Car indoor air pollution by volatile organic compounds and aldehydes in Japan

Masahiro Tokumura; Rurika Hatayama; Kouichi Tatsu; Toshiyuki Naito; Tetsuya Takeda; Mohammad Raknuzzaman; Md. Habibullah-Al-Mamun; Shigeki Masunaga; Masahiro Tokumura; Rurika Hatayama; Kouichi Tatsu; Toshiyuki Naito; Tetsuya Takeda; Mohammad Raknuzzaman; Md. Habibullah-Al-Mamun; Shigeki Masunaga

doi:10.3934/environsci.2016.3.362

AIMS Environmental Science

2016, Volume 3, Issue 3: 362-381. doi: 10.3934/environsci.2016.3.362

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Research article

Car indoor air pollution by volatile organic compounds and aldehydes in Japan

1.
Faculty of Environment and Information Sciences, Yokohama National University, Kanagawa 240-8501 Japan
2.
Graduate School of Environment and Information Sciences, Yokohama National University, Kanagawa 240-8501 Japan
3.
Isuzu Advanced Engineering Center, LTD., Kanagawa 252-8501 Japan
4.
Department of Fisheries, University of Dhaka, Dhaka 1000, Bangladesh

Received: 13 April 2016 Accepted: 16 June 2016 Published: 21 June 2016

Fifty-five organic substances including volatile organic compounds (VOCs) and aldehydes present in indoor air were measured from 24 car cabins in Japan. A screening-level risk assessment was also performed. Acetaldehyde (3.81–36.0 μg/m³), formaldehyde (3.26–26.7 μg/m³), n-tetradecane (below the method quantification limit (<MQL) to 47.7 μg/m³), toluene (4.23–78.3 μg/m³), and n-undecane (<MQL to 6.24 μg/m³) concentrations exceeded method detection limits in all the investigated car cabins. Ratios between indoor and outdoor concentrations revealed that most organic compounds originated from the car interior materials. Total volatile organic compound (TVOC) concentrations in 14 car cabins (58% of all car cabins) exceeded the advisable values established by the Ministry of Health, Labour and Welfare of Japan (400 μg/m³). The highest TVOC concentration (1136 μg/m³) was found in a new car (only one month since its purchase date). Nevertheless, TVOC concentrations exceeded the advisable value even for cars purchased over 10 years ago. Hazard quotients (HQs) for formaldehyde obtained using measured median and highest concentrations in both exposure scenarios for occupational use (residential time in a car cabin was assumed to be 8 h) were higher than that expected, a threshold indicative of potential adverse effects. Under the same exposure scenarios, HQ values for all other organic compounds remained below this threshold.

Keywords:

Citation: Masahiro Tokumura, Rurika Hatayama, Kouichi Tatsu, Toshiyuki Naito, Tetsuya Takeda, Mohammad Raknuzzaman, Md. Habibullah-Al-Mamun, Shigeki Masunaga. Car indoor air pollution by volatile organic compounds and aldehydes in Japan[J]. AIMS Environmental Science, 2016, 3(3): 362-381. doi: 10.3934/environsci.2016.3.362

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Abstract

1. Introduction

The Bessel function has immense applications in the field of engineering, physics, and applied mathematics. Baricz ^[1], Generalized Bessel functions of the first kind in $(2010)$ , which discussed the geometric properties, functional inequalities of generalized Bessel function and also the inequalities involving circular and hyperbolic functions to Bessel function and modified Bessel functions. Tumakov ^[2] investigated the numerical algorithms for fast computations of the Bessel functions of an integer order with the required accuracy. Choi and Agarwal^[3], Abramowitz and Stegun ^[4], Heymans and Podlubny ^[5], Watson ^[6] and Purohit et al. ^[7] studied the following Bessel function (Bf) defined by

$\begin{align} J_{\alpha}(y) = \sum^{\infty}_{n = 0}\frac{(-1)^{n}(y/2)^{\alpha+2n}}{n!\Gamma(\alpha+n+1)}. \end{align}$

(1.1)

Edward Maitland Wright ^[8] introduced the generalized form of Bessel function with the name of Bessel-Maitland function (B-M1)

$\begin{align} J^{\alpha}_{\beta}(y) = \sum^{\infty}_{n = 0}\frac{(-y)^{n}}{n!\Gamma(\alpha n+\beta+1)}. \end{align}$

(1.2)

The properties of generalised Bessel function can be found in the work of Srivastava and Singh ^[9]. Suthar et al. ^[10,11] discussed the various properties of Bessel-Maitland function. Ali et al. ^[12] established some fractional operators with the generalized Bessel-Maitland function.

Waseem et al. ^[13] established the generalized Bessel-Maitland function (B-M11) and discuss the numerous integral formulas for $y\in\mathbb{C}/(-\infty, 0]$ ; $\alpha, \beta, \gamma \in \mathbb{C}$ , $\Re(\alpha)\geq0$ , $\Re(\beta)\geq-1$ , $\Re(\gamma)\geq0$ , $k\in(0, 1)\cup\mathbb{N}$ defined by

$\begin{align} J^{\alpha, \gamma}_{\beta, k}(y) = \sum^{\infty}_{n = 0}\frac{(\gamma)_{k n}(-y)^{n}}{n!\Gamma(\alpha n+\beta+1)}. \end{align}$

(1.3)

Suthar et al. ^[14] studied the following generalized multi-index Bessel function (Gm-Bf) defined by

$\begin{align} J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(y) = \sum^{\infty}_{n = 0}\frac{(\gamma)_{k n}(-y)^{n}}{n!\prod^{m}_{j = 1}\Gamma(\alpha_{j} n+\beta_{j}+1)}. \end{align}$

(1.4)

Recently, fractional integrals are widely applied in different branches of mathematics, physics, engineering due to their wide applications (see e.g., ^{[5,15,16,17,18]}).

