Etiology of dental implant complication and failure—an overview

Anirudh Gupta; Bhairavi Kale; Deepika Masurkar; Priyanka Jaiswal; Anirudh Gupta; Bhairavi Kale; Deepika Masurkar; Priyanka Jaiswal

doi:10.3934/bioeng.2023010

AIMS Bioengineering

2023, Volume 10, Issue 2: 141-152. doi: 10.3934/bioeng.2023010

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Review Special Issues

Etiology of dental implant complication and failure—an overview

1.
Sharad Pawar Dental College, Datta Meghe Institute Of Higher Education and Research (Deemed To Be University) Sawangi (Meghe) Wardha-442001 (Maharashtra), India
2.
Department of Periodontology, Sharad Pawar Dental College, Datta Meghe Institute Of Higher Education and Research (Deemed To Be University) Sawangi (Meghe) Wardha-442001 (Maharashtra), India

Received: 20 December 2022 Revised: 21 April 2023 Accepted: 22 April 2023 Published: 17 May 2023

Dental implant treatment is turning into a widely accepted and popular treatment option for patients. With the growing era of dental implant therapy, complications and failures have also become common. Such intricacies are becoming a vexing issue for clinicians as well as patients. Implant failures can be due to mechanical or biological reasons. Failure of osseointegration of the implant falls under biological failures, whereas mechanical complications include fracture of the implant, framework or prosthetic components. Diligently observing the implant after placement is the first step in managing the declining circumstances. It is important to have a thorough understanding of how and why implants fail to achieve successful treatment outcomes in the long run. In dentistry, nanoparticles are used to make antibacterial chemicals that improve dental implants. They can be used in conjunction with acrylic resins for fabricating removable dentures during prosthetic treatments, composite resins for direct restoration during restorative treatments, endodontic irrigants and obturation materials during endodontic procedures, orthodontic adhesives and titanium coating during dental implant procedures. This article aimed to review the etiological factors that lead to implant failure and their solutions.

Keywords:

Citation: Anirudh Gupta, Bhairavi Kale, Deepika Masurkar, Priyanka Jaiswal. Etiology of dental implant complication and failure—an overview[J]. AIMS Bioengineering, 2023, 10(2): 141-152. doi: 10.3934/bioeng.2023010

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Abstract

1. Introduction

The Bessel function ^{[1,2,3,4,5,6,7,8]} has great importance in the field of mathematics, physics and engineering due to its applications. Researchers and mathematicians developed a new class of Bessel functions in the sense of multi-index functions, which motivate the future research work in the field of special functions and fractional calculus. The theory of multi-index multivariate Bessel function discussed by Dattoli et al. ^[9] in $1997$ .

Generalized multi-index Mittag-Leffler function was defined by Choi et al. in ^[10]. Kamarujjama et al. ^[11] introduced and studied the extended multi-index Bessel function. Suthar et al. ^[12] discussed a large number of results for the generalized multi-index Bessel function. Recently, many authors worked on generalized multi-index Bessel functions ^[13,14,15]. We describe extension of extended generalized multi-index Bessel function (E $^{1}$ GMBF) which is generalized version of generalized multi-index Bessel function.

Definition 1.1. ^[11] Kamarujjama et al. introduced and studied the extended generalized multi-index Bessel function, defined as:

$\begin{equation} \mathbb{J}_{(\beta_{j})_{m}, k, b, \delta}^{(\alpha_{j})_{m}, \gamma , c}(z) = \sum^{\infty}_{n = 0}\frac{ (\gamma)_{kn}(-cz)^{n}}{(\delta)_{n}\prod^{m}_{j = 1}\Gamma(\alpha_j n+\beta_j+\frac{1+b}{2})}, \; \; \; \; \; \; \; \; \; \; \; \; \; m\in\mathbb{N}. \end{equation}$

(1.1)

where $\alpha_{j}, \beta_{j}, b, \delta, \gamma, c \in \mathbb{C}$ $(j = 1, 2 \cdots m)$ be such that $\sum^m_{j = 1}\Re(\alpha_j) > max\{0; \Re(k)-1\}$ ; $k > 0$ , $\Re(\beta_j) > -1$ , $\Re(\gamma) > 0$ , $\Re(\delta) > 0$ .

Definition 1.2. ^[16] Generalized fractional integral operator is defined for $\alpha, \acute{\alpha}, \beta, \grave{\beta}, \lambda \in \mathbb{C}$ , and $x > 0$ as follows:

$\begin{eqnarray} I^{\alpha, \acute{\alpha}, \beta, \acute{\beta}, \lambda}_{0^{+}}f(t) = \frac{x^{-\alpha}}{\Gamma(\lambda)}\int^{x}_{0}(x-t)^{\lambda-1}t^{-\acute{\alpha}}F_{3}(\alpha, \acute{\alpha}, \beta, \acute{\beta}; \lambda; 1-\frac{t}{x}; 1-\frac{x}{t})f(t)dt, \end{eqnarray}$

(1.2)

and

$\begin{eqnarray} I^{\alpha, \acute{\alpha}, \beta, \acute{\beta}, \lambda}_{-}f(t) = \frac{x^{-\acute{\alpha}}}{\Gamma(\lambda)}\int^{\infty}_{x}(t-x)^{\lambda-1}t^{-\alpha}F_{3}(\alpha, \acute{\alpha}, \beta, \acute{\beta}; \lambda; 1-\frac{x}{t}; 1-\frac{t}{x})f(t)dt. \end{eqnarray}$

(1.3)

where $F_{3}$ is the Appell function.

Definition 1.3. ^[17] Appell function $F_{3}$ also called the (Horn function) and defined for $\alpha, \acute{\alpha}, \beta, \grave{\beta}, \lambda \in \mathbb{C}$ , as follows:

$\begin{equation} F_{3}(\alpha, \acute{\alpha}, \beta, \acute{\beta}; \lambda; x; y) = \sum\limits^{\infty}_{m, n = 0}\frac{(\alpha)_{m}(\acute{\alpha})_{n}(\beta)_{m}(\acute{\beta})_{n}}{(\lambda)_{m+n}m!n!}x^{m}y^{n}, \; \; \; \; \; \; \; max\{|x|, |y|\} \lt 1 \end{equation}$

(1.4)

Definition 1.4. ^[18,19] The integral representation of gamma function is defined for $\Re(s) > 0$ , as follows:

$\begin{equation} \Gamma(s) = \int^{\infty}_{0}u^{s-1}e^{-u}du. \end{equation}$

(1.5)

Definition 1.5. ^[18,19] Classical beta function is defined for $\Re(x) > 0$ and $\Re(y) > 0$ , as follows:

$\begin{align} B (x, y)& = \int^{1}_{0}t^{x-1}(1-t)^{y-1}dt \end{align}$

(1.6)

$\begin{align} & = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}. \end{align}$

(1.7)

Definition 1.6. ^[20,21] Extended beta function is defined for $\Re(x) > 0$ , $\Re(y) > 0$ , $\Re(p) > 0$ as follows:

$\begin{equation} B_{p} (x, y) = \int^{1}_{0}t^{x-1}(1-t)^{y-1}\exp\bigg(\frac{-p}{t(1-t)}\bigg)dt, \end{equation}$

(1.8)

if $p = 0$ , then extended beta function $B_{p} (x, y)$ reduces into the classical beta function.

Definition 1.7. ^[22] Generalized Wright type hypergeometric function is defined as follows:

$\begin{equation} _{r}\psi_{s}(z) = {_{r}\psi_{s}} \left[ \begin{array}{c} (y_{j}, h_{j})_{1, r}\\ (x_{i}, q_{i})_{1, s} \end{array}\middle|z \right]\equiv\sum^{\infty}_{n = 0}\frac{\prod^{r}_{j = 1}\Gamma(y_{j}+h_{j}n)}{\prod^{s}_{i = 1}\Gamma(x_{i}+q_{i}n)}\frac{z^{n}}{n!}. \end{equation}$

(1.9)

where $z\in \mathbb{C}$ , $y_{j}, x_{i}\in \mathbb{C}$ and $h_{j}, q_{i} \in \Re$ $(j = 1, 2 \cdots r; i = 1, 2 \cdots s)$ .

Definition 1.8. ^[23] Laplace transform is defined $\Re(s) > 0$ , as follows:

$\begin{equation} Ł[f(t)] = f(s) = \int^{\infty}_{0}e^{-st}f(t)dt. \end{equation}$

(1.10)

Definition 1.9. ^[24] Euler transform of a function $f(z)$ is defined as follows:

$\begin{align} B\{f(z);a, b\} = \int^{1}_{0}z^{a-1}(1-z)^{b-1}f(z)dz\; \; \; \; \; \; (\Re(a) \gt 0, \Re(b) \gt 0).\\ \end{align}$

(1.11)

Definition 1.10. ^[24] Mellin transform of the function $f(z)$ is defined as follows:

$\begin{eqnarray} \mathfrak{M}\{f(z); s\} = \int^{\infty}_{0}z^{s-1}f(z)dz = f^{*}(s), \; \; \; \; \; \Re(s) \gt 0, \\ \end{eqnarray}$

(1.12)

then inverse Mellin transform

$\begin{align} f(z) = \mathfrak{M}^{-1}[f^{*}(s); z] = \frac{1}{2\pi i}\int^{\lambda+i\infty}_{\lambda-i\infty}f^{*}z^{-s}ds, \; \; \; \; \; \; \lambda \gt 0.\\ \end{align}$

(1.13)

Definition 1.11. The Pochhammer symbol defined as

$\begin{align} (\delta)_{n} = \left\{ \begin{array}{ll} 1, & \hbox{$n = 0$} \\ \delta(\delta+1)(\delta+2) \cdots (\delta+n-1), & \hbox{$n = 1, 2 \cdots $} \end{array} \right. \end{align}$

(1.14)

$\begin{align} &(\delta)_{n} = \frac{\Gamma(\delta+n)}{\Gamma(\delta)} \end{align}$

(1.15)

$\begin{align} &(\delta)_{kn} = \frac{\Gamma(\delta+kn)}{\Gamma(\delta)}, \end{align}$

(1.16)

where $\delta \in \mathbb{C}$ and $n, k\in \mathbb{N}$ .

Definition 1.12. The E $^{1}$ GMBF $\mathbb{J}_{(\beta_{j})_{m}, k, b, \delta}^{(\alpha_{j})_{m}, \gamma, c}(z)$ is defined in the following way:

$\begin{align} \mathbb{J}^{c, b, \delta}_{(\gamma, d); k}[(\alpha_{j}, \beta_{j})_{m}; (z; p)]& = \mathbb{J}^{k, c, (\gamma, d); k}_{\delta, b, (\alpha_{j}, \beta_{j})_{m}}(z; p)\\ & = \sum^{\infty}_{n = 0}\frac{B_{p}(\gamma+kn, d-\gamma)}{B(\gamma, d-\gamma)}\frac{c^{n}(d)_{kn}(-z)^{n}}{(\delta)_{n}\prod^{m}_{j = 1}\Gamma(\alpha_j n+\beta_j+\frac{1+b}{2})}. \end{align}$

(1.17)

where $\alpha_{j}, \beta_{j}, b, d, \delta, \gamma, c \in \mathbb{C}$ $(j = 1, 2 \cdots m)$ , $p\geq0$ be such that $\sum^m_{j = 1}\Re(\alpha_j) > max\{0; \Re(k)-1\}$ ; $k > 0$ , $\Re(\beta_j) > -1$ , $\Re(d) > 0$ , $\Re(\gamma) > 0$ , $\Re(\delta) > 0$ .

