) = \int_{x}^{s}(\varrho-x)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)} (\varpi_1(\varrho-x)^{a_1},\cdots,\varpi_j(\varrho-x)^{a_j})h(\varrho)d\varrho, (x < s), \end{align}$ $

(4.2)

respectively.

Remark 4.1. If we consider $ p = 0 $, then the operators defined in (4.1) and (4.2) will take the form defined earlier by ^[34]. Similarly, if we consider $ p = 0 $ and $ j = 1 $, then the operators defined in (4.1) and (4.2) will take the form defined by ^[22]. If we take $ j = 1 $, then the work done in this paper will lead to the work presented by ^[28]. Also, if we consider one of $ \varpi_i = 0 $, for $ i = 1, 2, \cdots, j $, then the operators defined in (4.1) and (4.2) will take the form of the classical operators.

Next, we prove the following properties of integral operator defined in (4.1).

Theorem 4.1. Suppose that $ b, a_i, \lambda, c_i, \varpi_i\in\mathbb{C} $, $ \Re(a_i) > 0 $, $ \Re(b) > 0 $, $ \Re(\lambda) > 0 $, $ p\geq0 $ and $ \Re(c_i) > 0 $ for $ i = 1, 2, \cdots, j $, then the following result holds true:

$\begin{align*} \left(\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}[(\varrho-r)^{\lambda-1}] \right) (x) = (x-r)^{\lambda+b-1}\Gamma(\lambda)\varepsilon_{(a_i),b+\lambda}^{(c_i);p}(\varpi_1(x-r)^{a_1},\cdots,\varpi_j(x-r)^{a_j}). \end{align*}$ $

Proof. By the use of definition (4.1), we have

$\begin{equation*} \left( \mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}h \right)(x) = \int_{r}^{x}(x-\varrho)^{b-1}\varepsilon_{(a_i),b}^{(c_i)}(\varpi_1(x-\varrho)^{a_1},\cdots,\varpi_j(x-\varrho)^{a_j})h(\varrho)d\varrho. \end{equation*}$ $

Therefore, we get

$\begin{align*} &\left( \mathfrak{R}_{r+;(a_i), b;p}^{(\varpi_i);(c_i)}[(\varrho-r)^{\lambda-1}] \right) (x) = \int_{r}^{x}(x-\varrho)^{b-1}(\varrho-r)^{\lambda-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(x-\varrho)^{a_1},\cdots,\varpi_j(x-\varrho)^{a_j})d\varrho\\ = &\sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1;p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j+b))}\frac{\varpi_1^{l_1}\cdots\varpi_j^{l_j}} {l_1!\cdots l_j!} \left( \int_{r}^{x}(\varrho-r)^{\lambda-1}(x-\varrho)^{\lambda+a_1l_1+ \cdots+a_jl_j-1}d\varrho \right)\\ = &\sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1;p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j+b))}\frac{\varpi_1^{l_1}\cdots\varpi_j^{l_j}} {l_1!\cdots l_j!} \mathfrak{I}_{r+}^{a_1l_1+\cdots+a_jl_j+b}[(\varrho-r)^{\lambda-1}]\\ & = (x-r)^{b+\lambda-1} \sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1;p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j+b))} \frac{[\varpi_1(x-r)^{a_1l_1}\cdots\varpi_j(x-r)^{a_jl_j}]}{l_1!\cdots l_j!}\\ \times& \frac{\Gamma(\lambda)\Gamma(a_1l_1+\cdots+a_jl_j +b)}{\Gamma(a_1l_1+\cdots+a_jl_j+b+\lambda)}\\ = &(x-r)^{b+\lambda-1}\Gamma(\lambda)\varepsilon_{(a_i),b+\lambda;p}^{(c_i)}(\varpi_1(x-r)^{a_1},\cdots,\varpi_j(x-r)^{a_j}), \end{align*}$ $

which gives the desired proof.