Riemann-Liouville fractional integral operators for $\Re(\rho) > 0$ are defined by

$\begin{align} &I^{\rho}_{a^{+}}h(u) = \frac{1}{\Gamma(\rho)}\int^{y}_{a} (y-u)^{\rho-1}h(u)du, a < y \end{align}$

(1.5)

$\begin{align} &I^{\rho}_{b}h(u) = \frac{1}{\Gamma(\rho)}\int^{b}_{y}(u-y)^{\rho-1}h(u)du, y < b. \end{align}$

(1.6)

Riemann-Liouville fractional differentials operators (RLDO) ^[12,19] for $\Re(\rho) > 0$ ; $n = [\Re(n)-1]$

$\begin{align} &D^{\rho}_{a^{+}}h(u) = (d/dy)^{n}I^{n-\rho}_{a^{+}}h(y) \end{align}$

(1.7)

$\begin{align} &D^{\rho}_{b}h(u) = (-d/dy)^{n}I^{n-\rho}_{b}h(y). \end{align}$

(1.8)

Srivastava and Singh ^[9] defined the following fractional integral operator for $\alpha_{1}, \beta_{1}, r\in \mathbb{C}$ , $\Re(\alpha_{1}) > 0$ , $\Re(\beta_{1})\geq-1$ by

$\begin{align} h(y) = ^{def}\int^{y}_{0}(y-t)^{\beta_{1}}J^{\alpha_{1}}_{\beta_{1}}(r(y-t)^{\alpha_{1}})h(u)du. \end{align}$

(1.9)

Srivastava and Tomovski ^[20] established the fractional integral operator (FIO) having Mittag-Leffler function as a kernel, discuss its boundedness and convergence of integral and also derive the product of FIO with Riemann-Liouville fractional integral operator defined for $r, \gamma \in \mathbb{C}$ , $\Re(\alpha_{1}) > \max\{0, \Re(k)-1\}$ ; $\min\{\Re(\beta_{1}), \Re(k)\} > 0$

$\begin{align} (\mathscr{E}^{r; \gamma, k}_{a^{+}; \alpha_{1}, \beta_{1}}h)(y) = \int^{y}_{a}(y-t)^{\beta_{1}-1}E^{\gamma, k}_{\alpha_{1}, \beta_{1}}(r(y-t)^{\alpha_{1}})h(t)dt. \end{align}$

(1.10)

Prabhakar fractional integral operators for $\gamma, \beta_{1} \in \mathbb{C}$ , $\Re(\alpha_{1}) > 0$ are defined in ^[21] by

$\begin{align} &\mathfrak{E}^{*}(\alpha_{1}, \beta_{1}; \gamma; r)h(y) = ^{\circ}h(y) = \int^{y}_{a}(y-t)^{\beta_{1}-1}E^{\gamma}_{\alpha_{1}, \beta_{1}}(r(y-t)^{\alpha_{1}})h(t)dt, a < y, \end{align}$

(1.11)

$\begin{align} &\mathfrak{E}^{*}(\alpha_{1}, \beta_{1}; \gamma; r)h(y) = \int^{b}_{y}(t-y)^{\beta_{1}-1}E^{\gamma}_{\alpha_{1}, \beta_{1}}(r(t-y)^{\alpha_{1}})h(t)dt, y < b. \end{align}$

(1.12)

Tilahun et al. ^[22] derived the generalized FIO for $\Re(\beta_{1}) > 0$ , $\Re(\alpha_{1}) > 0$ and $r, \gamma\in\mathbb{C}$ as

$\begin{align} \bigg(\mathfrak{I}^{r; \alpha_{1}, \gamma}_{a^{+}, \beta_{1}, k}h\bigg)(y) = \int^{y}_{a}(y-t)^{\beta_{1}}J^{\gamma, k}_{\alpha_{1}, \beta_{1}}(r(y-t)^{\alpha_{1}}; p)h(t)dt, a < y \end{align}$

(1.13)

and

$\begin{align} \bigg(\mathfrak{I}^{r; \alpha_{1}, \gamma}_{a^{+}, \beta_{1}, k}h\bigg)(y) = \int^{b}_{y}(y-t)^{\beta_{1}}J^{\gamma, k}_{\alpha_{1}, \beta_{1}}(r(y-t)^{\alpha_{1}};p)h(t)dt, \end{align}$

(1.14)

where $y < b$ .

Definition 1.1. (FIO)Fractional integral operator with generalized multi-index Bessel function (Gm-Bf) kernel for $r, \alpha_{j}, \beta_{j}, \gamma \in \mathbb{C}$ , $(j = 1, 2 \cdots m)$ , $\Re(\alpha_{j}) > 0$ , $\Re(\beta_{j}) > -1$ , $\sum^{m}_{j = 1}\Re(\alpha)_{j} > \max\{0; \Re(k)-1\}$ , $k > 0$ .

$\begin{align} &\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j})_{m}}h(y) = \prod^{m}_{j = 1}\int^{y}_{a}(y-u)^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(y-u)^{\alpha_{j}})h(u)du, a < y \end{align}$

(1.15)

$\begin{align} &\mathcal{I}^{r; \gamma, k}_{b; (\alpha_{j}, \beta_{j})_{m}}h(y) = \prod^{m}_{j = 1}\int^{b}_{y}(u-y)^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(u-y)^{\alpha_{j}})h(u)du, y < b. \end{align}$

(1.16)

Dirichlet formula (Fubini's theorem) Samko et al. in ^[23] and Kelelaw et al. in ^[22] is defined as

$\begin{align} \int^{d}_{b}dy\int^{y}_{b}h(y, z)dz = \int^{d}_{b}dz\int^{d}_{z}h(y, z)dy. \end{align}$

(1.17)

Kilbas ^[24] analyzed the generalized Wright function for $\xi_{i}, \zeta_{j}\in\mathbb{R}$ , $(i = 1, 2 \cdots r), (j = 1, 2 \cdots s)$ and $b_{i}, c_{j}\in \mathbb{C}$ as

$\begin{align} {_{r}\psi_{s}}(y)& = \sum^{\infty}_{n = 0}\frac{\prod^{r}_{i = 1}\Gamma(b_{i}+\xi_{i}n)}{\prod^{s}_{j = 1}\Gamma(c_{j}+\zeta_{j}n)}\frac{y^{n}}{n!}\\ & = {_{r}\psi_{s}}\left[ \begin{array}{c} (b_{i}, \xi_{i})_{1, r}\\ (c_{j}, \zeta_{j})_{1, s} \end{array}\middle|y \right]. \end{align}$

(1.18)

The integral representation of beta function ^[25,26] for $\Re(y) > 0$ , $\Re(z) > 0$ and also in gamma form appearance of beta function is defined as follows

$\begin{align} B(y, z) = \int^{1}_{0}u^{y-1}(1-u)^{z-1}du = \frac{\Gamma(y)\Gamma(z)}{\Gamma(y+z)}. \end{align}$

(1.19)

Pochhammer symbol and its properties can be found ^[25,26,27] as

$\begin{align} (\gamma)_n& = \left\{ \begin{array}{ll} \gamma(\gamma+1)(\gamma+2) \cdots (\gamma+n-1), & \hbox{for $ n\geq1 $} \\ 1, & \hbox{for $n = 0$, $\gamma\neq0$} \end{array} \right. \end{align}$

(1.20)

$\begin{align} & = \frac{\Gamma(\gamma+n)}{\Gamma(\gamma)} \,\,\ \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; and \; \; \; \; \; \; \; \; \; \; \; \,\,\ (\gamma)_{kn} = \frac{\Gamma(\gamma+kn)}{\Gamma(\gamma)} \,\,\ \; \; \; \; \; (k > 0). \end{align}$