Remark 1.1. The E $^{1}$ GMBF can also be write as

$\begin{align} \mathbb{J}^{c, b, \delta}_{(\gamma, d); k}[(\alpha_{j}, \beta_{j})_{m}; (z; p)] = \mathbb{J}^{b, \delta}_{(\gamma, d); k}[(\alpha_{j}, \beta_{j})_{m}; (cz; p)]. \end{align}$

(1.18)

2. Particular special cases

In this section, we establish some particular special cases of E $^{1}$ GMBF as below

● if we set $p = 0$ , then E $^{1}$ GMBF reduce into extended multi-index Bessel function

$\begin{equation} \mathbb{J}^{c, b, \delta}_{\gamma; k}[(\alpha_{j}, \beta_{j})_{m}; (z)] = \mathbb{J}_{(\beta_{j})_{m}, k, b, \delta}^{(\alpha_{j})_{m}, \gamma , c}(z) = \sum\limits^{\infty}_{n = 0}\frac{(c)^n (\gamma)_{kn}(-z)^{n}}{(\delta)_{n}\prod^{m}_{j = 1}\Gamma(\alpha_j n+\beta_j+\frac{1+b}{2})}. \end{equation}$

(2.1)

● when $p = 0$ , $c = b = \delta = 1$ , then

$\begin{equation} \mathbb{J}^{1, 1, 1}_{\gamma; k}[(\alpha_{j}, \beta_{j})_{m}; (z)] = J^{(\alpha_{j})_{m}, \gamma}_{(\beta_{j})_{m}, k}(z) = \sum\limits^{\infty}_{n = 0}\frac{ (\gamma)_{kn}(-z)^{n}}{n!\prod^{m}_{j = 1}\Gamma(\alpha_j n+\beta_j+1)}. \end{equation}$

(2.2)

● if we put $p = 0$ , $c = b = \delta = m = 1$ , then E $^{1}$ GMBF reduce to the generalized Bessel-Maitland function as,

$\begin{equation} \mathbb{J}^{1, 1, 1}_{\gamma; k}[(\alpha, \beta); (z)] = \mathrm{J}^{\alpha, \gamma}_{\beta, k}(z) = \sum\limits^{\infty}_{n = 0}\frac{ (\gamma)_{kn}(-z)^{n}}{n!\Gamma(\alpha n+\beta+1)}. \end{equation}$

(2.3)

● when $p = 0$ , $k = 0$ , $\delta = c = b = 1$ , then E $^{1}$ GMBF reduce to the Bessel-Maitland function as given below

$\begin{equation} \mathbb{J}^{1, 1, 1}_{\gamma}[(\alpha, \beta); (z)] = \mathrm{J}^{\alpha}_{\beta}(z) = \sum\limits^{\infty}_{n = 0}\frac{(-z)^{n}}{n!\Gamma(\alpha n+\beta+1)}. \end{equation}$

(2.4)

● if we put $p = 0$ , $c = \delta = 1$ , $z = -z$ and set $\beta_{j} = \beta_{j}-1$ , then E $^{1}$ GMBF reduce to the multi-index Mittag Leffler function as given below

$\begin{equation} \mathbb{J}^{1, b, 1}_{\gamma; k}[(\alpha_{j}, \beta_{j})_{m}; (-z)] = E_{\gamma, k}[(\alpha_{j}, \beta)_{j})_{j = 1}^{m}] = \sum\limits^{\infty}_{n = 0}\frac{(\gamma)_{kn}}{\prod^{m}_{j = 1}(\alpha_{j}n +\beta_{j})}\frac{z^{n}}{n!}. \end{equation}$

(2.5)

● if we set $p = k = 0$ , $b = c = m = 1$ , $\alpha_{1} = \delta = 1$ , $\beta_{1} = \nu$ and replace $z = \frac{z^{2}}{4}$ then E $^{1}$ GMBF reduce into Bessel function of fist kind

$\begin{equation} \mathbb{J}^{1, 1, 1}_{\gamma; 0}[(1, \nu)_{m}; (\frac{z^{2}}{4})] = \sum\limits^{\infty}_{n = 0}\frac{(-z)^{n}}{n!\Gamma(n+\nu+1)}. \end{equation}$

(2.6)

3. Results of E $^{1}$ GMBF

In this section, we investigate the E $^{1}$ GMBF, and studied some important observations. Moreover, we develop integral and differential of E $^{1}$ GMBF in the form of theorems.

Theorem 3.1. The E $^{1}$ GMBF can be able to represent with $\alpha_{j}, \beta_{j}, b, \delta, \gamma, c \in \mathbb{C}$ $(j = 1, 2 \cdots m)$ be such that $\sum^m_{j = 1}\Re(\alpha_j) > max\{0; \Re(k)-1\}$ ; $k > 0$ , $\Re(\beta_j) > -1$ , $\Re(\gamma) > 0$ , $\Re(\delta) > 0$ then following relation holds

$\begin{eqnarray} \mathbb{J}^{k, c, (\gamma, d); k}_{\delta, b, (\alpha_{j}, \beta_{j})_{m}}(z; p) = \frac{1}{B(\gamma, d-\gamma)}\int^{1}_{0}t^{\gamma-1}(1-t)^{^{d-\gamma-1}}e^{\frac{-p}{t(1-t)}}\mathbb{J}_{(\beta_{j})_{m}, k, b, \delta}^{(\alpha_{j})_{m}, \gamma , c}(t^{k}z)dt. \end{eqnarray}$

(3.1)

Proof. Using the definition of Eq (1.8) in (1.17), we obtain

$\begin{align} \mathbb{J}^{k, c, (\gamma, d); k}_{\delta, b, (\alpha_{j}, \beta_{j})_{m}}(z; p) = &\sum^{\infty}_{n = 0}\bigg\{\int^{1}_{0}t^{\gamma+kn-1}(1-t)^{d-\gamma-1}e^{\frac{-p}{t(1-t)}}\bigg\}\\ \times&\frac{c^{n}(d)_{kn}(-z)^{n}}{B({\gamma, d-\gamma})(\delta)_{n}\prod^{m}_{j = 1}\Gamma(\alpha_j n+\beta_j+\frac{1+b}{2})}dt. \end{align}$

(3.2)

Changing the order of summation and integration, and after simplification of Eq (3.2), we get

$\begin{align} \mathbb{J}^{k, c, (\gamma, d); k}_{\delta, b, (\alpha_{j}, \beta_{j})_{m}}&(z; p)\\ = &\frac{1}{B({\gamma, d-\gamma})}\int^{1}_{0}t^{\gamma-1}(1-t)^{d-\gamma-1}e^{\frac{-p}{t(1-t)}} \sum^{\infty}_{n = 0} \frac{c^{n}(d)_{kn}(-{t^{k}}z)^{n}}{(\delta)_{n}\prod^{m}_{j = 1}\Gamma(\alpha_j n+\beta_j+\frac{1+b}{2})}dt.\\ \end{align}$

(3.3)

Using Eq (1.1) in Eq (3.3), we obtain the desired result in theorem 3.1.

Corollary 3.1. Let $\alpha_{j}, \beta_{j}, b, \delta, \gamma, c \in \mathbb{C}$ $(j = 1, 2 \cdots m)$ be such that $\sum^m_{j = 1}\Re(\alpha_j) > max\{0; \Re(k)-1\}$ ; $k > 0$ , $\Re(\beta_j) > -1$ , $\Re(\gamma) > 0$ , $\Re(\delta) > 0$ . Taking $t = \frac{r}{1+r}$ in theorem 3.1, then following relation holds

$\begin{eqnarray} \mathbb{J}^{k, c, (\gamma, d); k}_{\delta, b, (\alpha_{j}, \beta_{j})_{m}}(z; p) = \frac{1}{B(\gamma, d-\gamma)}\int^{\infty}_{0}\frac{r^{\gamma-1}}{(1+r)^{d}}e^{\frac{-p(1+r)^{2}}{r}}\mathbb{J}_{(\beta_{j})_{m}, k, b, \delta}^{(\alpha_{j})_{m}, \gamma , c}\big(\frac{r^{k} z}{(1+r)^{k}}\big)dr.\\ \end{eqnarray}$

(3.4)

Corollary 3.2. Let $\alpha_{j}, \beta_{j}, b, \delta, \gamma, c \in \mathbb{C}$ $(j = 1, 2 \cdots m)$ be such that $\sum^m_{j = 1}\Re(\alpha_j) > max\{0; \Re(k)-1\}$ ; $k > 0$ , $\Re(\beta_j) > -1$ , $\Re(\gamma) > 0$ , $\Re(\delta) > 0$ and consider $t = \cos^{2}\theta$ in theorem 3.1, then following relation holds

$\begin{align} \mathbb{J}^{k, c, (\gamma, d); k}_{\delta, b, (\alpha_{j}, \beta_{j})_{m}}(z; p) = &\frac{2}{B(\gamma, d-\gamma)}\int^{\frac{\pi}{2}}_{0}(\cos \theta)^{2\gamma-1}(\sin \theta)^{2d-2\gamma-1}\exp{\big(\frac{-p}{\sin^{2} \theta \cos^{2} \theta}\big)}\\ \times&\mathbb{J}_{(\beta_{j})_{m}, k, b, \delta}^{(\alpha_{j})_{m}, \gamma , c}\big({z\cos^{2k}\theta}\big)d\theta.\\ \end{align}$

(3.5)

Theorem 3.2. Let $\alpha, \beta, b, \delta, \gamma, c \in \mathbb{C}$ be such that $\Re(\alpha) > max\{0; \Re(k)-1\}$ ; $k > 0$ , $\Re(\beta) > -1$ , $\Re(\gamma) > 0$ , $\Re(\delta) > 0$ , then the following recurrence relation holds in the definition (1.17) for $j = 1$ as

$\begin{eqnarray} \mathbb{J}^{k, c, (\gamma, d); k}_{\delta, b, \alpha, \beta}(z; p) = (\beta+\frac{b+1}{2}) \mathbb{J}^{k, c, (\gamma, d); k}_{\delta, b, \alpha, \beta+1}(z; p)+\alpha z\frac{d}{dz}\mathbb{J}^{k, c, (\gamma, d); k}_{\delta, b, (\alpha, \beta+1)}(z; p).\\ \end{eqnarray}$

(3.6)

Proof. Consider the definition of (1.17) for $j = 1$ , and the right side of the Eq (3.6), we get