Theorem 4.2. Suppose that $ c_i, a_i, b, \varpi_i \in\mathbb{C} $, $ \Re(a_i) > 0 $, $ \Re(b) > 0, $ $ p\geq0 $ for $ i = 1, 2, \cdots, j $, then the following result holds true:

$\begin{equation*} \|\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}\Phi\|_1\leq K\|\Phi\|_1. \end{equation*}$ $

Where

$\begin{align*} &K: = (s-r)^{\mathfrak{Re}(b)}\sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{|(c_1;p)_{l_1}\cdots(c_j)_{l_j}|}{\Gamma(a_1l_1+\cdots+a_jl_j +b)(\Re(b)+\Re(a_1)l_1+\cdots+\Re(a_j)l_j)}\\ \times&\frac{|\varpi_1^{l_1}(s-r)^{a_1l_1}\cdots\varpi_j^{l_j}(s-r)^{a_jl_j}|}{l_1!\cdots l_j!}. \end{align*}$ $

Proof. By the use of (1.13) and (4.1) and by interchanging integration and summation order, we have

$\begin{align*} \|\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i) }\Phi\|_1& = \int\limits_{r}^{s}|\int_{r}^{x}(x-\varrho)^{b-1} \varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(x-\varrho)^{a_1},\cdots,\varpi_j(x-\varrho)^{a_j}) \Phi(\varrho)d\varrho|dx\\ \leq&\int_{r}^{s}\left[ \int_{\varrho}^{x} (x-\varrho)^{\Re(b)-1}|\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(x-\varrho)^{a_1},\cdots,\varpi_j(x-\varrho)^{a_j})|dx \right]|\Phi(\varrho)|d\varrho\\ = &\int_{r}^{s}\left[ \int_{0}^{x-\varrho} u^{\Re(b)-1}|\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1 u^{a_1},\cdots,\varpi_j u^{a_j})|du \right]|\Phi(\varrho)|d\varrho, \end{align*}$ $

by setting $ u = x-\varrho. $ After simplification, we obtain

$\begin{align*} &\|\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}\Phi\|_1\\ \leq&\int_{r}^{s} \Bigr[ \sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{|(c_1;p)_{l_1}\cdots(c_j)_{l_j}|}{\Gamma(a_1l_1+\cdots+a_jl_j+b)}\frac{|\varpi_1^{a_1}\cdots\varpi_j^{l_j}|} {l_1!\cdots l_j!}\\ \times& \left( \frac{(u)^{\Re(b)+\Re(a_1)l_1+\cdots+\Re(a_j)l_j}}{(\Re(b)+\Re(a_1)l_1+\cdots+\Re(a_j)l_j)} \right)_{0}^{s-r} \Bigr] |\Phi(\varrho)|d\varrho. \end{align*}$ $

It follows that

$\begin{align*} &\|\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}\Phi\|_1\leq \Bigr\{ (s-r)^{\Re(b)} \sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{|(c_1;p)_{l_1}\cdots(c_j)_{l_j}|}{\Gamma(a_1l_1+\cdots+a_jl_j +b)(\Re(b)+\Re(a_1)l_1+\cdots+\Re(a_j)l_j)}\\ \times&\frac{|\varpi_1^{l_1}(s-r)^{a_1l_1}\cdots\varpi_j^{l_j}(s-r)^{a_jl_j}|}{l_1!\cdots l_j!} \Bigr\} \int\limits_{r}^{s}|\Phi(\varrho)|d\varrho\\ = &K||\Phi||_1, \end{align*}$ $

where

$\begin{align*} &K = (s-r)^{\mathfrak{Re}(b)}\sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{|(c_1;p)_{l_1}\cdots(c_j)_{l_j}|}{\Gamma(a_1l_1+\cdots+a_jl_j +b)(\Re(b)+\Re(a_1)l_1+\cdots+\Re(a_j)l_j)}\\ \times&\frac{|\varpi_1^{l_1}(s-r)^{a_1l_1}\cdots\varpi_j^{l_j}(s-r)^{a_jl_j}|}{l_1!\cdots l_j!}, \end{align*}$ $

which gives the desired result.