(1.21)

The space of Lebesgue measurable for complex and real valued functions defined by Kelelaw et al. ^[22] as follows

$\begin{align} L(a, y) = \bigg\{h: ||h||_{1}: = \int^{y}_{a}\big|h(u)\big|du < \infty\bigg\}. \end{align}$

(1.22)

1.1. Conditions of fractional integral operator

The following some conditions of fractional integral operators can be obtained by setting the integrals according to requirements:

$1).$ Setting $r = 0$ , $j = 1 = m$ and $\beta_{1} = \beta_{1}-1$ in (FIO) defined in Eqs (1.15) and (1.16), we get the Riemann-Liouville fractional integral operator defined in ^[28] as

$\begin{align} &\mathcal{I}^{0; \gamma, k}_{a^{+}; (\alpha_{1}, \beta_{1}-1)_{m}}h(y) = I^{\beta_{1}}_{a^{+}}h(y) \end{align}$

(1.23)

$\begin{align} &\mathcal{I}^{0; \gamma, k}_{b; (\alpha_{1}, \beta_{1}-1)_{m}}h(y) = I^{\beta_{1}}_{b}h(y). \end{align}$

(1.24)

$2).$ Setting $j = m = 1$ , $\beta_{1} = \beta_{1}-1$ in Eq (1.15), we have a fractional integral defined in Eq (1.10) as

$\begin{align} &\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{1}, \beta_{1}-1)_{m}}h(y) = (\mathscr{E}^{r; \gamma, k}_{a^{+}; \alpha_{1}, \beta_{1}}h)(y). \end{align}$

(1.25)

$3).$ Setting $j = m = 1$ , $k = 1$ , $\beta_{1} = \beta_{1}-1$ , in Eqs (1.15) and (1.16), we get the FIO defined in Eqs (1.11) and (1.12) respectively

$\begin{align} &\mathcal{I}^{r; \gamma, 1}_{a^{+}; (\alpha_{1}, \beta_{1}-1)_{m}}h(y) = \mathfrak{E}^{*}(\alpha_{1}, \beta_{1}; \gamma; r)h(y) = ^{\circ}h(y) \end{align}$

(1.26)

$\begin{align} &\mathcal{I}^{r; \gamma, 1}_{b; (\alpha_{1}, \beta_{1}-1)_{m}}h(y) = \mathfrak{E}^{*}(\alpha_{1}, \beta_{1}; \gamma; r)h(y). \end{align}$

(1.27)

$4).$ Setting $j = m = 1$ , $k = 0$ and limits from $[0, y]$ in Eq (1.15), we get a fractional integral defined in Eq (1.9) as

$\begin{align} &\mathcal{I}^{r; \gamma, 0}_{a^{+}; (\alpha_{1}, \beta_{1})_{m}}h(y) = \int^{y}_{0}(y-t)^{\beta_{1}}J^{\alpha_{1}}_{\beta_{1}}(r(y-t)^{\alpha_{1}})h(u)du = h(y). \end{align}$

(1.28)

$5).$ Setting $j = m = 1$ in Eq (1.15) then, we get the generalized fractional integral operator defined in Eq (1.13) as

$\begin{align} &\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{1}, \beta_{1})}h(y) = \bigg(\mathfrak{I}^{r; \alpha_{1}, \gamma}_{a^{+}, \beta_{1}, k}h\bigg)(y). \end{align}$

(1.29)

Lemma 1.1. Consider the Riemann-Liouville fractional integral operator with multi-index power function for $\alpha_{j}, \beta_{j}, \gamma \in \mathbb{C}$ , $(j = 1, 2 \cdots m)$ , $\Re(\alpha_{j}) > 0$ , $\Re(\beta_{j}) > -1$ , $\sum^{m}_{j = 1}\Re(\alpha)_{j} > \max\{0; \Re(k)-1\}$ , $k > 0$ and $\Re(\rho) > 0$ as

$\begin{align} \prod^{m}_{j = 1}I^{\rho}_{a^{+}}[(u-a)^{\beta_{j}+\alpha_{j}n}](y)& = \prod^{m}_{j = 1} \Big[\frac{ \Gamma(\beta_{j}+\alpha_{j}n+1)}{\Gamma(\rho+\beta_{j}+\alpha_{j}n+1)}(y-a)^{\rho+\beta_{j}+\alpha_{j}n}\Big]. \end{align}$

(1.30)

Remark 1.1. Setting $j = m = 1$ in lemma 1.1 then we obtain the result that defined the Mathai Haubold ^[29] and Kelelaw et al. ^[22] as

$\begin{align} I^{\rho}_{a^{+}}[(u-a)^{\beta_{1}+\alpha_{1}n}](y)& = (y-a)^{\rho+\beta_{1}+\alpha_{1}n}\frac{ \Gamma(\beta_{1}+\alpha_{1}n+1)}{\Gamma(\rho+\beta_{1}+\alpha_{1}b+1)}. \end{align}$

(1.31)

2. Some properties of generalized multi-index Bessel function

The preliminary results for generalized multi-index Bessel function which used to proceed the new results is given in this section. We calculate the $nth$ -differential and also develop some results with the coordination of Riemann-Liouville fractional operator and (Gm-Bf).

Theorem 2.1. Consider the $nth$ -differential of generalized multi-index Bessel function with power function for $r, \alpha_{j}, \beta_{j}, \gamma \in \mathbb{C}$ , $(j = 1, 2 \cdots m)$ , $\Re(\alpha_{j}) > 0$ , $\Re(\beta_{j}) > -1$ , $\sum^{m}_{j = 1}\Re(\alpha)_{j} > \max\{0; \Re(k)-1\}$ , $k > 0$ , $n\in\mathbb{N}$ as

$\begin{align} (d/dy)^{n}[(y-u)^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(y-u)^{\alpha_{j}})] = (y-u)^{\beta_{j}-n}J^{(\alpha_{j}, \beta_{j}-n)_{m}}_{\gamma, k}(r(y-u)^{\alpha_{j}}). \end{align}$

(2.1)

Proof. Let the $nth$ -differential of generalized multi-index Bessel function with power function as

$\begin{align} (d/dy)^{n}[(y-u)^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(y-u)^{\alpha_{j}})], \end{align}$