$\begin{align} (\beta+\frac{b+1}{2}) &\mathbb{J}^{k, c, (\gamma, d); k}_{\delta, b, (\alpha, \beta+1)}(z; p)+\alpha z\frac{d}{dz}\mathbb{J}^{k, c, (\gamma, d); k}_{\delta, b, (\alpha, \beta+1)}(z; p)\\ = & (\beta+\frac{b+1}{2})\sum^{\infty}_{n = 0}\frac{B_{p}(\gamma+kn, d-\gamma)}{B(\gamma, d-\gamma)}\frac{c^{n}(d)_{kn}(-z)^{n}}{(\delta)_{n} \Gamma(\alpha n+\beta+1+\frac{1+b}{2})}\\ +&\alpha z\frac{d}{dz}\sum^{\infty}_{n = 0}\frac{B_{p}(\gamma+kn, d-\gamma)}{B(\gamma, d-\gamma)}\frac{c^{n}(d)_{kn}(-z)^{n}}{(\delta)_{n} \Gamma(\alpha n+\beta+1+\frac{1+b}{2})}\\ = &\sum^{\infty}_{n = 0}\frac{B_{p}(\gamma+kn, d-\gamma)}{B(\gamma, d-\gamma)}\frac{c^{n}(d)_{kn}}{(\delta)_{n}}\\ \times&\bigg[\frac{(\beta+\frac{b+1}{2})(-z)^{n}}{\Gamma(\alpha n+\beta+1+\frac{1+b}{2})}+\frac{\alpha z\frac{d}{dz}(-z)^{n}}{\Gamma(\alpha n+\beta+1+\frac{1+b}{2})}\bigg]\\ = &\sum^{\infty}_{n = 0}\frac{B_{p}(\gamma+kn, d-\gamma)}{B(\gamma, d-\gamma)}\frac{c^{n}(d)_{kn}(-z)^{n}(\alpha n+\beta+\frac{1+b}{2})}{(\delta)_{n}\Gamma(\alpha n+\beta+1+\frac{1+b}{2})}\\ = &\sum^{\infty}_{n = 0}\frac{B_{p}(\gamma+kn, d-\gamma)}{B(\gamma, d-\gamma)}\frac{c^{n}(d)_{kn}(-z)^{n}}{(\delta)_{n} \Gamma(\alpha n+\beta+\frac{1+b}{2})}\\ = &\mathbb{J}^{k, c, (\gamma, d); k}_{\delta, b, (\alpha, \beta)}(z; p) \end{align}$

(3.7)

Theorem 3.3. For the E $^{1}$ GMBF we have the following higher derivative formula for $\delta = 1$ , is given below

$\begin{eqnarray} \frac{d^{n}}{dz^{n}}\mathbb{J}^{k, c, (\gamma, d); k}_{1, b, (\alpha_{j}, \beta_{j})_{m}}(z; p) = (-c)^{n}(d)_{k}(d+k)_{k} \cdots (d+(n-1)k)_{k}\mathbb{J}^{k, c, (\gamma+kn, d+kn); k}_{1, b, (\alpha_{j}, \beta_{j}+\alpha_{j} n)_{m}}(z; p).\\ \end{eqnarray}$

(3.8)

where $\alpha_{j}, \beta_{j}, b, \gamma, c \in \mathbb{C}$ $(j = 1, 2 \cdots m)$ be such that $\sum^m_{j = 1}\Re(\alpha_j) > max\{0; \Re(k)-1\}$ ; $k > 0$ , $\Re(\beta_j) > -1$ , $\Re(\gamma) > 0$ .

Proof. Differentiation with respect to $z$ in Eq (1.17), we get

$\begin{align} &\frac{d}{dz}\mathbb{J}^{k, c, (\gamma, d); k}_{1, b, (\alpha_{j}, \beta_{j})_{m}}(z; p) = \sum^{\infty}_{n = 1}\frac{B_{p}(\gamma+kn, d-\gamma)}{B(\gamma, d-\gamma)}\frac{c^{n}(d)_{kn}(-1)^{n}n z^{n-1}}{n!\prod^{m}_{j = 1}\Gamma(\alpha_j n+\beta_j+\frac{1+b}{2})}\\ & = \sum^{\infty}_{n = 1}\frac{B_{p}(\gamma+k(n-1)+k, d-\gamma)}{B(\gamma, d-\gamma)}\frac{(-c)^{n}(d)_{k(n-1)+k}n z^{n-1}}{n(n-1)!\prod^{m}_{j = 1}\Gamma(\alpha_j n+\beta_j+\frac{1+b}{2})} \end{align}$

(3.9)

we can write the pochhammer symbols as

$\begin{align} (d)_{k(n-1)+k}& = \frac{\Gamma(d+k(n-1)+k)}{\Gamma(d)}\\ & = \frac{\Gamma(d+k(n-1)+k)}{\Gamma(d+k)}\frac{\Gamma(d+k)}{\Gamma(d)}\\ & = (d+k)_{(n-1)k}(d)_{k}. \end{align}$

(3.10)

Now, using the Eq (3.10) in Eq (3.9), we have

$\begin{align} &\frac{d}{dz}\mathbb{J}^{k, c, (\gamma, d); k}_{1, b, (\alpha_{j}, \beta_{j})_{m}}(z; p)\\ & = (-c)(d)_{k}\sum^{\infty}_{n = 1}\frac{B_{p}(\gamma+k+k(n-1), d-\gamma)(-c)^{n-1}(d+k)_{k(n-1)} z^{n-1}}{B(\gamma, d-\gamma)(n-1)!\prod^{m}_{j = 1}\Gamma(\alpha_j (n-1)+\alpha_{j}+\beta_j+\frac{1+b}{2})}\\ & = (-c)(d)_{k}\mathbb{J}^{k, c, (\gamma+k, d+k); k}_{1, b, (\alpha_{j}, \beta_{j}+\alpha_{j})_{m}}(z; p). \end{align}$

(3.11)

Again differentiation with respect to $z$ in Eq (3.9), we have

$\begin{eqnarray} \frac{d^{2}}{dz^{2}}\mathbb{J}^{k, c, (\gamma, d); k}_{1, b, (\alpha_{j}, \beta_{j})_{m}}(z; p) = (-c)^{2}(d)_{k}(d+k)_{k}\mathbb{J}^{k, c, (\gamma+2k, d+2k); k}_{1, b, (\alpha_{j}, \beta_{j}+2\alpha_{j})_{m}}(z; p), \end{eqnarray}$

continue this technique up to $n$ times, we obtain the desired result which state in the theorem 3.3.

Theorem 3.4. Let $\alpha_{j}, \beta_{j}, d, \gamma, c, \lambda \in \mathbb{C}$ $(j = 1, 2 \cdots m)$ , $p\geq0$ be such that $\sum^m_{j = 1}\Re(\alpha_j) > max\{0; \Re(k)-1\}$ ; $k > 0$ , $\Re(\beta_j) > 0$ , $\Re(d) > 0$ , $\Re(\gamma) > 0$ , $\Re(\delta) > 0$ then the following relation holds as:

$\begin{eqnarray} \frac{d^{n}}{dz^{n}}\big\{z^{\beta_{1} \cdots \beta_{m}-1}\mathbb{J}^{k, c, (\gamma, d); k}_{1, -1, (\alpha_{j}, \beta_{j})_{m}}(\lambda z^{\alpha_{1} \cdots \alpha_{m}}; p)\big\} = \frac{\mathbb{J}^{k, c, (\gamma, d); k}_{1, -1, (\alpha_{j}, \beta_{j}-n)_{m}}(\lambda z^{\alpha_{1} \cdots \alpha_{m}}; p)}{z^{n-\beta_{1} \cdots -\beta_{m}+1}}.\\ \end{eqnarray}$

(3.12)

Proof. Replacing $z$ by $\lambda z^{\alpha_{j} \cdots \alpha_{j}}$ , $b = -1$ and $\delta = 1$ in Eq (1.17), take its product $z^{\beta_{1} \cdots \beta_{j}}$ , and after taking differentiation with respect to $z$ up to $n$ times, we obtain our required result.

4. Integral transforms of E $^{1}$ GMBF

In this section, we establish some integral transforms (Euler, Mellin and Laplace transform) of E $^{1}$ GMBF in the form of theorems, and also discuss its sub cases.

Theorem 4.1. Euler transform of E $^{1}$ GMBF holds for $\alpha_{j}, \beta_{j}, b, \delta, \gamma, c \in \mathbb{C}$ $(j = 1, 2 \cdots m)$ be such that $\sum^m_{j = 1}\Re(\alpha_j) > max\{0; \Re(k)-1\}$ ; $k > 0$ , $\Re(\beta_j) > -1$ , $\Re(\gamma) > 0$ , $\Re(\delta) > 0$ .

$\begin{eqnarray} B\{\mathbb{J}^{k, c, (\gamma, d); k}_{\delta, -1, (\alpha_{j}, \beta_{j})_{m}}(\lambda z^{\alpha_{j}}; p); \beta_{1} \cdots \beta_{m}, 1\} = \mathbb{J}^{k, c, (\gamma, d); k}_{\delta, -1, (\alpha_{j}, \beta_{j}+1)_{m}}(\lambda; p). \end{eqnarray}$

(4.1)

Proof. Apply the definition of Euler transform (1.9) in Eq (1.17), we get

$\begin{multline} B\{\mathbb{J}^{k, c, (\gamma, d); k}_{\delta, -1, (\alpha_{j}, \beta_{j})_{m}}(\lambda z^{\alpha_{j}}; p); \beta_{1} \cdots \beta_{m}, 1\}\\ = \int^{1}_{0}z^{\beta_{1} \cdots \beta_{m}-1}(1-z)^{1-1}\sum^{\infty}_{n = 0}\frac{B_{p}(\gamma+kn, d-\gamma)}{B(\gamma, d-\gamma)}\\ \times\frac{c^{n}(d)_{kn}(-1)^{n}(\lambda z^{\alpha_{j}})^{n}}{(\delta)_{n}\prod^{m}_{j = 1}\Gamma(\alpha_j n+\beta_j)}dz.\\ \end{multline}$

(4.2)

Interchanging the order of summations and integration in Eq (4.2), we get

$\begin{multline} B\{\mathbb{J}^{k, c, (\gamma, d); k}_{\delta, -1, (\alpha_{j}, \beta_{j})_{m}}(\lambda z^{\alpha_{j}}; p); \beta_{1} \cdots \beta_{m}, 1\}\\ = \sum^{\infty}_{n = 0}\frac{B_{p}(\gamma+kn, d-\gamma)}{B(\gamma, d-\gamma)}\frac{c^{n}(d)_{kn}(-\lambda)^{n} }{(\delta)_{n}\prod^{m}_{j = 1}\Gamma(\alpha_j n+\beta_j)}\\ \times\int^{1}_{0}z^{\beta_{1} \cdots \beta_{m}+\alpha_j n-1}(1-z)^{1-1}dz.\\ \end{multline}$

(4.3)

Using the Eq (1.6) and Eq (1.7) in Eq (4.3), then we obtain

$\begin{align} B&\{\mathbb{J}^{k, c, (\gamma, d); k}_{\delta, -1, (\alpha_{j}, \beta_{j})_{m}}(\lambda z^{\alpha_{j}}; p); \beta_{1} \cdots \beta_{m}, 1\}\\ = &\sum^{\infty}_{n = 0}\frac{B_{p}(\gamma+kn, d-\gamma)c^{n}(d)_{kn}(-\lambda)^{n}}{B(\gamma, d-\gamma)(\delta)_{n}\prod^{m}_{j = 1}\Gamma(\alpha_j n+\beta_j+1)}\\ = &\mathbb{J}^{k, c, (\gamma, d); k}_{\delta, -1, (\alpha_{j}, \beta_{j}+1)_{m}}(\lambda; p). \end{align}$

(4.4)