Corollary 4.1. If we take $ a_i, b, c_i, \varpi_i \in \mathbb{C} $, $ \Re(a_i) > 0 $, $ \Re(b) > 0 $, $ \Re(c_i) > 0 $ with $ i = 1, 2, \cdots, j $, then the following result holds true:

$\begin{align*} \left(\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}1\right)(x) = (x-r)^{b}\varepsilon_{(a_i),b+1;p}^{(c_i)}(\varpi_1(x-r)^{a_1},\cdots,\varpi_j(x-r)^{a_j}). \end{align*}$ $

Proof. Consider

$\begin{align*} &\left(\mathfrak{R}_{r+;(a_i),b}^{(\varpi_i);(c_i)}1\right)(x) = \int_{r}^x(x-\varrho)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(x-\varrho)^{a_1},\cdots,\varpi_j(x-r)^{a_j})d\varrho\\ = &\int_{r}^x(x-\varrho)^{b-1}\left(\sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1;p)_{l_1}\cdots(c_j)_{l_j}\varpi_1^{l_1}(x-\varrho)^{a_1l_1}\cdots\varpi_j^{l_j}(x-\varrho)^{a_jl_j}} {\Gamma(a_1l_1+\cdots+a_jl_j+b)l_1!\cdots l_j!}\right)d\varrho. \end{align*}$ $

It follows that

$\begin{align*} \left(\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}1\right)(x) & = \sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1;p)_{l_1}\cdots(c_j)_{l_j}\varpi_1^{l_1}\cdots\varpi_j^{l_j}} {\Gamma(a_1l_1+\cdots+a_jl_j+b)l_1!\cdots l_j!} \int_{r}^x(x-\varrho)^{b+a_1l_1+\cdots+a_jl_j-1}d\varrho\\ = &(x-r)^{b}\sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1;p)_{l_1}\cdots(c_j)_{l_j}\varpi_1^{l_1}(x-r)^{a_1l_1}\cdots\varpi_j^{l_j}(x-r)^{a_jl_j}} {\Gamma(a_1l_1+\cdots+a_jl_j+b)(a_1l_1+\cdots+a_jl_j+b)l_1!\cdots l_j!}\\ = &(x-r)^{b}\,\varepsilon_{(a_i),b+1;p}^{(c_i)}(\varpi_1(x-r)^{a_1},\cdots,\varpi_j(x-r)^{a_j}), \end{align*}$ $

which gives the desired proof.

Theorem 4.3. The generalized fractional operator can be represented in term of Riemann–Liouville fractional integrals for $ c_i $, $ a_i $, $ b $, $ \varpi_i\in\mathbb{C} $ with $ \Re(a_i) > 0 $, $ \Re(b) > 0 $, $ \Re(c_i) > 0 $ for $ i = 1, 2, \cdots, j $, $ p\geq0 $ and $ x > r $ as follows:

$\begin{align*} \left(\mathfrak{R}_{r+;(a_i),b}^{(\varpi_i);(c_i)}h\right)(x) = \sum\limits_{l_1 = \cdots = l_j = 0}^\infty \frac{\Gamma(c_1+l_1;p)(c_2)_{l_2}\cdots(c_j)_{l_j}\varpi_1^{a_1}\cdots\varpi_j^{a_j}} {\Gamma(c_1)l_1!\cdots l_j!}\mathfrak{I}_{r+}^{a_1l_1+\cdots+a_jl_j +b}h(x). \end{align*}$ $

Proof. By utilizing (2.1) in (4.1) and then interchanging the order of summation and integration, we have