(2.2)

using the behavior of (1.4), we take as

$\begin{align} &(d/dy)^{n}[(y-u)^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(y-u)^{\alpha_{j}})]\\ & = (d/dy)^{n}\bigg[(y-u)^{\beta_{j}}\sum^{\infty}_{n = 0}\frac{(\gamma)_{k n}(-r(y-u)^{\alpha_{j}})^{n}}{n!\prod^{m}_{j = 1}\Gamma(\alpha_{j} n+\beta_{j}+1)}\bigg]\\ & = \sum^{\infty}_{n = 0}\frac{(\gamma)_{k n}(-r)^{n}}{n!\prod^{m}_{j = 1}\Gamma(\alpha_{j} n+\beta_{j}+1)}(d/dy)^{n}\bigg[(y-u)^{\beta_{j}+\alpha_{j}n}\bigg]\\ & = \sum^{\infty}_{n = 0}\frac{(\gamma)_{k n}(-r)^{n}}{n!\prod^{m}_{j = 1}\Gamma(\alpha_{j} n+\beta_{j}+1)}(d/dy)^{n}\bigg[(y-u)^{(\beta_{1}+\beta_{2}+ \cdots +\beta_{m})+(\alpha_{1}n+\alpha_{2}n+ \cdots +\alpha_{m}n)}\bigg]\\ & = \sum^{\infty}_{n = 0}\frac{(\gamma)_{k n}(-r)^{n}}{n!\prod^{m}_{j = 1}\Gamma(\alpha_{j} n+\beta_{j}+1)}(d/dy)^{n}\bigg[(y-u)^{(\beta_{1}+\alpha_{1}n)+(\beta_{2}+\alpha_{2}n)+ \cdots +(\beta_{m}+\alpha_{m}n)}\bigg]\\ & = \sum^{\infty}_{n = 0}\frac{(\gamma)_{k n}(-r)^{n}}{n!\prod^{m}_{j = 1}\Gamma(\alpha_{j} n+\beta_{j}+1)}\\ &\times(d/dy)^{n}\bigg[(y-u)^{(\beta_{1}+\alpha_{1}n)}(y-u)^{(\beta_{2}+\alpha_{2}n)} \cdots (y-u)^{(\beta_{m}+\alpha_{m}n)}\bigg]. \end{align}$

(2.3)

Using the identity result for simplification of (2.3), we get

$\begin{align} &(d/dy)^{n}y^{\theta} = \frac{\Gamma(\theta+1)}{\Gamma(\theta-n+1)}y^{\theta-n}, \theta\geq n \end{align}$

(2.4)

$\begin{align} &(d/dy)^{n}[(y-u)^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(y-u)^{\alpha_{j}})] = \sum^{\infty}_{n = 0}\frac{(\gamma)_{k n}(-r)^{n}}{n!\prod^{m}_{j = 1}\Gamma(\alpha_{j} n+\beta_{j}+1)}\\ &\times\frac{\Gamma(\beta_{1}+\alpha_{1}n+1)\Gamma(\beta_{2}+\alpha_{2}n+1) \cdots \Gamma(\beta_{m}+\alpha_{m}n+1)(y-u)^{\alpha_{j}n+\beta_{j}-n}}{\Gamma(\beta_{1}+\alpha_{1}n-n+1)\Gamma(\beta_{2}+\alpha_{2}n-n+1) \cdots \Gamma(\beta_{m}+\alpha_{m}n-n+1)}\\ & = \sum^{\infty}_{n = 0}\frac{(\gamma)_{k n}(-r)^{n}}{n!\prod^{m}_{j = 1}\Gamma(\alpha_{j} n+\beta_{j}+1)}\frac{\prod^{m}_{j = 1}\Gamma(\alpha_{j} n+\beta_{j}+1)}{\prod^{m}_{j = 1}\Gamma(\alpha_{j} n+\beta_{j}-n+1)}(y-u)^{\alpha_{j}n+\beta_{j}-n}\\ & = (y-u)^{\beta_{j}-n}\sum^{\infty}_{n = 0}\frac{(\gamma)_{k n}(-r(y-u)^{\alpha_{j}})^{n}}{n!\prod^{m}_{j = 1}\Gamma(\alpha_{j} n+\beta_{j}-n+1)}\\ & = (y-u)^{\beta_{j}-n}J^{(\alpha_{j}, \beta_{j}-n)_{m}}_{\gamma, k}(r(y-u)^{\alpha_{j}}). \end{align}$

(2.5)

Corollary 2.1. Suppose that $\alpha_{j}, \beta_{j}, \gamma \in \mathbb{C}$ , $(j = 1, 2 \cdots m)$ , $\Re(\alpha_{j}) > 0$ , $\Re(\beta_{j}) > -1$ , $\sum^{m}_{j = 1}\Re(\alpha)_{j} > \max\{0; \Re(k)-1\}$ , $k > 0$ , $n\in\mathbb{N}$ then theorem 2.1 can be expressed as

$\begin{align} &(d/dy)^{n}[(y-u)^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(y-u)^{\alpha_{j}})]\\ & = \frac{1}{\Gamma(\gamma)}(y-u)^{\beta_{j}-n} {_{2}\psi_{m+1}}\left\{ \begin{array}{c} (\gamma, k)(\beta_{j}+1, \alpha_{j})\\ (\beta_{j}-n+1, \alpha_{j})(\beta_{j}+1, \alpha_{j})|^{m}_{j = 1} \end{array}\middle|r(y-u)^{\alpha_{j}} \right\}. \end{align}$

(2.6)

Corollary 2.2. Suppose that $r, \alpha_{j}, \beta_{j}, \gamma \in \mathbb{C}$ , $(j = 1, 2 \cdots m)$ , $\Re(\alpha_{j}) > 0$ , $\Re(\beta_{j}) > -1$ , $\sum^{m}_{j = 1}\Re(\alpha)_{j} > \max\{0; \Re(k)-1\}$ , $k > 0$ , $n\in\mathbb{N}$ and setting that $\beta_{j} = \beta_{j}-1$ , $(r(y-u)^{\alpha_{j}}) = (-r(y-u)^{\alpha_{j}})$ in theorem 2.1 we see that

$\begin{align} &(d/dy)^{n}[(y-u)^{\beta_{j}-1}J^{(\alpha_{j}, \beta_{j}-1)_{m}}_{\gamma, k}(r(y-u)^{\alpha_{j}})]\\ & = (y-u)^{\beta_{j}-1-n}E^{(\alpha_{j}, \beta_{j}-n)_{m}}_{\gamma, k}(r(y-u)^{\alpha_{j}}), \end{align}$

(2.7)

where $E^{(\alpha_{j}, \beta_{j}-n)_{m}}_{\gamma, k}(.)$ is generalized multi-index Mittag-Leffler function.