Theorem 4.2. The Mellin transform of E $^{1}$ GMBF is given by for $\alpha_{j}, \beta_{j}, b, \delta, \gamma, c \in \mathbb{C}$ $(j = 1, 2 \cdots m)$ be such that $\sum^m_{j = 1}\Re(\alpha_j) > max\{0; \Re(k)-1\}$ ; $k > 0$ , $\Re(\beta_j) > -1$ , $\Re(\gamma) > 0$ , $\Re(\delta) > 0$ . Then the following relation holds

$\begin{align} & \mathfrak{M}\{\mathbb{J}^{k, c, (\gamma, d); k}_{\delta, b, (\alpha_{j}, \beta_{j})_{m}}(z; p); s\}\\ & = \frac{\Gamma(s)\Gamma(\delta)\Gamma(d)\Gamma(d-\gamma+s)}{[\Gamma(\gamma)]^{2}\Gamma(d-\gamma)}{_{3}\psi_{m+2}}\left[ \begin{array}{c} (\gamma, k)(\gamma+s, k)(1, 1) \\ (\delta, 1)(d+2s, k)(\beta_j+\frac{1+b}{2}, \alpha_j)|^{m}_{j = 1} \\ \end{array}\middle|-cz \right]. \end{align}$

Proof. By applying the definition of the Mellin transform to the E $^{1}$ GMBF, we have

$\begin{eqnarray} \mathfrak{M}\{\mathbb{J}^{k, c, (\gamma, d); k}_{\delta, b, (\alpha_{j}, \beta_{j})_{m}}(z; p); s\} = \int^{\infty}_{0}p^{s-1}\mathbb{J}^{k, c, (\gamma, d); k}_{\delta, b, (\alpha_{j}, \beta_{j})_{m}}(z; p)dp.\\ \end{eqnarray}$

(4.5)

Using theorem 3.1 in right side of Eq (4.5), we get

$\begin{multline} \mathfrak{M}\{\mathbb{J}^{k, c, (\gamma, d); k}_{\delta, b, (\alpha_{j}, \beta_{j})_{m}}(z; p); s\}\\ = \frac{1}{B(\gamma, d-\gamma)}\int^{\infty}_{0}p^{s-1}\bigg\{\int^{1}_{0}t^{\gamma-1}(1-t)^{^{d-\gamma-1}}e^{\frac{-p}{t(1-t)}}\mathbb{J}_{(\beta_{j})_{m}, k, b, \delta}^{(\alpha_{j})_{m}, \gamma , c}(t^{k}z)dt\bigg\}dp.\\ \end{multline}$

(4.6)

Interchanging the order of integration in Eq (4.6), then we have

$\begin{multline} \mathfrak{M}\{\mathbb{J}^{k, c, (\gamma, d); k}_{\delta, b, (\alpha_{j}, \beta_{j})_{m}}(z; p); s\}\\ = \frac{1}{B(\gamma, d-\gamma)}\int^{1}_{0}t^{\gamma-1}(1-t)^{^{d-\gamma-1}}\mathbb{J}_{(\beta_{j})_{m}, k, b, \delta}^{(\alpha_{j})_{m}, \gamma , c}(t^{k}z)\bigg\{\int^{\infty}_{0}p^{s-1}e^{\frac{-p}{t(1-t)}}dp\bigg\}dt.\\ \end{multline}$

(4.7)

Now, putting $\frac{p}{t(1-t)} = u$ in Eq (4.7), and applying the mathematical formula of Eq (1.5), we get

$\begin{align} \mathfrak{M}\{\mathbb{J}^{k, c, (\gamma, d); k}_{\delta, b, (\alpha_{j}, \beta_{j})_{m}}(z; p); s\} = \frac{\Gamma(s)}{B(\gamma, d-\gamma)}\int^{1}_{0}t^{\gamma+s-1}(1-t)^{^{d-\gamma+s-1}}\mathbb{J}_{(\beta_{j})_{m}, k, b, \delta}^{(\alpha_{j})_{m}, \gamma , c}(t^{k}z)dt.\\ \end{align}$

(4.8)

Using Eq (1.1), and interchanging the order of integration and summation in Eq (4.8), we obtain

$\begin{multline} \mathfrak{M}\{\mathbb{J}^{k, c, (\gamma, d); k}_{\delta, b, (\alpha_{j}, \beta_{j})_{m}}(z; p); s\}\\ = \frac{\Gamma(s)}{B(\gamma, d-\gamma)}\sum^{\infty}_{n = 0}\frac{(c)^n (\gamma)_{kn}(-z)^{n}}{(\delta)_{n}\prod^{m}_{j = 1}\Gamma(\alpha_j n+\beta_j+\frac{1+b}{2})}\int^{1}_{0}t^{\gamma+s+kn-1}(1-t)^{^{d-\gamma+s-1}}dt.\\ \end{multline}$

(4.9)

Using Eq (1.6) and Eq (1.7) in Eq (4.9), we get

$\begin{multline} \mathfrak{M}\{\mathbb{J}^{k, c, (\gamma, d); k}_{\delta, b, (\alpha_{j}, \beta_{j})_{m}}(z; p); s\}\\ = \frac{\Gamma(s)}{B(\gamma, d-\gamma)}\sum^{\infty}_{n = 0}\frac{(c)^n (\gamma)_{kn}(-z)^{n}}{(\delta)_{n}\prod^{m}_{j = 1}\Gamma(\alpha_j n+\beta_j+\frac{1+b}{2})}\frac{\Gamma(\gamma+s+kn)\Gamma(d-\gamma+s)}{\Gamma(2s+kn+d)}.\\ \end{multline}$

(4.10)

After simplification in Eq (4.10), we get

$\begin{align*} &\mathfrak{M}\{\mathbb{J}^{k, c, (\gamma, d); k}_{\delta, b, (\alpha_{j}, \beta_{j})_{m}}(z; p); s\}\\ = &\frac{\Gamma(s)\Gamma(\delta)\Gamma(d)\Gamma(d-\gamma+s)}{[\Gamma(\gamma)]^{2}\Gamma(d-\gamma)}\sum^{\infty}_{n = 0}\frac{\Gamma(\gamma+kn)(-cz)^{n}}{\Gamma(\delta+n)\prod^{m}_{j = 1}\Gamma(\alpha_j n+\beta_j+\frac{1+b}{2})}\frac{\Gamma(\gamma+s+kn)}{\Gamma(2s+kn+d)}\\ = &\frac{\Gamma(s)\Gamma(\delta)\Gamma(d)\Gamma(d-\gamma+s)}{[\Gamma(\gamma)]^{2}\Gamma(d-\gamma)}{_{3}\psi_{m+2}}\left[ \begin{array}{c} (\gamma, k)(\gamma+s, k)(1, 1) \\ (\delta, 1)(d+2s, k)(\beta_j+\frac{1+b}{2}, \alpha_j)|^{m}_{j = 1} \\ \end{array}\middle|-cz \right]. \end{align*}$

Corollary 4.1. Let $\alpha_{j}, \beta_{j}, b, \delta, \gamma, c \in \mathbb{C}$ $(j = 1, 2 \cdots m)$ be such that $\sum^m_{j = 1}\Re(\alpha_j) > max\{0; \Re(k)-1\}$ ; $k > 0$ , $\Re(\beta_j) > -1$ , $\Re(\gamma) > 0$ , $\Re(\delta) > 0$ . Taking $s = 1$ in theorem 4.2, then the following relation holds

$\begin{eqnarray} \int^{\infty}_{0}\mathbb{J}^{k, c, (\gamma, d); k}_{\delta, b, (\alpha_{j}, \beta_{j})_{m}}(z; p)dp = \frac{(d-\gamma)\Gamma(\delta)}{\Gamma(\gamma)}{_{3}\psi_{m+2}}\left[ \begin{array}{c} (\gamma, k)(\gamma+1, k)(1, 1) \\ (\delta, 1)(d+2, k)(\beta_j+\frac{1+b}{2}, \alpha_j)|^{m}_{j = 1} \\ \end{array}\middle|-cz \right]. \end{eqnarray}$

(4.11)

Corollary 4.2. Let $\alpha_{j}, \beta_{j}, b, \delta, \gamma, c \in \mathbb{C}$ $(j = 1, 2 \cdots m)$ be such that $\sum^m_{j = 1}\Re(\alpha_j) > max\{0; \Re(k)-1\}$ ; $k > 0$ , $\Re(\beta_j) > -1$ , $\Re(\gamma) > 0$ , $\Re(\delta) > 0$ . Applying the inverse Mellin transform on left and right side of Eq (1.17), we gain the important complex integral representation as follows:

$\begin{align} \mathfrak{M}^{-1}\{\mathbb{J}^{k, c, (\gamma, d); k}_{\delta, b, (\alpha_{j}, \beta_{j})_{m}}(z; p); s\} = &\frac{1}{2\pi i\Gamma(\gamma)\Gamma(d-\gamma)}\int^{\lambda+i\infty}_{\lambda-i\infty}\Gamma(s)\Gamma(\delta)\Gamma(d-\gamma+s)\\ \times&{_{3}\psi_{m+2}}\left[ \begin{array}{c} (\gamma, k)(\gamma+s, k)(1, 1) \\ (\delta, 1)(d+2s, k)(\beta_j+\frac{1+b}{2}, \alpha_j)|^{m}_{j = 1} \\ \end{array}\middle|-cz \right]p^{-s}ds. \end{align}$

(4.12)

Theorem 4.3. The Laplace transform of E $^{1}$ GMBF is given as for $\alpha_{j}, \beta_{j}, b, \delta, \gamma, c \in \mathbb{C}$ $(j = 1, 2 \cdots m)$ be such that $\sum^m_{j = 1}\Re(\alpha_j) > max\{0; \Re(k)-1\}$ ; $k > 0$ , $\Re(\beta_j) > -1$ , $\Re(\gamma) > 0$ , $\Re(\delta) > 0$ .

$\begin{eqnarray} Ł(\mathbb{J}^{k, c, (\gamma, d); k}_{1, b, (\alpha_{j}, \beta_{j})_{m}}(z; p)) = \frac{1}{s}\mathbb{J}^{k, c, (\gamma, d); k}_{1, b, (\alpha_{j}, \beta_{j})_{m}}(\frac{1}{s}; p). \end{eqnarray}$

(4.13)

Proof. Using the definition of Laplace transform (1.8) in Eq (1.17), we have

$\begin{align} Ł(\mathbb{J}^{k, c, (\gamma, d); k}_{1, b, (\alpha_{j}, \beta_{j})_{m}}(z; p))& = \int^{\infty}_{0}e^{-st}\sum^{\infty}_{n = 0}\frac{B_{p}(\gamma+kn, d-\gamma)}{B(\gamma, d-\gamma)}\frac{c^{n}(d)_{kn}(-t)^{n}}{n!\prod^{m}_{j = 1}\Gamma(\alpha_j n+\beta_j+\frac{1+b}{2})}dt\\ & = \sum^{\infty}_{n = 0}\frac{B_{p}(\gamma+kn, d-\gamma)}{B(\gamma, d-\gamma)}\frac{c^{n}(d)_{kn}(-1)^{n}}{n!\prod^{m}_{j = 1}\Gamma(\alpha_j n+\beta_j+\frac{1+b}{2})}\int^{\infty}_{0}e^{-st}t^{n}dt\\ & = \sum^{\infty}_{n = 0}\frac{B_{p}(\gamma+kn, d-\gamma)}{B(\gamma, d-\gamma)}\frac{c^{n}(d)_{kn}(-1)^{n}}{n!\prod^{m}_{j = 1}\Gamma(\alpha_j n+\beta_j+\frac{1+b}{2})}\frac{n!}{s^{n+1}}\\ & = \frac{1}{s}\sum^{\infty}_{n = 0}\frac{B_{p}(\gamma+kn, d-\gamma)}{B(\gamma, d-\gamma)}\frac{c^{n}(d)_{kn}(-s)^{n}}{\prod^{m}_{j = 1}\Gamma(\alpha_j n+\beta_j+\frac{1+b}{2})}\\ & = \frac{1}{s}\mathbb{J}^{k, c, (\gamma, d); k}_{1, b, (\alpha_{j}, \beta_{j})_{m}}(\frac{1}{s}; p). \end{align}$

(4.14)

5. Relation of the E $^{1}$ GMBF with the Laguerre polynomial and Whittaker function

In this section, the authors represent the E $^{1}$ GMBF in terms of Laguerre polynomial, and Whittaker function in the form of theorems.