$\begin{align*} \left(\mathfrak{R}_{r+;(a_i),b}^{(\varpi_i);(c_i)}h\right)(x)& = \int_{r}^{x}(x-\varrho)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)} (\varpi_1(x-\varrho)^{a_1},\cdots,\varpi_j(x-\varrho)^{a_j})h(\varrho)d\varrho\\ = &\int_{r}^{x}(x-\varrho)^{b-1}\sum\limits_{l_1 = \cdots = l_j = 0}^\infty\frac{\Gamma(c_1+l_1;p)(c_2)_{l_2}\cdots(c_j)_{l_j}\varpi_1^{l_1} (x-\varrho)^{a_1l_1}\cdots\varpi_j^{l_j} (x-\varrho)^{a_jl_j}} {\Gamma(c_1)\Gamma(a_1l_1+\cdots+a_jl_j+b)l_1!\cdots l_j!}h(\varrho)d\varrho\\ = &\sum\limits_{l_1 = \cdots = l_j = 0}^\infty\frac{\Gamma(c_1+l_1;p)(c_2)_{l_2}\cdots(c_j)_{l_j}\varpi_1^{a_1l_1}\cdots\varpi_j^{a_jl_j} }{\Gamma(c_1)l_1!\cdots l_j!} \frac{1}{\Gamma(a_1l_1+\cdots+a_jl_j +b)}\\ \times&\int_{r}^{x}(x-\varrho)^{a_1l_1+\cdots+a_jl_j+b-1}h(\varrho)d\varrho\\ = &\sum\limits_{l_1 = \cdots = l_j = 0}^\infty\frac{\Gamma(c_1+l_1;p)(c_2)_{l_2}\cdots(c_j)_{l_j}\varpi_1^{a_1l_1}\cdots\varpi_j^{a_jl_j} }{\Gamma(c_1)l_1!\cdots l_j!}\mathfrak{I}_{r+}^{a_1l_1 +\cdots+a_jl_j+b}h(x), \end{align*}$ $

which gives the desired proof.

Theorem 4.4. For $ \lambda $, $ c_i $, $ a_i $, $ b $, $ \varpi_i\in\mathbb{C} $ with $ \Re(a_i) > 0 $, $ \Re(b) > 0 $, $ \Re(c_i) > 0 $, $ \Re(\lambda) > 0 $, for $ i = 1, 2, \cdots, j $, $ p\geq0 $ and $ x > r $, then the following result holds true:

$\begin{equation} (\mathfrak{I}_{r+}^{\lambda}[\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}h])(x) = (\mathfrak{R}_{r+;(a_i),b+\lambda}^{(\varpi_i);(c_i)}h)(x) = (\mathfrak{R}_{r+;(a_i),b}^{(\varpi_i);(c_i)}[\mathfrak{I}_{r+}^{\lambda}h])(x), \end{equation}$ $

(4.3)

where $ h\in L(r, s) $.

Proof. By employing (1.14) and (4.1), we have

$\begin{align*} &(\mathfrak{I}_{r+}^{\lambda}[\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}h])(x)\\ = &\frac{1}{\Gamma(\lambda)}\int_{r}^{x}\frac{[(\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}h)(\varrho)]}{(x-\varrho) ^{1-\lambda}}d\varrho\\ = &\frac{1}{\Gamma(\lambda)}\int_{r}^{x}(x-\varrho)^{\lambda-1} \left[ \int_{r}^{\varrho} (\varrho-u)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(\varrho-u)^{a_1},\cdots,\varpi_j(\varrho-u)^{a_j})h(u)du \right]d\varrho. \end{align*}$ $

It follows that

$\begin{align*} &(\mathfrak{I}_{r+}^{\lambda}[\mathfrak{R}_{r+;(a_i),b}^{(\varpi_i);(c_i)}h])(x) \\ = &\int_{r}^{x} \left[ \frac{1}{\Gamma(\lambda)}\int_{u}^{x} (x-\varrho)^{\lambda-1}(\varrho-u)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i) }(\varpi_1(\varrho-u)^{a_1},\cdots,\varpi_j(\varrho-u)^{a_j})d\varrho \right]h(u)du. \end{align*}$ $

By considering $ \varrho-u = \theta $, we get

$\begin{align*} \begin{aligned} &(\mathfrak{I}_{r+}^{\lambda}[\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}h])(x)\\ = &\int_{r}^{x} \left[ \frac{1}{\Gamma(\lambda)}\int_{0}^{x-u} (x-u-\theta)^{\lambda-1}\theta^{b-1}\varepsilon_{(a_i),b;p}^{(c_i) } (\varpi_1\theta^{a_1},\cdots,\varpi_j\theta^{a_j})d\theta \right]h(u)du\\ = &\int_{r}^{x} \left[ \frac{1}{\Gamma(\lambda)}\int_{0}^{x-u} \frac{\theta^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1\theta^{a_1},\cdots,\varpi_j\theta^{a_j})}{(x-u-\theta)^{1-\lambda}}d\theta \right]h(u)du.\end{aligned} \end{align*}$ $