Theorem 2.2. Consider the Riemann-Liouville fractional integral operator defined in Eq (1.5) with generalized multi-index Bessel function for $r, \alpha_{j}, \beta_{j}, \gamma \in \mathbb{C}$ , $(j = 1, 2 \cdots m)$ , $\Re(\alpha_{j}) > 0$ , $\Re(\beta_{j}) > -1$ , $\sum^{m}_{j = 1}\Re(\alpha)_{j} > \max\{0; \Re(k)-1\}$ , $k > 0$ and $y > a$ , $a\in\Re_{+}(0, \infty)$ , $\Re(\rho) > 0$ as

(2.8)

Proof. Let (RLIO) with (Gm-Bf) is defined in Eq (1.4), we have

$\begin{align} &\prod^{m}_{j = 1}I^{\rho}_{a^{+}}[(u-a)^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(u-a)^{\alpha_{j}})\bigg](y)\\ & = \prod^{m}_{j = 1}I^{\rho}_{a^{+}}[(u-a)^{\beta_{j}}\sum^{\infty}_{n = 0}\frac{(\gamma)_{k n}(-r(u-a)^{\alpha_{j}})^{n}}{n!\prod^{m}_{j = 1}\Gamma(\alpha_{j}n+\beta_{j}+1)}]\\ & = \sum^{\infty}_{n = 0}\frac{(\gamma)_{k n}(-r)^{n}}{n!\prod^{m}_{j = 1}\Gamma(\alpha_{j}n+\beta_{j}+1)}\prod^{m}_{j = 1}I^{\rho}_{a^{+}}[(u-a)^{\beta_{j}+\alpha_{j}n}]. \end{align}$

(2.9)

By using Lemma 1.1 in Eq (2.9) then we attain the equation as

$\begin{align} &\prod^{m}_{j = 1}I^{\rho}_{a^{+}}[(u-a)^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(u-a)^{\alpha_{j}})](y)\\ & = \sum^{\infty}_{n = 0}\frac{(\gamma)_{k n}(-r)^{n}}{n!\prod^{m}_{j = 1}\Gamma(\alpha_{j}n+\beta_{j}+1)} \prod^{m}_{j = 1}\frac{\Gamma(\alpha_{j}n+\beta_{j}+1)(y-a)^{\alpha_{j}n+\rho+\beta_{j}}} {\Gamma(\alpha_{j}n+\beta_{j}+\rho+1)}\\ & = \prod^{m}_{j = 1}(y-a)^{\rho+\beta_{j}}\sum^{\infty}_{n = 0}\frac{(\gamma)_{k n}(-r(y-a)^{\alpha_{j}})^{n}}{n!\prod^{m}_{j = 1}\Gamma(\alpha_{j}n+\beta_{j}+\rho+1)}\\ & = \prod^{m}_{j = 1}(y-a)^{\beta_{j}+\rho}J^{(\alpha_{j}, \beta_{j}+\rho)_{m}}_{\gamma, k}(r(y-a)^{\alpha_{j}}). \end{align}$

(2.10)

Corollary 2.3. Consider the right-sided Riemann-Liouville fractional integral operator with generalized multi-index Bessel function for $r, \alpha_{j}, \beta_{j}, \gamma \in \mathbb{C}$ , $(j = 1, 2 \cdots m)$ , $\Re(\alpha_{j}) > 0$ , $\Re(\beta_{j}) > -1$ , $\sum^{m}_{j = 1}\Re(\alpha)_{j} > \max\{0; \Re(k)-1\}$ , $k > 0$ and $y > a$ , $a\in\Re_{+}(0, \infty)$ , $\Re(\rho) > 0$ as

$\begin{align} &\prod^{m}_{j = 1}I^{\rho}_{b}[(b-u)^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(b-u)^{\alpha_{j}})](y)\\ & = \prod^{m}_{j = 1}(b-y)^{\beta_{j}+\rho}J^{(\alpha_{j}, \beta_{j}+\rho)_{m}}_{\gamma, k}(r(b-y)^{\alpha_{j}}). \end{align}$

(2.11)

Corollary 2.4. Consider the Riemann-Liouville fractional differential operator defined in Eq (1.7) with generalized multi-index Bessel function for $r, \alpha_{j}, \beta_{j}, \gamma \in \mathbb{C}$ , $(j = 1, 2 \cdots m)$ , $\Re(\alpha_{j}) > 0$ , $\Re(\beta_{j}) > -1$ , $\sum^{m}_{j = 1}\Re(\alpha)_{j} > \max\{0; \Re(k)-1\}$ , $k > 0$ and $y > a$ , $a\in\Re_{+}(0, \infty)$ , $\Re(\rho) > 0$ as

$\begin{align} &\prod^{m}_{j = 1}D^{\rho}_{a^{+}}[(u-a)^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(u-a)^{\alpha_{j}})](y)\\ & = \prod^{m}_{j = 1}(y-a)^{\beta_{j}-\rho}J^{(\alpha_{j}, \beta_{j}-\rho)_{m}}_{\gamma, k}(r(y-a)^{\alpha_{j}}). \end{align}$

(2.12)

Corollary 2.5. Consider the right-sided Riemann-Liouville fractional differential operator with generalized multi-index Bessel function for $r, \alpha_{j}, \beta_{j}, \gamma \in \mathbb{C}$ , $(j = 1, 2 \cdots m)$ , $\Re(\alpha_{j}) > 0$ , $\Re(\beta_{j}) > -1$ , $\sum^{m}_{j = 1}\Re(\alpha)_{j} > \max\{0; \Re(k)-1\}$ , $k > 0$ and $y > a$ , $a\in\Re_{+}(0, \infty)$ , $\Re(\rho) > 0$ as

$\begin{align} &\prod^{m}_{j = 1}D^{\rho}_{b}[(b-u)^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(b-u)^{\alpha_{j}})](y)\\ & = \prod^{m}_{j = 1}(b-y)^{\beta_{j}-\rho}J^{(\alpha_{j}, \beta_{j}-\rho)_{m}}_{\gamma, k}(r(b-y)^{\alpha_{j}}). \end{align}$

(2.13)

3. Properties of generalized fractional integral with non singular function as a kernel

In this section, we discuss some properties of the generalized fractional integrals with non singular function as a kernel.