Theorem 5.1. Let $\alpha_{j}, \beta_{j}, b, d, \delta, \gamma, c \in \mathbb{C}$ $(j = 1, 2 \cdots m)$ , $p\geq0$ be such that $\sum^m_{j = 1}\Re(\alpha_j) > max\{0; \Re(k)-1\}$ ; $k > 0$ , $\Re(\beta_j) > 0$ , $\Re(d) > 0$ , $\Re(\gamma) > 0$ , $\Re(\delta) > 0$ , then the E $^{1}$ GMBF holds

$\begin{eqnarray} &&e^{2p}\mathbb{J}^{k, c, (\gamma, d); k}_{\delta, b, (\alpha_{j}, \beta_{j})_{m}}(z; p) = \frac{\Gamma(\delta)\sum^{\infty}_{a, b = 0}L_{b}(p)L_{a}(p)\Gamma(b+d-\gamma+1)}{\Gamma(\gamma)B(\gamma, d-\gamma)}\\ &&\times{_{3}\psi_{m+2}}\left[ \begin{array}{c} (\gamma, k)(a+\gamma+1, k)(1, 1) \\ (\delta, 1)(a+b+d+2, k)(\beta_j+\frac{1+b}{2}, \alpha_j)|^{m}_{j = 1} \\ \end{array}\middle|-cz \right].\\ \end{eqnarray}$

(5.1)

Proof. We being recalling the valuable identity ^[25] which is

$\begin{align} e^{\frac{-p}{t(1-t)}} = e^{-2p}\sum^{\infty}_{a, b = 0}L_{b}(p)L_{a}(p)t^{a+1}(1-t)^{b+1}, \; \; \; \; \; (0 \lt t \lt 1).\\ \end{align}$

(5.2)

Applying Eq (5.2) in theorem 3.1, we get

$\begin{align} &\mathbb{J}^{k, c, (\gamma, d); k}_{\delta, b, (\alpha_{j}, \beta_{j})_{m}}(z; p)\\ & = \frac{1}{B(\gamma, d-\gamma)}\int^{1}_{0}t^{\gamma-1}(1-t)^{d-\gamma-1}e^{-2p} \sum^{\infty}_{a, b = 0}L_{b}(p)L_{a}(p)t^{a+1}(1-t)^{b+1}\mathbb{J}_{(\beta_{j})_{m}, k, b, \delta}^{(\alpha_{j})_{m}, \gamma , c}(t^{k}z)dt\\ & = \frac{1}{B(\gamma, d-\gamma)}\int^{1}_{0}t^{\gamma-1}(1-t)^{d-\gamma-1}e^{-2p}\sum^{\infty}_{a, b = 0}L_{b}(p)L_{a}(p)t^{a+1}(1-t)^{b+1}\\ &\times\sum^{\infty}_{n = 0}\frac{(c)^n (\gamma)_{kn}(-t^{k}z)^{n}}{(\delta)_{n}\prod^{m}_{j = 1}\Gamma(\alpha_j n+\beta_j+\frac{1+b}{2})}dt.\\ \end{align}$

(5.3)

Interchanging the order of integration and summations in Eq (5.3), we obtain

$\begin{align} &\mathbb{J}^{k, c, (\gamma, d); k}_{\delta, b, (\alpha_{j}, \beta_{j})_{m}}(z; p)\\ & = \frac{e^{-2p}}{B(\gamma, d-\gamma)}\sum^{\infty}_{a, b, n = 0}\frac{L_{b}(p)L_{a}(p)(\gamma)_{kn}(-cz)^{n}}{(\delta)_{n}\prod^{m}_{j = 1}\Gamma(\alpha_j n+\beta_j+\frac{1+b}{2})}\int^{1}_{0}t^{a+kn+\gamma}(1-t)^{b+d-\gamma}dt. \end{align}$

(5.4)

Using Eq (1.6) and Eq (1.7) in Eq (5.4), then we have

$\begin{align} &e^{2p}\mathbb{J}^{k, c, (\gamma, d); k}_{\delta, b, (\alpha_{j}, \beta_{j})_{m}}(z; p)\\ & = \frac{1}{B(\gamma, d-\gamma)}\sum^{\infty}_{a, b, n = 0}\frac{L_{b}(p)L_{a}(p)(\gamma)_{kn}(-cz)^{n}}{(\delta)_{n}\prod^{m}_{j = 1}\Gamma(\alpha_j n+\beta_j+\frac{1+b}{2})}\frac{\Gamma(a+kn+\gamma+1)\Gamma(b+d-\gamma+1)}{\Gamma(a+b+d+kn+2)}\\ & = \frac{\Gamma(\delta)\sum^{\infty}_{a, b = 0}L_{b}(p)L_{a}(p)\Gamma(b+d-\gamma+1)}{\Gamma(\gamma)B(\gamma, d-\gamma)}\\ &\times{_{3}\psi_{m+2}}\left[ \begin{array}{c} (\gamma, k)(a+\gamma+1, k)(1, 1) \\ (\delta, 1)(a+b+d+2, k)(\beta_j+\frac{1+b}{2}, \alpha_j)|^{m}_{j = 1} \\ \end{array}\middle|-cz \right].\\ \end{align}$

(5.5)

Theorem 5.2. For the E $^{1}$ GMBF with $\alpha_{j}, \beta_{j}, b, \delta, \gamma, c \in \mathbb{C}$ $(j = 1, 2 \cdots m)$ be such that $\sum^m_{j = 1}\Re(\alpha_j) > max\{0; \Re(k)-1\}$ ; $k > 0$ , $\Re(\beta_j) > -1$ , $\Re(\gamma) > 0$ , $\Re(\delta) > 0$ , we have

$\begin{align} e^{\frac{3p}{2}}\mathbb{J}^{k, c, (\gamma, d); k}_{\delta, b, (\alpha_{j}, \beta_{j})_{m}}(z; p)& = \frac{e^{-p}}{B(\gamma, d-\gamma)}\sum^{\infty}_{a, n = 0}\frac{L_{a}(p)(\delta)_{kn}(-cz)^{n}}{(\delta)_{n}\prod^{m}_{j = 1}\Gamma(\alpha_j n+\beta_j+\frac{b+1}{2})}\\ &\times\Gamma(d-\gamma+1)p^{\frac{\gamma+kn}{2}}\mathrm{W}_{\frac{-1+\gamma-2d-kn}{2}, \frac{\gamma+kn}{2}}.\\ \end{align}$

(5.6)

Proof. Allowing for the following equality $e^{\frac{-p}{t(1-t)}} = e^{(\frac{-p}{1-t})}e^{(\frac{-p}{t})}$ and via generating function related to the Laguerre polynomial ^[25], we obtain

$\begin{eqnarray} e^{\frac{-p}{t(1-t)}} = e^{-p}e^{\frac{-p}{t}}(1-t)\sum^{\infty}_{a = 0}L_{a}(p)t^{n}.\\ \end{eqnarray}$

(5.7)

Using Eq (5.7) in Eq (1.17), we have

$\begin{align} &\mathbb{J}^{k, c, (\gamma, d); k}_{\delta, b, (\alpha_{j}, \beta_{j})_{m}}(z; p)\\ & = \frac{1}{B(\gamma, d-\gamma)}\int^{1}_{0}t^{\gamma-1}(1-t)^{d-\gamma-1}e^{-p}e^{\frac{-p}{t}}(1-t) \sum^{\infty}_{a = 0}L_{a}(p)t^{n}\mathbb{J}_{(\beta_{j})_{m}, k, b, \delta}^{(\alpha_{j})_{m}, \gamma , c}(t^{k}z)dt\\ & = \frac{e^{-p}}{B(\gamma, d-\gamma)}\sum^{\infty}_{a, n = 0}\frac{L_{a}(p)(\delta)_{kn}(-cz)^{n}}{(\delta)_{n}\prod^{m}_{j = 1}\Gamma(\alpha_j n+\beta_j+\frac{b+1}{2})}\int^{1}_{0}t^{\gamma+kn-1}(1-t)^{d-\gamma}e^{\frac{-p}{t}}dt.\\ \end{align}$

(5.8)

Now, integral representation of Whittaker function is defined ^[26] as follows

$\begin{eqnarray} \int^{1}_{0}t^{\mu-1}(1-t)^{\nu-1}e^{\frac{-p}{t}}dt = \Gamma(\nu)p^{\frac{\mu-1}{2}}e^{\frac{-p}{2}}\mathrm{W}_{\frac{1-\mu-2\nu}{2}, \frac{\mu}{2}}(p).\\ \end{eqnarray}$

(5.9)

Using Eq (5.9) in Eq (5.8), then we have

$\begin{align} \mathbb{J}^{k, c, (\gamma, d); k}_{\delta, b, (\alpha_{j}, \beta_{j})_{m}}(z; p)& = \frac{e^{-p}}{B(\gamma, d-\gamma)}\sum^{\infty}_{a, n = 0}\frac{L_{a}(p)(\delta)_{kn}(-cz)^{n}}{(\delta)_{n}\prod^{m}_{j = 1}\Gamma(\alpha_j n+\beta_j+\frac{b+1}{2})}\\ &\times\Gamma(d-\gamma+1)p^{\frac{\gamma+kn}{2}}e^{\frac{-p}{2}}\mathrm{W}_{\frac{-1+\gamma-2d-kn}{2}, \frac{\gamma+kn}{2}}.\\ \end{align}$

(5.10)

Theorem 5.3. Let $\alpha_{j}, \beta_{j}, b, d, \delta, \gamma, \sigma, \eta, c \in \mathbb{C}$ $(j = 1, 2 \cdots m)$ , $p\geq0$ be such that $\sum^m_{j = 1}\Re(\alpha_j) > max\{0; \Re(k)-1\}$ ; $k > 0$ , $\Re(\beta_j) > 0$ , $\Re(d) > 0$ , $\Re(\gamma) > 0$ , $\Re(\delta) > 0$ the E $^{1}$ GMBF holds