Hence, from (1.14) and applying (3.1), we obtain

$\begin{align*} &(\mathfrak{I}_{r+}^{\lambda}[\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}h])(x)\\ = &\int_{r}^{x}\left[ \theta^{\lambda+b-1}\varepsilon_{(a_i),b+\lambda;p}^{(c_i)}(\varpi_1\theta^{a_1},\cdots,\varpi_j\theta^{a_j}) \right]h(u)du\\ = &\int_{r}^{x} (x-u)^{\lambda+b-1}\varepsilon_{(a_i),b+\lambda}^{(c_i)}(\varpi_1(x-u)^{a_1},\cdots,\varpi_j(x-u)^{a_j})h(u)du. \end{align*}$ $

Thus, we have

$\begin{equation} (\mathfrak{I}_{r+}^{\lambda}[\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}h])(x) = \left( \mathfrak{R}_{r+;(a_i),b+\lambda}^{(\varpi_i);(c_i)}h \right)(x). \end{equation}$ $

(4.4)

Next, consider the right hand side of (4.3) and employing (4.1) to derive the second part, we have

$\begin{align*} &(\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}[\mathfrak{I}_{r+}^{\lambda}h])(x)\\ = &\int_{r}^{x}(x-\varrho)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(x-\varrho)^{a_1},\cdots,\varpi_j(x-\varrho)^{a_j}) [\mathfrak{I}_{r+}^{\lambda}h] (\varrho)d\varrho\\ = &\int_{r}^{x}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(x-\varrho)^{a_1},\cdots,\varpi_j(x-\varrho)^{a_j})\left( \frac{1}{\Gamma(\lambda)}\int\limits_{r}^{\varrho}\frac{h(u)}{(\varrho-u)^{1-\lambda}}du \right)d\varrho. \end{align*}$ $

It follows that

$\begin{align*} &(\mathfrak{R}_{r+;(a_i),b}^{(\varpi_i);(c_i)}[\mathfrak{I}_{r+}^{\lambda}h])(x)\\ = &\int\limits_{r}^{x}\frac{1}{\Gamma(\lambda)} \left[ \int_{u}^{x}(x-\varrho)^{b-1}(\varrho-u)^{\lambda-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(x-\varrho)^{a_1},\cdots,\varpi_j(x-\varrho)^{a_j})d\varrho \right] h(u)du. \end{align*}$ $

By setting $ x-\varrho = \theta $, we get

$\begin{align*} &(\mathfrak{R}_{r+;(a_i),b}^{(\varpi_i);(c_i)}[\mathfrak{I}_{r+}^{\lambda}h])(x)\\ = &\int_{r}^{x}\frac{1}{\Gamma(\lambda)} \left[ \int_{x-u}^{0}\theta^{b-1}(x-\theta-u)^{\lambda-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1\theta^{a_1},\cdots,\varpi_j\theta^{a_j}) (-d\theta) \right] h(u)du\\ = &\int\limits_{r}^{x}\frac{1}{\Gamma(\lambda)} \left[ \int_{0}^{x-u}\theta^{b-1}(x-\theta-u)^{\lambda-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1\theta^{a_1},\cdots,\varpi_j\theta^{a_j})d\theta \right] h(u)du. \end{align*}$ $

Further, by using (1.14) and applying (3.1), we obtain

$\begin{equation} (\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}[\mathfrak{I}_{r+}^{\lambda}h])(x) = \left( \mathfrak{R}_{r+;(a_i),b+\lambda}^{(\varpi_i);(c_i)}h \right)(x). \end{equation}$ $

(4.5)

Thus, (4.4) and (4.5) gives the desired proof.