Theorem 3.1. Fractional integral operator having generalized multi-index Bessel function (Gm-Bf) kernel for $\vartheta, \rho, \sigma, r, \alpha_{j}, \beta_{j}, \gamma \in \mathbb{C}$ , $(j = 1, 2 \cdots m)$ , $\Re(\alpha_{j}) > 0$ , $\Re(\beta_{j}) > -1$ , $\sum^{m}_{j = 1}\Re(\alpha)_{j} > \max\{0; \Re(k)-1\}$ , $k > 0$ , when $h(u) = (u-a)^{\frac{\vartheta+\rho}{\sigma}-1}$ , then

$\begin{align} [\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j})_{m}}&(u-a)^{\frac{\vartheta+\rho}{\sigma}-1}](y)\\ & = \prod^{m}_{j = 1}(y-a)^{\frac{\vartheta+\rho}{\sigma}+\beta_{j}}\Gamma(\frac{\vartheta+\rho}{\sigma})J^{(\alpha_{j}, \beta_{j}+\frac{\vartheta+\rho}{\sigma})_{m}}_{\gamma, k}(r(y-a)^{\alpha_{j}}). \end{align}$

(3.1)

Proof. (FIO) defined in Eq (1.15) with Eq (1.4), we obtain as

$\begin{align} &[\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j})_{m}}(u-a)^{\frac{\vartheta+\rho}{\sigma}-1}](y)\\ & = \prod^{m}_{j = 1}\int^{y}_{a}(y-u)^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(y-u)^{\alpha_{j}})(u-a)^{\frac{\vartheta+\rho}{\sigma}-1}du\\ & = \sum^{\infty}_{n = 0}\frac{(\gamma)_{k n}(-r)^{n}}{n!\prod^{m}_{j = 1}\Gamma(\alpha_{j}n+\beta_{j}+1)}\prod^{m}_{j = 1}\int^{y}_{a}(u-a)^{\frac{\vartheta+\rho}{\sigma}-1} (y-u)^{\beta_{j}+\alpha_{j}n}du\\ & = \sum^{\infty}_{n = 0}\frac{(\gamma)_{k n}(-r)^{n}}{n!}\frac{1}{\prod^{m}_{j = 1}\Gamma(\alpha_{j}n+\beta_{j}+1)}\prod^{m}_{j = 1}\int^{y}_{a}(u-a)^{\frac{\vartheta+\rho}{\sigma}-1} (y-u)^{\beta_{j}+\alpha_{j}n}du\\ \end{align}$

(3.2)

Substituting $u = y-z(u-a)$ and using the definition of beta function and the following relation in Eq (3.2), we get

$\begin{align*} \mathcal{I}^{\lambda}_{a^{+}}[(y-a)^{u-1}](y) = \frac{\Gamma(u)}{\Gamma(\lambda+u)}(y-a)^{\lambda+u-1}. \end{align*}$

$\begin{align} &[\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j})_{m}}(u-a)^{\frac{\vartheta+\rho}{\sigma}}](y)\\ & = \prod^{m}_{j = 1}\sum^{\infty}_{n = 0}\frac{(\gamma)_{k n}(-r)^{n}}{n!}\frac{\Gamma(\frac{\vartheta+\rho}{\sigma})}{\prod^{m}_{j = 1}\Gamma(\frac{\vartheta+\rho}{\sigma}+\alpha_{j}n+\beta_{j}+1)} (y-a)^{\frac{\vartheta+\rho}{\sigma}+\beta_{j}+\alpha_{j}n}\\ & = \prod^{m}_{j = 1}(y-a)^{\frac{\vartheta+\rho}{\sigma}+\beta_{j}}\Gamma(\frac{\vartheta+\rho}{\sigma})\sum^{\infty}_{n = 0}\frac{(\gamma)_{k n}(-r(y-a)^{\alpha_{j}})^{n}}{n!\prod^{m}_{j = 1}\Gamma(\frac{\vartheta+\rho}{\sigma}+\alpha_{j}n+\beta_{j}+1)}\\ & = \prod^{m}_{j = 1}(y-a)^{\frac{\vartheta+\rho}{\sigma}+\beta_{j}}\Gamma(\frac{\vartheta+\rho}{\sigma})J^{(\alpha_{j}, \beta_{j}+\frac{\vartheta+\rho}{\sigma})_{m}}_{\gamma, k}(r(y-a)^{\alpha_{j}}). \end{align}$

(3.3)

Corollary 3.1. Fractional integral operator having generalized multi-index Bessel function (Gm-Bf) kernel for $\vartheta, \rho, \sigma, r, \alpha_{j}, \beta_{j}, \gamma \in \mathbb{C}$ , $(j = 1, 2 \cdots m)$ , $\Re(\alpha_{j}) > 0$ , $\Re(\beta_{j}) > -1$ , $\sum^{m}_{j = 1}\Re(\alpha)_{j} > \max\{0; \Re(k)-1\}$ , $k > 0$ , when $h(u) = (a-u)^{\frac{\vartheta+\rho}{\sigma}}$ , then

$\begin{align} [\mathcal{I}^{r; \gamma, k}_{b; (\alpha_{j}, \beta_{j})_{m}}&(b-u)^{\frac{\vartheta+\rho}{\sigma}}](y)\\ & = \prod^{m}_{j = 1}(b-y)^{\frac{\vartheta+\rho}{\sigma}+\beta_{j}+1}\Gamma(\frac{\vartheta+\rho}{\sigma}+1)J^{(\alpha_{j}, \beta_{j}+\frac{\vartheta+\rho}{\sigma}+1)_{m}}_{\gamma, k}(r(b-y)^{\alpha_{j}}). \end{align}$

(3.4)

Theorem 3.2. Consider the composition of Riemann-Liouville fractional integral operator with (FIO) for $r, \alpha_{j}, \beta_{j}, \gamma \in \mathbb{C}$ , $(j = 1, 2 \cdots m)$ , $\Re(\alpha_{j}) > 0$ , $\Re(\beta_{j}) > -1$ , $\sum^{m}_{j = 1}\Re(\alpha)_{j} > \max\{0; \Re(k)-1\}$ , $k > 0$ , $\Re(\rho) > 0$ then

$\begin{align} \bigg\{I^{\rho}_{a^{+}}[\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j})_{m}}h]\bigg\}(y) = \bigg\{{\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j}+\rho)_{m}}}\bigg\}(y) = \bigg\{{\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j})_{m}}}[I^{\rho}_{a^{+}}h]\bigg\}(y). \end{align}$

(3.5)

Proof. Let the left side of Eq (3.5), we seen as

$\begin{align} &\bigg\{I^{\rho}_{a^{+}}[\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j})_{m}}h]\bigg\}(y)\\ & = \frac{1}{\Gamma(\rho)}\int^{y}_{a}(y-u)^{\rho-1}[\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j})_{m}}h](u)du\\ & = \frac{1}{\Gamma(\rho)}\int^{y}_{a}(y-u)^{\rho-1}\bigg[\prod^{m}_{j = 1}\int^{u}_{a}(u-t)^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(u-t)^{\alpha_{j}})h(t)dt\bigg]du. \end{align}$

(3.6)

By interchanging the order of integrations and using the Eq (1.17) in Eq (3.6), we attain as

$\begin{align} &\bigg\{I^{\rho}_{a^{+}}[\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j})_{m}}h]\bigg\}(y)\\ & = \int^{y}_{a}\bigg[\frac{1}{\Gamma(\rho)}\prod^{m}_{j = 1}\int^{y}_{t}(y-u)^{\rho-1}(u-t)^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(u-t)^{\alpha_{j}})du\bigg] \times h(t)dt. \end{align}$

(3.7)

