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.
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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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.
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 (E1GMBF) 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:
J(αj)m,γ,c(βj)m,k,b,δ(z)=∑∞n=0(γ)kn(−cz)n(δ)n∏mj=1Γ(αjn+βj+1+b2),m∈N. | (1.1) |
where αj,βj,b,δ,γ,c∈C (j=1,2⋯m) be such that ∑mj=1ℜ(αj)>max{0;ℜ(k)−1}; k>0, ℜ(βj)>−1, ℜ(γ)>0, ℜ(δ)>0.
Definition 1.2. [16] Generalized fractional integral operator is defined for α,ˊα,β,ˊβ,λ∈C, and x>0 as follows:
Iα,ˊα,β,ˊβ,λ0+f(t)=x−αΓ(λ)∫x0(x−t)λ−1t−ˊαF3(α,ˊα,β,ˊβ;λ;1−tx;1−xt)f(t)dt, | (1.2) |
and
Iα,ˊα,β,ˊβ,λ−f(t)=x−ˊαΓ(λ)∫∞x(t−x)λ−1t−αF3(α,ˊα,β,ˊβ;λ;1−xt;1−tx)f(t)dt. | (1.3) |
where F3 is the Appell function.
Definition 1.3. [17] Appell function F3 also called the (Horn function) and defined for α,ˊα,β,ˊβ,λ∈C, as follows:
F3(α,ˊα,β,ˊβ;λ;x;y)=∞∑m,n=0(α)m(ˊα)n(β)m(ˊβ)n(λ)m+nm!n!xmyn,max{|x|,|y|}<1 | (1.4) |
Definition 1.4. [18,19] The integral representation of gamma function is defined for ℜ(s)>0, as follows:
Γ(s)=∫∞0us−1e−udu. | (1.5) |
Definition 1.5. [18,19] Classical beta function is defined for ℜ(x)>0 and ℜ(y)>0, as follows:
B(x,y)=∫10tx−1(1−t)y−1dt | (1.6) |
=Γ(x)Γ(y)Γ(x+y). | (1.7) |
Definition 1.6. [20,21] Extended beta function is defined for ℜ(x)>0, ℜ(y)>0, ℜ(p)>0 as follows:
Bp(x,y)=∫10tx−1(1−t)y−1exp(−pt(1−t))dt, | (1.8) |
if p=0, then extended beta function Bp(x,y) reduces into the classical beta function.
Definition 1.7. [22] Generalized Wright type hypergeometric function is defined as follows:
rψs(z)=rψs[(yj,hj)1,r(xi,qi)1,s|z]≡∑∞n=0∏rj=1Γ(yj+hjn)∏si=1Γ(xi+qin)znn!. | (1.9) |
where z∈C, yj,xi∈C and hj,qi∈ℜ (j=1,2⋯r;i=1,2⋯s).
Definition 1.8. [23] Laplace transform is defined ℜ(s)>0, as follows:
Ł[f(t)]=f(s)=∫∞0e−stf(t)dt. | (1.10) |
Definition 1.9. [24] Euler transform of a function f(z) is defined as follows:
B{f(z);a,b}=∫10za−1(1−z)b−1f(z)dz(ℜ(a)>0,ℜ(b)>0). | (1.11) |
Definition 1.10. [24] Mellin transform of the function f(z) is defined as follows:
M{f(z);s}=∫∞0zs−1f(z)dz=f∗(s),ℜ(s)>0, | (1.12) |
then inverse Mellin transform
f(z)=M−1[f∗(s);z]=12πi∫λ+i∞λ−i∞f∗z−sds,λ>0. | (1.13) |
Definition 1.11. The Pochhammer symbol defined as
(δ)n={1,n=0δ(δ+1)(δ+2)⋯(δ+n−1),n=1,2⋯ | (1.14) |
or
(δ)n=Γ(δ+n)Γ(δ) | (1.15) |
(δ)kn=Γ(δ+kn)Γ(δ), | (1.16) |
where δ∈C and n,k∈N.
Definition 1.12. The E1GMBF J(αj)m,γ,c(βj)m,k,b,δ(z) is defined in the following way:
Jc,b,δ(γ,d);k[(αj,βj)m;(z;p)]=Jk,c,(γ,d);kδ,b,(αj,βj)m(z;p)=∞∑n=0Bp(γ+kn,d−γ)B(γ,d−γ)cn(d)kn(−z)n(δ)n∏mj=1Γ(αjn+βj+1+b2). | (1.17) |
where αj,βj,b,d,δ,γ,c∈C (j=1,2⋯m), p≥0 be such that ∑mj=1ℜ(αj)>max{0;ℜ(k)−1}; k>0, ℜ(βj)>−1, ℜ(d)>0, ℜ(γ)>0, ℜ(δ)>0.