5. Conclusions

Nowadays, the theories are developed very rapidly. The scientists are introducing more advanced and generalized forms of the classical ones. In this present study, we introduced a generalized form of the multivariate M-L function (2.1) by employing the generalized Pochhammer symbol and its properties. By using this more extended form of M-L, we introduced a fractional integral operator and studied some of the basic properties of this operator. The special cases of the main results are if we take $ p = 0 $, then the operators defined in (4.1) and (4.2) will reduce to the work done by ^[34]. Similarly, if we take $ j = 1 $ and $ p = 0 $, then the operators defined in (4.1) and (4.2) will lead to the work done by ^[22]. If we take $ j = 1 $, then the work done in this paper will lead to the work presented by ^[28]. Moreover, if we consider one of $ \varpi_i = 0 $, for $ i = 1, 2, \cdots, j $, then the operators defined in (4.1) and (4.2) will reduce to the classical R-L operators. We believe that our proposed operator will be more applicable in the fields of fractional integral inequalities and operator theory.

Acknowledgments

The author T. Abdeljawad would like to thank Prince Sultan University for supporting through TAS research lab. Manar A. Alqudah: Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2023R14), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Conflict of interest

The authors declare no conflict of interest.

References

[1]	Bedrosian E (1963) A product theorem for Hilbert transforms.Proc IEEE 51: 868-869.
[2]	Bochner S (1994) Group invariance of Cauchy formula in several variables.Annals Math 45: 686-707.
[3]	Brown J (1974) Analytic signals and product theorems for Hilbert transforms.IEEE Trans Circuits and Systems CAS-21: 790-792.
[4]	Butzer P, Nessel R (1971) Fourier analysis and approximation.Birkhauser Verlag, Bassel and New York:Academic Press. .
[5]	Cerejeiras P, Chen Q and K$\rm \ddot{a}$ehler U (2010) A necessary and sufficient condition for a Bedrosian identity.Math Methods Appl Sci 33: 493-500.
[6]	Cerejeiras P, Chen Q and K$\rm \ddot{a}$ehler U (2012) Bedrosian identity in Blaschke product case.Complex Anal Oper Theory 6: 275-300.
[7]	Chen Q, Huang N, Riemenschneider S and Xu Y (2006) A B-spline approach for empirical mode decompositions.Adv Comput Math 24: 171-195.
[8]	Chen Q and Micchelli C (2012) The Bedrosian identity for functions analytic in a neighborhood of the unit circle.Complex Anal Oper Theory 6: 781-798.
[9]	Cohen L (1995) Time-Frequency Analysis: Theory and Applications.Englewood Cliffs, NJ: Prentice-Hall. .
[10]	Deng G (2010) Complex Analysis.Beijing Norm University Press (In Chinese) .
[11]	Donoghue W (1969) Distributions and Fourier Transforms.New York: Academic Press .
[12]	Fei M (2014) Hp space on Tube.PHD thesis of Beijing Norm University (In press) .
[13]	Gabor D (1946) Theory of communication.J Inst Electr Eng 93: 429-457.
[14]	Gasquet C andWitomski P (1999) Fourier Analysis and Applications..New York: Springer-Verlag .
[15]	Huang N, Shen Z, Long S, Wu M, Shih E, Zheng Q, Tung C and Liu H (1998) The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis.Proc R Soc Lond Ser A Math Phys Eng Sci 454: 903-995.
[16]	Huang N and Attoh-Okine N (2005) The Hilbert-Huang Transform in Engineering.CRC/Taylor & Francis, Boca Raton. .
[17]	Huang N and Shen S (2005) Hilbert-Huang transform and its Applications..London: World Scientific .
[18]	Huang N, Shen Z and Long S (1999) A new view of nonlinear water waves: the Hilbert spectrum.Annu Rev Fluid Mech 31: 417-457.
[19]	Lin R and Zhang H (2014) Existence of the Bedrosian identity for singular integral operators.1407-0861.
[20]	Luszczki Z and Ziele$\acute{z}$ny Z (1961) Distributionen der R$\ddot{a}$ume D0 Lp als Randverteilungen analytischer Funktionen.Colloq Math 8: 125-131.
[21]	Nuttall A and Bedrosian E (1966) On the quadrature approximation for the Hilbert transform of modulated signals.Proceedings of the IEEE 54: 1458-1459.
[22]	Oppenheim A and Lim J (1981) The importance of phase in signal.Proc IEEE 69: 529-541.
[23]	Pandey J (1996) The Hibert transform of schwartz distributions and applications..New York: John Wiley & Sons, Inc. .
[24]	Picibono B (1997) On instantaneous amplitude and phase of signals.IEEE Trans Signal Process 45: 552-560.
[25]	Qian T, Chen Q and Li L (2005) Analytic unit quadrature signals with nonlinear phase.Phys D 203: 80-87.
[26]	Qian T, Xu Y, Yan D, Yan L and Yu B (2009) Fourier spectrum characterization of Hardy spaces and applications.Proc Am Math Soc 137: 971-980.
[27]	Schwartz L (1978) Theorie des distributions..Hermann, Paris. .
[28]	Stein E and Weiss G (1971) Introduction to Fourier analysis on Euclidean space..Princeton, New Jersey: Princeton University Press. .
[29]	Stein E, Weiss G and Weiss M (1964) $H^{p}$ class of holomophic functions in tube domains..Proc Nat Acad Sci USA 52: 1035-1039.
[30]	Tan L, Shen L and Yang L (2010) Rational orthogonal bases satisfying the Bedrosian identity..Adv Comput Math 33: 285-303.
[31]	Tan L, Yang L and Huang D (2009) Necessary and sufficient conditions for the Bedrosian identity..J Integral Equations Appl 21: 77-94.
[32]	Tan L, Yang L, Huang D (2010) Construction of periodic analytic signals satisfying the circular Bedrosian identity..IMA J. Appl. Math 75: 246-256.
[33]	Venouziou M and Zhang H (2008) Characterzing the Hilbert transform by the Bedrosian theorem..J Math anal Appl 338: 1477-1481.
[34]	Wang S (2009) Simple proofs of the Bedrosian equality for the Hilbert transform..Sci China Ser A 52: 507-510.
[35]	Xu Y and Yan D (2006) The Bedrosian identity for the Hilbert transform of product functions..Proc Am Math Soc 134: 2719-2728.
[36]	Yang L and Zhang H (2008) The Bedrosian identity for $H^p$ functions.J Math Anal Appl 345: 975-984.
[37]	Yu B and Zhang H (2008) The Bedrosian identity and homogeneous semi-convolution equations..J Integral Equations Appl 20: 527-568.
[38]	Zhang H (2014) Multidimensional analytic signals and the Bedrosian identity..Integral Equations Operator Theory 78: 301-321.