Setting $u-t = \eta$ , we have

$\begin{align} &\bigg\{I^{\rho}_{a^{+}}[\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j})_{m}}h]\bigg\}(y)\\ & = \int^{y}_{a}\bigg[\prod^{m}_{j = 1}\frac{1}{\Gamma(\rho)}\int^{y-t}_{0}(y-t-\eta)^{\rho-1}\eta^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(\eta)^{\alpha_{j}})d\eta\bigg] \times h(t)dt\\ & = \int^{y}_{a}\prod^{m}_{j = 1}I^{\rho}_{0^{+}}\bigg[\eta^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(\eta)^{\alpha_{j}})\bigg](y-t) \times h(t)dt. \end{align}$

(3.8)

applying theorem 2.2, we see

$\begin{align} &\bigg\{I^{\rho}_{a^{+}}[\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j})_{m}}h]\bigg\}(y)\\ & = \prod^{m}_{j = 1}\int^{y}_{a}\bigg[(y-t)^{\beta_{j}+\rho}J^{(\alpha_{j}, \beta_{j}+\rho)_{m}}_{\gamma, k}(r(y-t)^{\alpha_{j}})\bigg] h(t)dt\\ & = \bigg\{{\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j}+\rho)_{m}}}\bigg\}(y). \end{align}$

(3.9)

We start the right side to determine the second part of (3.5) with (FIO) defined in Eq (1.15) as

$\begin{align} &\bigg\{{\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j})_{m}}}[I^{\rho}_{a^{+}}h]\bigg\}(y)\\ & = \prod^{m}_{j = 1}\int^{y}_{a}(y-u)^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(y-u)^{\alpha_{j}})[I^{\rho}_{a^{+}}h](u)du\\ & = \prod^{m}_{j = 1}\int^{y}_{a}(y-u)^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(y-u)^{\alpha_{j}})\bigg[\frac{1}{\Gamma(\rho)}\int^{u}_{a}(u-t)^{\rho-1}h(t)dt\bigg]du. \end{align}$

(3.10)

Interchanging the order of integration and using Eq (1.17), we get

(3.11)

Setting $y-u = x$ and using theorem (2.2) then

$\begin{align} &\bigg\{{\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j})_{m}}}[I^{\rho}_{a^{+}}h]\bigg\}(y)\\ & = \prod^{m}_{j = 1}\int^{y}_{a}\bigg[\frac{1}{\Gamma(\rho)}\int^{0}_{y-t}(x)^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(x)^{\alpha_{j}})(y-x-t)^{\rho-1}(-dx)\bigg]h(t)dt\\ & = \prod^{m}_{j = 1}\int^{y}_{a}\bigg[\frac{1}{\Gamma(\rho)}\int_{0}^{y-t}(x)^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(x)^{\alpha_{j}})(y-t-x)^{\rho-1}dx\bigg]h(t)dt\\ & = \prod^{m}_{j = 1}\int^{y}_{a}I^{\rho}_{0^{+}}\bigg[(x)^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(x)^{\alpha_{j}})\bigg](y-t)\times h(t)dt\\ & = \prod^{m}_{j = 1}\int^{y}_{a}(y-t)^{\beta_{j}+\rho}J^{(\alpha_{j}, \beta_{j}+\rho)_{m}}_{\gamma, k}(r(y-t)^{\alpha_{j}})h(t)dt\\ & = \bigg\{{\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j}+\rho)_{m}}}\bigg\}(y). \end{align}$

(3.12)

Thus, we obtain the desired results by combining Eqs (3.9) and (3.12).

CCorollary 3.2. Composition of right-sided (FIO) with right-sided Riemann-Liouville fractional integral for $r, \alpha_{j}, \beta_{j}, \gamma \in \mathbb{C}$ , $(j = 1, 2 \cdots m)$ , $\Re(\alpha_{j}) > 0$ , $\Re(\beta_{j}) > -1$ , $\sum^{m}_{j = 1}\Re(\alpha)_{j} > \max\{0; \Re(k)-1\}$ , $k > 0$ , $\Re(\rho) > 0$ then

$\begin{align} \bigg\{I^{\rho}_{b}[\mathcal{I}^{r; \gamma, k}_{b; (\alpha_{j}, \beta_{j})_{m}}h]\bigg\}(y) = \bigg\{{\mathcal{I}^{r; \gamma, k}_{b; (\alpha_{j}, \beta_{j}+\rho)_{m}}}\bigg\}(y) = \bigg\{{\mathcal{I}^{r; \gamma, k}_{b; (\alpha_{j}, \beta_{j})_{m}}}[I^{\rho}_{b}h]\bigg\}(y). \end{align}$

(3.13)

Corollary 3.3. Consider the composition of Riemann-Liouville fractional differential operator with (FIO) for $r, \alpha_{j}, \beta_{j}, \gamma \in \mathbb{C}$ , $(j = 1, 2 \cdots m)$ , $\Re(\alpha_{j}) > 0$ , $\Re(\beta_{j}) > -1$ , $\sum^{m}_{j = 1}\Re(\alpha)_{j} > \max\{0; \Re(k)-1\}$ , $k > 0$ , $\Re(\rho) > 0$ then

$\begin{align} \bigg\{D^{\rho}_{a^{+}}[\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j})_{m}}h]\bigg\}(y) = \bigg\{{\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j}-\rho)_{m}}}\bigg\}(y) = \bigg\{{\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j})_{m}}}[D^{\rho}_{a^{+}}h]\bigg\}(y). \end{align}$

(3.14)

Theorem 3.3. If $r, \alpha_{j}, \beta_{j}, \gamma \in \mathbb{C}$ , $(j = 1, 2 \cdots m)$ , $\Re(\alpha_{j}) > 0$ , $\Re(\beta_{j}) > -1$ , $\sum^{m}_{j = 1}\Re(\alpha)_{j} > \max\{0; \Re(k)-1\}$ , $k > 0$ then

$\begin{align} &||{\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j})_{m}}h}||_{1}\leq A||h||_{1}, \end{align}$

(3.15)

where

$\begin{align} A = \prod^{m}_{j = 1}\sum^{\infty}_{n = 0}\frac{|(y-a)^{\Re(\beta_{j})+1}||(\gamma)_{k n}||(-r (y-a)^{\Re(\alpha_{j})})^{n}|}{n!|\prod^{m}_{j = 1} \Gamma(\alpha_{j}n+\beta_{j}+1)|\Re(\alpha_{j})n+\Re(\beta_{j})+1}. \end{align}$

(3.16)