$\begin{align*} &(I^{\alpha, \acute{\alpha}, \beta, \acute{\beta}, \lambda}_{0^{+}}\mathbb{J}^{c, b, \delta}_{(\gamma, d); k}[(\alpha_{j}, \beta_{j})_{m}; (t^{-\sigma+\eta}; p)])(x)\\ & = \sum^{\infty}_{n = 0}\frac{B_{p}(\gamma+kn, d-\gamma)(-c)^{n}}{x^{\sigma n-\eta n+\alpha-\lambda+\acute{\alpha}}}\Gamma\bigg[ \begin{array}{ll} (d+kn)(\delta)(\lambda-\acute{\alpha}-\sigma n+\eta n-\alpha-\beta+1) \\ (\gamma)(d-\gamma)(\delta+n)(\lambda-\acute{\alpha}-\sigma n+\eta n-\beta+1) \\ \end{array}\nonumber\\ &\times \begin{array}{ll} (1-\acute{\alpha}-\sigma n+\eta n+\acute{\beta})(1-\sigma n+\eta n)\\ (1-\sigma n+\eta n+\acute{\beta})(\lambda-\acute{\alpha}-\sigma n+\eta n-\alpha+1)(\alpha_j n+\beta_j+\frac{1+b}{2})|^{m}_{j = 1} \end{array}\nonumber \bigg].\\ \end{align*}$

Proof. Consider the composition of generalized fractional integral operator having Appell function as its kernel with the E $^{1}$ GMBF,

$\begin{align} &(I^{\alpha, \acute{\alpha}, \beta, \acute{\beta}, \lambda}_{0^{+}}\mathbb{J}^{c, b, \delta}_{(\gamma, d); k}[(\alpha_{j}, \beta_{j})_{m}; (t^{-\sigma+\eta}; p)])(x)\\ & = \frac{x^{-\alpha}}{\Gamma(\lambda)}\int^{x}_{0}(x-t)^{\lambda-1}t^{-\acute{\alpha}}F_{3}(\alpha, \acute{\alpha}, \beta, \acute{\beta}; \lambda; 1-\frac{t}{x}; 1-\frac{x}{t})\sum^{\infty}_{n = 0}\frac{B_{p}(\gamma+kn, d-\gamma)}{B(\gamma, d-\gamma)}\\ &\times\frac{c^{n}(d)_{kn}(-1)^{n}t^{- \sigma n+\eta n}}{(\delta)_{n}\prod^{m}_{j = 1}\Gamma(\alpha_j n+\beta_j+\frac{1+b}{2})}dt\\ & = \frac{x^{-\alpha+\lambda-1}}{\Gamma(\lambda)}\mathbb{J}^{k, c, (\gamma, d); k}_{\delta, b, (\alpha_{j}, \beta_{j})_{m}}(1; p)|^{\infty}_{n = 0}\int^{x}_{0}(1-\frac{t}{x})^{\lambda-1}t^{-\acute{\alpha}-\sigma n+\eta n}\sum^{\infty}_{m, s = 0}\frac{(\alpha)_{m}(\acute{\alpha})_{s}(\beta)_{m}(\acute{\beta})_{s}}{\lambda_{m+s}m!s!}\\ &\times(1-\frac{t}{x})^{m}(1-\frac{x}{t})^{s}dt\\ & = \frac{x^{-\alpha+\lambda-1}}{\Gamma(\lambda)}\mathbb{J}^{k, c, (\gamma, d); k}_{\delta, b, (\alpha_{j}, \beta_{j})_{m}}(1; p)|^{\infty}_{n = 0}\sum^{\infty}_{m, s = 0}\frac{(\alpha)_{m}(\acute{\alpha})_{s}(\beta)_{m}(\acute{\beta})_{s}}{\lambda_{m+s}m!s!} \int^{x}_{0}(1-\frac{t}{x})^{m+\lambda-1}\\ &\times(1-\frac{x}{t})^{s}t^{-\acute{\alpha}-\sigma n+\eta n}dt.\\ \end{align}$

(5.11)

Putting these values $\frac{t}{x} = \tau$ $\Rightarrow$ $d\tau = x dt$ , $t = x \Rightarrow \tau = 1$ and $t = 0 \Rightarrow \tau = 0$ in Eq (5.11), then we have

(5.12)

Using Eqs (1.6) and (1.7) in Eq (5.12), we obtain

$\begin{align} &(I^{\alpha, \acute{\alpha}, \beta, \acute{\beta}, \lambda}_{0^{+}}\mathbb{J}^{c, b, \delta}_{(\gamma, d); k}[(\alpha_{j}, \beta_{j})_{m}; (t^{-\sigma+\eta}; p)])(x)-x^{-\alpha+\lambda-\acute{\alpha}}\mathbb{J}^{k, c, (\gamma, d); k}_{\delta, b, (\alpha_{j}, \beta_{j})_{m}}(\frac{x^{\eta}}{x^{\sigma}}; p)|^{\infty}_{n = 0}\\ & = \sum^{\infty}_{m, s = 0}\frac{(\alpha)_{m}(\acute{\alpha})_{s}(\beta)_{m}(\acute{\beta})_{s}(-1)^{s}}{\lambda_{m+s}m!s!}\frac{\Gamma(s+m+\lambda)\Gamma(-\acute{\alpha}-\sigma n+\eta n-s+1)}{\Gamma(\lambda)\Gamma(m+\lambda-\acute{\alpha}-\sigma n+\eta n+1)}\\ & = \frac{\Gamma(-\acute{\alpha}-\sigma n+\eta n+1)}{\Gamma(\lambda-\acute{\alpha}-\sigma n+\eta n+1)} \sum^{\infty}_{m = 0}\frac{(\alpha)_{m}(\beta)_{m}}{(\lambda-\acute{\alpha}-\sigma n+\eta n+1)_{m}m!} \sum^{\infty}_{s = 0}\frac{(\acute{\alpha})_{s}(\acute{\beta})_{s}}{(\acute{\alpha}+\sigma n-\eta n)_{s}s!}\\ & = \frac{\Gamma(-\acute{\alpha}-\sigma n+\eta n+1)\Gamma(\lambda-\acute{\alpha}-\sigma n+\eta n-\alpha-\beta+1)\Gamma(\acute{\alpha}+\sigma n-\eta n)\Gamma(\sigma n-\eta n-\acute{\beta})}{\Gamma(\lambda-\acute{\alpha}-\sigma n+\eta n-\alpha+1)\Gamma(\lambda-\acute{\alpha}-\sigma n+\eta n-\beta+1)\Gamma(\sigma n-\eta n)\Gamma(\acute{\alpha}+\sigma n-\eta n-\acute{\beta})}\\ & = \frac{\Gamma(\lambda-\acute{\alpha}-\sigma n+\eta n-\alpha-\beta+1)\Gamma(1-\acute{\alpha}-\sigma n+\eta n+\acute{\beta})\Gamma(1-\sigma n+\eta n)}{\Gamma(\lambda-\acute{\alpha}-\sigma n+\eta n-\alpha+1)\Gamma(\lambda-\acute{\alpha}-\sigma n+\eta n-\beta+1)\Gamma(1-\sigma n+\eta n+\acute{\beta})}.\\ \end{align}$

(5.13)

we have the required result

$\begin{align} &(I^{\alpha, \acute{\alpha}, \beta, \acute{\beta}, \lambda}_{0^{+}}\mathbb{J}^{c, b, \delta}_{(\gamma, d); k}[(\alpha_{j}, \beta_{j})_{m}; (t^{-\sigma+\eta}; p)])(x)\\ & = \sum^{\infty}_{n = 0}\frac{B_{p}(\gamma+kn, d-\gamma)(-c)^{n}}{x^{\sigma n-\eta n+\alpha-\lambda+\acute{\alpha}}}\Gamma\bigg[ \begin{array}{ll} (d+kn)(\delta)(\lambda-\acute{\alpha}-\sigma n+\eta n-\alpha-\beta+1) \\ (\gamma)(d-\gamma)(\delta+n)(\lambda-\acute{\alpha}-\sigma n+\eta n-\beta+1) \\ \end{array}\\ &\times \begin{array}{ll} (1-\acute{\alpha}-\sigma n+\eta n+\acute{\beta})(1-\sigma n+\eta n)\\ (1-\sigma n+\eta n+\acute{\beta})(\lambda-\acute{\alpha}-\sigma n+\eta n-\alpha+1)(\alpha_j n+\beta_j+\frac{1+b}{2})|^{m}_{j = 1} \end{array} \bigg]. \end{align}$

Theorem 5.4. Let $\alpha_{j}, \beta_{j}, b, d, \delta, \gamma, \sigma, \eta, c \in \mathbb{C}$ $(j = 1, 2 \cdots m)$ , $p\geq0$ be such that $\sum^m_{j = 1}\Re(\alpha_j) > max\{0; \Re(k)-1\}$ ; $k > 0$ , $\Re(\beta_j) > 0$ , $\Re(d) > 0$ , $\Re(\gamma) > 0$ , $\Re(\delta) > 0$ , then the E $^{1}$ GMBF holds true:

$\begin{align} &(I^{\alpha, \acute{\alpha}, \beta, \acute{\beta}, \lambda}_{-}\mathbb{J}^{c, b, \delta}_{(\gamma, d); k}[(\alpha_{j}, \beta_{j})_{m}; (t^{\frac{\sigma-d}{\eta+b}}; p)])(x)\\ & = \sum^{\infty}_{n = 0}\frac{B_{p}(\gamma+kn, d-\gamma)(-c)^{n}}{x^{\sigma n-\eta n+\alpha-\lambda+\acute{\alpha}}}\Gamma\bigg[ \begin{array}{ll} (d+kn)(\delta)(\frac{(d-\sigma)n}{(\eta+b)n}-\beta)(\frac{(d-\sigma)n}{(\eta+b)n}+\alpha-\lambda+\acute{\beta})\\ (\gamma)(d-\gamma)(\delta+n)(\frac{(d-\sigma)n}{(\eta+b)n})(\frac{(d-\sigma)n}{(\eta+b)n}+\alpha-\beta) \\ \end{array}\\ &\times \begin{array}{ll} (\frac{(d-\sigma)n}{(\eta+b)n}+\alpha-\lambda+\acute{\alpha})\\ (\frac{(d-\sigma)n}{(\eta+b)n}+\alpha-\lambda+\acute{\alpha}+\acute{\beta})(\alpha_j n+\beta_j+\frac{1+b}{2})|^{m}_{j = 1} \end{array} \bigg]. \end{align}$

Proof. Consider the composition of right side generalized fractional integral operator with the E $^{1}$ GMBF \newpage