Reader Comments

Your name:*

Email:*
© 2016 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

AIMS Mathematics

1.8 3.1

Metrics

Article views(5737) PDF downloads(1489) Cited by(0)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

AIMS Mathematics

The Bedrosian Identity for L^p Function and the Hardy Space on Tube

Related Papers:

Abstract

1. Introduction

2. The generalized multivariate of M-L function

3. Fractional integral and differential formulas of generalized M-L function

4. Generalization of fractional integral and its properties

5. Conclusions

Acknowledgments

Conflict of interest

References

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Other Articles By Authors

Catalog

Abstract

1. Introduction

2. The generalized multivariate of M-L function

3. Fractional integral and differential formulas of generalized M-L function

4. Generalization of fractional integral and its properties

5. Conclusions

Acknowledgments

Conflict of interest

References

AIMS Mathematics

The Bedrosian Identity for Lp Function and the Hardy Space on Tube

Related Papers:

Abstract

1. Introduction

2. The generalized multivariate of M-L function

3. Fractional integral and differential formulas of generalized M-L function

4. Generalization of fractional integral and its properties

5. Conclusions

Acknowledgments

Conflict of interest

References

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog

Abstract

1. Introduction

2. The generalized multivariate of M-L function

3. Fractional integral and differential formulas of generalized M-L function

4. Generalization of fractional integral and its properties

5. Conclusions

Acknowledgments

Conflict of interest

References

The Bedrosian Identity for L^p Function and the Hardy Space on Tube