Proof. Let $V_{n}$ be the nth-terms of (3.16); we have

$\begin{align} \bigg|\frac{V_{n+1}}{V_{n}}\bigg|& = \prod^{m}_{j = 1}\bigg|\frac{(\gamma)_{k n+k}|}{(\gamma)_{k n}}\bigg|\frac{n!}{(n+1)!} \bigg|\frac{\prod^{m}_{j = 1}\Gamma(\alpha_{j}n+\beta_{j}+1)}{\prod^{m}_{j = 1}\Gamma(\alpha_{j}n+\alpha_{j}+\beta_{j}+1)}\bigg|\\ &\times\frac{\Re(\alpha_{j})n+\Re(\beta_{j})+1}{\Re(\alpha_{j})n+ \Re(\alpha_{j})+\Re(\beta_{j})+1}\bigg|\frac{(-1)^{n+1}}{(-1)^{n}}\bigg| \bigg|r(y-a)^{\Re(\alpha_{j})}\bigg|\\ &\approx\prod^{m}_{j = 1}\frac{(kn)^{k}\big|-r(y-a)^{\Re(\alpha_{j})}\big|}{(n+1)\big|\prod^{m}_{j = 1}{|(\alpha_{j})|n}^{(\alpha_{j})}\big|}, \; \; \; as \; \; \; n \rightarrow \infty \end{align}$

(3.17)

hence, $\bigg|\frac{V_{n+1}}{V_{n}}\bigg| \rightarrow 0$ as $n \rightarrow \infty$ , and $k < \Re(\alpha_{j})$ which means that right hand side of (3.16) is convergent and finite under the given condition. The condition of boundedness of the integral operator $({\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j})_{m}}h})(y)$ is discussed in the space of Lebesgue measure $L(a, y)$ of a continuous function, where $y > a$ . Consider the Lebesgue measurable space (1.22) and FIO (1.15), we have

$\begin{align} ||{\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j})_{m}}h}||_{1}& = \int^{y}_{a}\bigg|{\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j})_{m}}h}\bigg|du\\ & = \int^{y}_{a}\bigg|\prod^{m}_{j = 1}\int^{u}_{a}(u-\tau)^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(u-\tau)^{\alpha_{j}})h(\tau)d\tau\bigg|du\\ &\leq\int^{y}_{a}\bigg[\prod^{m}_{j = 1} \int^{y}_{\tau}(u-\tau)^{\beta_{j}}\big|J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(u-\tau)^{\alpha_{j}})\big|du \bigg]|h(\tau)|d\tau. \end{align}$

(3.18)

Putting $u-\tau = \lambda, u = y \Rightarrow \lambda = y-\tau; u = \tau \Rightarrow \lambda = 0, du = d\lambda$ in Eq (3.18), we get

$\begin{align} ||{\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j})_{m}}h}||_{1}&\leq\int^{y}_{a}\bigg[\prod^{m}_{j = 1} \int^{y-\tau}_{0}\lambda^{\Re(\beta_{j})}\big|J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(\lambda)^{\alpha_{j}})\big|d\lambda \bigg]|h(\tau)|d\tau\\ &\leq\int^{y}_{a}\bigg[\prod^{m}_{j = 1} \int^{y-a}_{0}\lambda^{\Re(\beta_{j})}\big|J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(\lambda)^{\alpha_{j}})\big|d\lambda \bigg]|h(\tau)|d\tau, \end{align}$

where

$\begin{align} &\prod^{m}_{j = 1}\int^{y-a}_{0}\lambda^{\Re(\beta_{j})}\big|J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(\lambda)^{\alpha_{j}})\big|d\lambda \\ \leq&\prod^{m}_{j = 1}\int^{y-a}_{0}\lambda^{\Re(\beta_{j})}\bigg|\sum^{\infty}_{n = 0}\frac{(\gamma)_{k n}|(-r)^{n}(\lambda)^{\alpha_{j} b}|}{n!\prod^{m}_{j = 1}\Gamma(\alpha_{j}n+\beta_{j}+1)}\bigg|d\lambda\\ \leq&\sum^{\infty}_{n = 0}\frac{|(\gamma)_{k n}||(-r)^{n}|}{n!|\prod^{m}_{j = 1}\Gamma(\alpha_{j}n+\beta_{j}+1)|} \prod^{m}_{j = 1}\int^{y-a}_{0}\lambda^{\Re(\alpha_{j})n+\Re(\beta_{j})}d\lambda\\ \leq&\prod^{m}_{j = 1}\sum^{\infty}_{n = 0}\frac{|(y-a)^{\Re(\beta_{j})+1}||(\gamma)_{k n}||(-r (y-a)^{\Re(\alpha_{j})})^{n}|}{n!|\prod^{m}_{j = 1} \Gamma(\alpha_{j}n+\beta_{j}+1)|\Re(\alpha_{j})n+\Re(\beta_{j})+1}\\ = &A. \end{align}$

(3.19)

Therefore,

$\begin{align} ||{\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j})_{m}}h}||_{1}&\leq\int^{y}_{a}A|h(\tau)|d\tau\leq A||h||_{1}\\ &\Rightarrow||{\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j})_{m}}h}||_{1}\leq A||h||_{1} \end{align}$

Corollary 3.4. If $r, \alpha_{j}, \beta_{j}, \gamma \in \mathbb{C}$ , $(j = 1, 2 \cdots m)$ , $\Re(\alpha_{j}) > 0$ , $\Re(\beta_{j}) > -1$ , $\sum^{m}_{j = 1}\Re(\alpha)_{j} > \max\{0; \Re(k)-1\}$ , $k > 0$ then

$\begin{align} &||{\mathcal{I}^{r; \gamma, k}_{b; (\alpha_{j}, \beta_{j})_{m}}h}||_{1}\leq A||h||_{1}, \end{align}$

(3.20)

where

$\begin{align} A = \prod^{m}_{j = 1}(b-y)^{\Re(\beta_{j})+1}\sum^{\infty}_{n = 0}\frac{|(\gamma)_{k n}||(-r(b-y)^{\Re(\alpha_{j})})^{n}|}{n!|\prod^{m}_{j = 1} \Gamma(\alpha_{j}n+\beta_{j}+1)|\Re(\alpha_{j})n+\Re(\beta_{j})+1}. \end{align}$

(3.21)

4. Conclusions

The results we discussed in this paper is creating a chain of fractional operators with kernels, having convergence and boundedness, continuity, symmetric properties and composition with Riemann-Liouville operators ^{[9,12,20,21,22,25,30,31]} in fractional calculus. We constructed the fractional operator with generalized multi-index Bessel function as a kernel, and discussed its properties, continuity, and check the behaviour with Riemann-Liouville fractional operators. We analyzed the generalized multi-index Bessel function $nth$ -derivative and integral in the field of fractional calculus.

Acknowledgments

Taif University Researchers Supporting Project number (TURSP-2020/20), Taif University, Taif, Saudi Arabia.

Conflict of interest

The authors declare that they have no competing interests.

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