$\begin{align} &(I^{\alpha, \acute{\alpha}, \beta, \acute{\beta}, \lambda}_{-}\mathbb{J}^{c, b, \delta}_{(\gamma, d); k}[(\alpha_{j}, \beta_{j})_{m}; (t^{\frac{\sigma-d}{\eta+b}}; p)])(x)\\ & = \frac{x^{-\acute{\alpha}}}{\Gamma(\lambda)}\int^{\infty}_{x}(t-x)^{\lambda-1}t^{-\alpha}F_{3}(\alpha, \acute{\alpha}, \beta, \acute{\beta}; \lambda; 1-\frac{x}{t}; 1-\frac{t}{x})\sum^{\infty}_{n = 0}\frac{B_{p}(\gamma+kn, d-\gamma)}{B(\gamma, d-\gamma)}\\ &\times\frac{c^{n}(d)_{kn}(-1)^{n}t^{\frac{\sigma n-d n}{\eta n+b n}}}{(\delta)_{n}\prod^{m}_{j = 1}\Gamma(\alpha_j n+\beta_j+\frac{1+b}{2})}dt\\ & = \frac{x^{-\acute{\alpha}}}{\Gamma(\lambda)}\mathbb{J}^{k, c, (\gamma, d); k}_{\delta, b, (\alpha_{j}, \beta_{j})_{m}}(1; p)|^{\infty}_{n = 0}\int^{\infty}_{x}(1-\frac{x}{t})^{\lambda-1}t^{\frac{(\sigma-d)n}{(\eta+b)n}-\alpha+\lambda-1}\sum^{\infty}_{m, s = 0}\frac{(\alpha)_{m}(\acute{\alpha})_{s}(\beta)_{m}(\acute{\beta})_{s}}{\lambda_{m+s}m!s!}\\ &\times(1-\frac{x}{t})^{m}(1-\frac{t}{x})^{s}dt\\ & = \frac{x^{-\acute{\alpha}}}{\Gamma(\lambda)}\mathbb{J}^{k, c, (\gamma, d); k}_{\delta, b, (\alpha_{j}, \beta_{j})_{m}}(1; p)|^{\infty}_{n = 0}\sum^{\infty}_{m, s = 0}\frac{(\alpha)_{m}(\acute{\alpha})_{s}(\beta)_{m}(\acute{\beta})_{s}}{\lambda_{m+s}m!s!} \int^{\infty}_{x}(1-\frac{x}{t})^{\lambda+m-1}(1-\frac{t}{x})^{s}\\ &\times t^{\frac{(\sigma-d)n}{(\eta+b)n}-\alpha+\lambda-1}dt.\\ \end{align}$

(5.14)

Putting these values $\frac{x}{t} = u$ $\Rightarrow$ $\frac{-x}{u^{2}}du = dt$ , $t = x \Rightarrow u = 1$ and $t = \infty \Rightarrow u = 0$ in Eq (5.14), then we have

(5.15)

Using Eqs (1.6) and (1.7) in Eq (5.15), we have

$\begin{align*} &(I^{\alpha, \acute{\alpha}, \beta, \acute{\beta}, \lambda}_{-}\mathbb{J}^{c, b, \delta}_{(\gamma, d); k}[(\alpha_{j}, \beta_{j})_{m}; (t^{\frac{\sigma-d}{\eta+b}}; p)])(x)-x^{-\acute{\alpha}+\lambda-\alpha}\mathbb{J}^{k, c, (\gamma, d); k}_{\delta, b, (\alpha_{j}, \beta_{j})_{m}}(x^{{\frac{\sigma-d}{\eta+b}}}; p)|^{\infty}_{n = 0}\nonumber\\ & = \sum^{\infty}_{m, s = 0}\frac{(\alpha)_{m}(\acute{\alpha})_{s}(\beta)_{m}(\acute{\beta})_{s}(-1)^{s}}{\lambda_{m+s}m!s!} \frac{\Gamma(\lambda+m+s)\Gamma(\frac{(d-\sigma)n}{(\eta+b)n}+\alpha-\lambda-s)}{\Gamma(\lambda)\Gamma(\frac{(d-\sigma)n}{(\eta+b)n}+\alpha+m)}\nonumber\\ & = \frac{\Gamma(\frac{(d-\sigma)n}{(\eta+b)n}+\alpha-\lambda)}{\Gamma(\frac{(d-\sigma)n}{(\eta+b)n}+\alpha)}\sum^{\infty}_{m = 0}\frac{(\alpha)_{m} (\beta)_{m}}{ (\frac{(d-\sigma)n}{(\eta+b)n}+\alpha)_{m}m!}\sum^{\infty}_{s = o}\frac{(\acute{\alpha})_{s}(\acute{\beta})_{s}}{(1-\frac{(d-\sigma)n} {(\eta+b)n}-\alpha+\lambda)_{s}s!}\nonumber\\ & = \frac{\Gamma(\frac{(d-\sigma)n}{(\eta+b)n}+\alpha-\lambda)\Gamma(\frac{(d-\sigma)n}{(\eta+b)n}-\beta)}{\Gamma(\frac{(d-\sigma)n}{(\eta+b)n}) \Gamma(\frac{(d-\sigma)n}{(\eta+b)n}+\alpha-\beta)} \frac{\Gamma(1-\frac{(d-\sigma)n}{(\eta+b)n}-\alpha+\lambda)\Gamma(1-\frac{(d-\sigma)n} {(\eta+b)n}-\alpha+\lambda-\acute{\alpha}-\acute{\beta})}{\Gamma(1-\frac{(d-\sigma)n}{(\eta+b)n}-\alpha+\lambda-\acute{\alpha})\Gamma(1-\frac{(d-\sigma)n} {(\eta+b)n}-\alpha+\lambda-\acute{\beta})}\nonumber\\ & = \frac{\Gamma(\frac{(d-\sigma)n}{(\eta+b)n}-\beta)\Gamma(\frac{(d-\sigma)n}{(\eta+b)n}+\alpha-\lambda+\acute{\beta})\Gamma(\frac{(d-\sigma)n} {(\eta+b)n}+\alpha-\lambda+\acute{\alpha})}{\Gamma(\frac{(d-\sigma)n}{(\eta+b)n})\Gamma(\frac{(d-\sigma)n}{(\eta+b)n}+\alpha-\beta) \Gamma(\frac{(d-\sigma)n}{(\eta+b)n}+\alpha-\lambda+\acute{\alpha}+\acute{\beta})}.\nonumber\\ \end{align*}$

We have a desired result

$\begin{align*} &(I^{\alpha, \acute{\alpha}, \beta, \acute{\beta}, \lambda}_{-}\mathbb{J}^{c, b, \delta}_{(\gamma, d); k}[(\alpha_{j}, \beta_{j})_{m}; (t^{\frac{\sigma-d}{\eta+b}}; p)])(x)\nonumber\\ & = \sum^{\infty}_{n = 0}\frac{B_{p}(\gamma+kn, d-\gamma)(-c)^{n}}{x^{\sigma n-\eta n+\alpha-\lambda+\acute{\alpha}}}\Gamma\bigg[ \begin{array}{ll} (d+kn)(\delta)(\frac{(d-\sigma)n}{(\eta+b)n}-\beta)(\frac{(d-\sigma)n}{(\eta+b)n}+\alpha-\lambda+\acute{\beta})\\ (\gamma)(d-\gamma)(\delta+n)(\frac{(d-\sigma)n}{(\eta+b)n})(\frac{(d-\sigma)n}{(\eta+b)n}+\alpha-\beta) \\ \end{array}\nonumber\\ &\times \begin{array}{ll} (\frac{(d-\sigma)n}{(\eta+b)n}+\alpha-\lambda+\acute{\alpha})\\ (\frac{(d-\sigma)n}{(\eta+b)n}+\alpha-\lambda+\acute{\alpha}+\acute{\beta})(\alpha_j n+\beta_j+\frac{1+b}{2})|^{m}_{j = 1} \end{array}\nonumber \bigg]. \end{align*}$

6. Conclusions

In this research, we described extension of extended generalized multi-index Bessel function (E $^{1}$ GMBF) and developed some results with the Laguerre polynomial and Whittaker function, integral representation, derivatives and solved integral transforms (beta transform, Laplace transform, Mellin transforms). Moreover, we discussed the composition of the generalized fractional integral operator having Appell function as a kernel with the E $^{1}$ GMBF and obtained results in terms of Wright functions.

Conflict of interest

The authors declare that they have no competing interests.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

The authors declare no conflict of interest.

References

[1]	Khungar PN, Dahane TM, Revankar RP, et al. (2020) Customized treatment option for malpositioned dental implant placed in aesthetic zone. J Evolution Med Dent Sci 9: 2930-2934. https://doi.org/10.14260/jemds/2020/642
[2]	Kochar SP, Reche A, Paul P (2022) The etiology and management of dental implant failure: a review. Cureus 14: e30455. https://doi.org/10.7759/cureus.30455
[3]	Moraschini V, Poubel LD, Ferreira VF, et al. (2015) Evaluation of survival and success rates of dental implants reported in longitudinal studies with a follow-up period of at least 10 years: a systematic review. Int J Oral Max Surg 44: 377-388. https://doi.org/10.1016/j.ijom.2014.10.023
[4]	Hjalmarsson L, Gheisarifar M, Jemt T (2016) A systematic review of survival of single implants as presented in longitudinal studies with a follow-up of at least 10 years. Eur J Oral Implantol 9: S155-S162.
[5]	Jung RE, Zembic A, Pjetursson BE, et al. (2012) Systematic review of the survival rate and the incidence of biological, technical, and aesthetic complications of single crowns on implants reported in longitudinal studies with a mean follow-up of 5 years. Clin Oral Implants Res 23: 2-21. https://doi.org/10.1111/j.1600-0501.2012.02547.x
[6]	Pjetursson BE, Thoma D, Jung R, et al. (2012) A systematic review of the survival and complication rates of implant-supported fixed dental prostheses (FDP s) after a mean observation period of at least 5 years. Clin Oral Implants Res 23: 22-38. https://doi.org/10.1111/j.1600-0501.2012.02546.x
[7]	Raikar S, Talukdar P, Kumari S, et al. (2017) Factors affecting the survival rate of dental implants: a retrospective study. J Int Soc Prev Community Dent 7: 351-355. https://doi.org/10.4103/jispcd.JISPCD_380_17
[8]	Do TA, Le HS, Shen YW, et al. (2020) Risk factors related to late failure of dental implant—A systematic review of recent studies. Int J Environ Res Public Health 17: 3931. https://doi.org/10.3390/ijerph17113931
[9]	Rail R, Tabassum A, Prema AG (2019) Implant failures–a review. Int J Sci Res 8: 547-552.
[10]	El Askary AS, Meffert RM, Griffin T (1999) Why do dental implants fail? Part I. Implant Dent 8: 173-185. https://doi.org/10.1097/00008505-199908020-00011
[11]	Truhlar RS (1998) Peri-implantitis: causes and treatment. Oral Maxil Surg Clin 10: 299-308. https://doi.org/10.1016/S1042-3699(20)30332-0
[12]	Misch K, Wang HL (2008) Implant surgery complications: etiology and treatment. Implant Dent 17: 159-168. https://doi.org/10.1097/ID.0b013e3181752f61
[13]	Prashanti E, Sajjan S, Reddy JM (2011) Failures in implants. Indian J Dent Res 22: 446-453. https://doi.org/10.4103/0970-9290.87069
[14]	Ardekian L, Dodson TB (2003) Complications associated with the placement of dental implants. Oral Maxil Surg Clin 15: 243-249. https://doi.org/10.1016/S1042-3699(03)00014-1
[15]	Jo KH, Yoon KH, Park KS, et al. (2011) Thermally induced bone necrosis during implant surgery: 3 case reports. J Korean Assoc Oral Maxillofac Surg 37: 406-414. https://doi.org/10.5125/jkaoms.2011.37.5.406
[16]	Patil SS, Arunachaleshwar S, Balkunde, et al. (2015) Complications of immediate implant placement and its management: a review article. Int J Curr Med Appl Sci 8: 78-80.
[17]	Annibali S, Ripari M, La Monaca G, et al. (2008) Local complications in dental implant surgery: prevention and treatment. Oral Implantol 1: 21-33.
[18]	Goodacre CJ, Bernal G, Rungcharassaeng K, et al. (2003) Clinical complications with implants and implant prostheses. J Prosthet Dent 90: 121-132. https://doi.org/10.1016/S0022-3913(03)00212-9
[19]	Dreyer H, Grischke J, Tiede C, et al. (2018) Epidemiology and risk factors of peri-implantitis: A systematic review. J Periodontal Res 53: 657-681. https://doi.org/10.1111/jre.12562
[20]	Gupta S, Gupta H, Tandan A (2015) Technical complications of implant-causes and management: a comprehensive review. Natl J Maxillofac Surg 6: 3-8. https://doi.org/10.4103/0975-5950.168233
[21]	Hanif A, Qureshi S, Sheikh Z, et al. (2017) Complications in implant dentistry. Eur J Dent 11: 135-140. https://doi.org/10.4103/ejd.ejd_340_16
[22]	Bakaeen LG, Winkler S, Neff PA (2001) The effect of implant diameter, restoration design, and occlusal table variations on screw loosening of posterior single-tooth implant restorations. J Oral Implantol 27: 63-72. https://doi.org/10.1563/1548-1336(2001)027<0063:TEOIDR>2.3.CO;2
[23]	Shinde DM, Godbole SR, Dhamande MM, et al. (2020) Aesthetic rehabilitation of maxillary anterior teeth with implant supported fixed partial prosthesis. J Evol Med Dent Sci 9: 3079-3082. https://doi.org/10.14260/jemds/2020/676
[24]	Blustein R, Jackson R, Rotskoff K, et al. (1986) Use of splint material in the placement of implants. Int J Oral Max Impl 1: 47-49.
[25]	Fragkioudakis I, Tseleki G, Doufexi AE, et al. (2021) Current concepts on the pathogenesis of peri-implantitis: a narrative review. Eur J Dent 15: 379-387. https://doi.org/10.1055/s-0040-1721903
[26]	Lee FK, Tan KB, Nicholls JI (2010) Critical bending moment of four implant-abutment interface designs. Int J Oral Max Impl 25: 744-751.
[27]	Misch CE, Suzuki JB, Misch-Dietsh FM, et al. (2005) A positive correlation between occlusal trauma and peri-implant bone loss: literature support. Implant Dent 14: 108-116. https://doi.org/10.1097/01.id.0000165033.34294.db
[28]	Neidlinger J, Lilien BA, Kalant DC (1993) Surgical implant stent: a design modification and simplified fabrication technique. J Prosthet Dent 69: 70-72. https://doi.org/10.1016/0022-3913(93)90243-h
[29]	Dubey A, Dangorekhasbage S, Bhowate R (2019) Assessment of maxillo-mandibular implant sites by digitized volumetric tomography. J Evolution Med Dent Sci 8: 3780-3784. https://doi.org/10.14260/jemds/2019/819
[30]	Ghoshal PK, Kamble RH, Shrivastav SS, et al. (2019) Radiographic evaluation of alveolar bone dimensions in the inter radicular area between maxillary central incisors as “safe zone” for the placement of mini-screw implants in different growth patterns--a digital volume tomographical study. J Evolution Med Dent Sci 8: 3836-3840. https://doi.org/10.14260/jemds/2019/831
[31]	Ribas BR, Nascimento EH, Freitas DQ, et al. (2020) Positioning errors of dental implants and their associations with adjacent structures and anatomical variations: a CBCT-based study. Imaging Sci Dent 50: 281-290. https://doi.org/10.5624/isd.2020.50.4.281
[32]	Mahale KM, Yeshwante BJ, Baig N, et al. (2013) Iatrogenic complications of implant surgery. J Dent Implant 3: 157-159. https://doi.org/10.4103/0974-6781.118857
[33]	Schimmel M, Srinivasan M, McKenna G, et al. (2018) Effect of advanced age and/or systemic medical conditions on dental implant survival: A systematic review and meta-analysis. Clin Oral Implant Res 29: 311-330. https://doi.org/10.1111/clr.13288
[34]	Papi P, Letizia C, Pilloni A, et al. (2018) Peri-implant diseases and metabolic syndrome components: a systematic review. Eur Rev Med Pharmacol Sci 22: 866-875. https://doi.org/10.26355/eurrev_201802_14364
[35]	Bazli L, Chahardehi AM, Arsad H, et al. (2020) Factors influencing the failure of dental implants: a systematic review. J Compos Compd 2: 18-25. https://doi.org/10.29252/jcc.2.1.3
[36]	Esposito M, Hirsch JM, Lekholm U, et al. (1998) Biological factors contributing to failures of osseointegrated oral implants,(I). Success criteria and epidemiology. Eur J Oral Sci 106: 527-551. https://doi.org/10.1046/j.0909-8836..t01-2-.x
[37]	Albrektsson T, Dahlin C, Jemt T, et al. (2014) Is marginal bone loss around oral implants the result of a provoked foreign body reaction?. Clin Implant Dent R 16: 155-165. https://doi.org/10.1111/cid.12142
[38]	Zhang Y, Gulati K, Li Z, et al. (2021) Dental implant nano-engineering: advances, limitations and future directions. Nanomaterials 11: 2489. https://doi.org/10.3390/nano11102489
[39]	Somsanith N, Kim YK, Jang YS, et al. (2018) Enhancing of osseointegration with propolis-loaded TiO₂ nanotubes in rat mandible for dental implants. Materials 11: 61. https://doi.org/10.3390/ma11010061
[40]	Lee YH, Kim JS, Kim JE, et al. (2017) Nanoparticle mediated PPARγ gene delivery on dental implants improves osseointegration via mitochondrial biogenesis in diabetes mellitus rat model. Nanomed-Nanotechnol 13: 1821-1832. https://doi.org/10.1016/j.nano.2017.02.020
[41]	Lee SH, An SJ, Lim YM, et al. (2017) The efficacy of electron beam irradiated bacterial cellulose membranes as compared with collagen membranes on guided bone regeneration in peri-implant bone defects. Materials 10: 1018. https://doi.org/10.3390/ma10091018
[42]	Shin YS, Seo JY, Oh SH, et al. (2014) The effects of Erh BMP-2-/EGCG-coated BCP bone substitute on dehiscence around dental implants in dogs. Oral Dis 20: 281-287. https://doi.org/10.1111/odi.12109
[43]	Tahmasebi E, Alam M, Yazdanian M, et al. (2020) Current biocompatible materials in oral regeneration: A comprehensive overview of composite materials. J Mater Res Technol 9: 11731-11755. https://doi.org/10.1016/j.jmrt.2020.08.042
[44]	Hossain N, Islam MA, Chowdhury MA, et al. (2022) Advances of nanoparticles employment in dental implant applications. Appl Surf Sci Adv 12: 100341. https://doi.org/10.1016/j.apsadv.2022.100341
[45]	Yan X, Wan P, Tan L, et al. (2018) Corrosion and biological performance of biodegradable magnesium alloys mediated by low copper addition and processing. Mater Sci Eng 93: 565-581. https://doi.org/10.1016/j.msec.2018.08.013
[46]	Liu C, Fu X, Pan H, et al. (2016) Biodegradable Mg-Cu alloys with enhanced osteogenesis, angiogenesis, and long-lasting antibacterial effects. Sci Rep 6: 27374. https://doi.org/10.1038/srep27374
[47]	Ohkubo C, Hanatani S, Hosoi T (2008) Present status of titanium removable dentures–a review of the literature. J Oral Rehabil 35: 706-714. https://doi.org/10.1111/j.1365-2842.2007.01821.x
[48]	Memarzadeh K, Sharili AS, Huang J, et al. (2015) Nanoparticulate zinc oxide as a coating material for orthopedic and dental implants. J Biomed Mater Res A 103: 981-989. https://doi.org/10.1002/jbm.a.35241
[49]	Shayani Rad M, Kompany A, Khorsand Zak A, et al. (2013) Microleakage and antibacterial properties of ZnO and ZnO: Ag nanopowders prepared via a sol–gel method for endodontic sealer application. J Nanopart Res 15: 1925. https://doi.org/10.1007/s11051-013-1925-6
[50]	Hu C, Sun J, Long C, et al. (2019) Synthesis of nano zirconium oxide and its application in dentistry. Nanotechnol Rev 8: 396-404. https://doi.org/10.1515/ntrev-2019-0035
[51]	Gaihre B, Jayasuriya AC (2018) Comparative investigation of porous nano-hydroxyapaptite/chitosan, nano-zirconia/chitosan and novel nano-calcium zirconate/chitosan composite scaffolds for their potential applications in bone regeneration. Mater Sci Eng 91: 330-339. https://doi.org/10.1016/j.msec.2018.05.060
[52]	Kordbacheh Changi K, Finkelstein J, et al. (2019) Peri-implantitis prevalence, incidence rate, and risk factors: A study of electronic health records at a US dental school. Clin Oral Implant Res 30: 306-314. https://doi.org/10.1111/clr.13416
[53]	Zitzmann NU, Berglundh T (2008) Definition and prevalence of peri-implant diseases. J clin periodontol 35: 286-291. https://doi.org/10.1111/j.1600-051X.2008.01274.x
[54]	Thiebot N, Hamdani A, Blanchet F, et al. (2022) Implant failure rate and the prevalence of associated risk factors: a 6-year retrospective observational survey. J Oral Med Oral Surg 28: 19. https://doi.org/10.1051/mbcb/2021045
[55]	Pyare MAR, Lade AY, Manhas N, et al. (2022) Evaluation of prevalence of dental implants failures with various risk factors: A 15 years retrospective study. J Pharm Negat Results 2022: 1308-1312. https://doi.org/10.47750/pnr.2022.13.S06.171

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11.	Rana Safdar Ali, Humira Sif, Gauhar Rehman, Ahmad Aloqaily, Nabil Mlaiki, Significant Study of Fuzzy Fractional Inequalities with Generalized Operators and Applications, 2024, 8, 2504-3110, 690, 10.3390/fractalfract8120690

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© 2023 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)

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沈阳化工大学材料科学与工程学院沈阳 110142

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AIMS Bioengineering

Etiology of dental implant complication and failure—an overview

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Abstract

1. Introduction

2. Particular special cases

3. Results of E $^{1}$ GMBF

4. Integral transforms of E $^{1}$ GMBF

5. Relation of the E $^{1}$ GMBF with the Laguerre polynomial and Whittaker function

6. Conclusions

Conflict of interest

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AIMS Bioengineering

Etiology of dental implant complication and failure—an overview

Related Papers:

Abstract

1. Introduction

2. Particular special cases

3. Results of E1 ^{1} GMBF

4. Integral transforms of E1 ^{1} GMBF

5. Relation of the E1 ^{1} GMBF with the Laguerre polynomial and Whittaker function

6. Conclusions

Conflict of interest

References

This article has been cited by:

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通讯作者: 陈斌, bchen63@163.com

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Catalog

3. Results of E $^{1}$ GMBF

4. Integral transforms of E $^{1}$ GMBF

5. Relation of the E $^{1}$ GMBF with the Laguerre polynomial and Whittaker function