Remark 1.1. The E1GMBF can also be write as
Jc,b,δ(γ,d);k[(αj,βj)m;(z;p)]=Jb,δ(γ,d);k[(αj,βj)m;(cz;p)]. | (1.18) |
In this section, we establish some particular special cases of E1GMBF as below
● if we set p=0, then E1GMBF reduce into extended multi-index Bessel function
Jc,b,δγ;k[(αj,βj)m;(z)]=J(αj)m,γ,c(βj)m,k,b,δ(z)=∞∑n=0(c)n(γ)kn(−z)n(δ)n∏mj=1Γ(αjn+βj+1+b2). | (2.1) |
● when p=0, c=b=δ=1, then
J1,1,1γ;k[(αj,βj)m;(z)]=J(αj)m,γ(βj)m,k(z)=∞∑n=0(γ)kn(−z)nn!∏mj=1Γ(αjn+βj+1). | (2.2) |
● if we put p=0, c=b=δ=m=1, then E1GMBF reduce to the generalized Bessel-Maitland function as,
J1,1,1γ;k[(α,β);(z)]=Jα,γβ,k(z)=∞∑n=0(γ)kn(−z)nn!Γ(αn+β+1). | (2.3) |
● when p=0, k=0, δ=c=b=1, then E1GMBF reduce to the Bessel-Maitland function as given below
J1,1,1γ[(α,β);(z)]=Jαβ(z)=∞∑n=0(−z)nn!Γ(αn+β+1). | (2.4) |
● if we put p=0, c=δ=1, z=−z and set βj=βj−1, then E1GMBF reduce to the multi-index Mittag Leffler function as given below
J1,b,1γ;k[(αj,βj)m;(−z)]=Eγ,k[(αj,β)j)mj=1]=∞∑n=0(γ)kn∏mj=1(αjn+βj)znn!. | (2.5) |
● if we set p=k=0, b=c=m=1, α1=δ=1, β1=ν and replace z=z24 then E1GMBF reduce into Bessel function of fist kind
J1,1,1γ;0[(1,ν)m;(z24)]=∞∑n=0(−z)nn!Γ(n+ν+1). | (2.6) |
In this section, we investigate the E1GMBF, and studied some important observations. Moreover, we develop integral and differential of E1GMBF in the form of theorems.
Theorem 3.1. The E1GMBF can be able to represent with αj,βj,b,δ,γ,c∈C (j=1,2⋯m) be such that ∑mj=1ℜ(αj)>max{0;ℜ(k)−1}; k>0, ℜ(βj)>−1, ℜ(γ)>0, ℜ(δ)>0 then following relation holds
Jk,c,(γ,d);kδ,b,(αj,βj)m(z;p)=1B(γ,d−γ)∫10tγ−1(1−t)d−γ−1e−pt(1−t)J(αj)m,γ,c(βj)m,k,b,δ(tkz)dt. | (3.1) |
Proof. Using the definition of Eq (1.8) in (1.17), we obtain
Jk,c,(γ,d);kδ,b,(αj,βj)m(z;p)=∞∑n=0{∫10tγ+kn−1(1−t)d−γ−1e−pt(1−t)}×cn(d)kn(−z)nB(γ,d−γ)(δ)n∏mj=1Γ(αjn+βj+1+b2)dt. | (3.2) |
Changing the order of summation and integration, and after simplification of Eq (3.2), we get
Jk,c,(γ,d);kδ,b,(αj,βj)m(z;p)=1B(γ,d−γ)∫10tγ−1(1−t)d−γ−1e−pt(1−t)∞∑n=0cn(d)kn(−tkz)n(δ)n∏mj=1Γ(αjn+βj+1+b2)dt. | (3.3) |
Using Eq (1.1) in Eq (3.3), we obtain the desired result in theorem 3.1.
Corollary 3.1. Let αj,βj,b,δ,γ,c∈C (j=1,2⋯m) be such that ∑mj=1ℜ(αj)>max{0;ℜ(k)−1}; k>0, ℜ(βj)>−1, ℜ(γ)>0, ℜ(δ)>0. Taking t=r1+r in theorem 3.1, then following relation holds
Jk,c,(γ,d);kδ,b,(αj,βj)m(z;p)=1B(γ,d−γ)∫∞0rγ−1(1+r)de−p(1+r)2rJ(αj)m,γ,c(βj)m,k,b,δ(rkz(1+r)k)dr. | (3.4) |
Corollary 3.2. Let αj,βj,b,δ,γ,c∈C (j=1,2⋯m) be such that ∑mj=1ℜ(αj)>max{0;ℜ(k)−1}; k>0, ℜ(βj)>−1, ℜ(γ)>0, ℜ(δ)>0 and consider t=cos2θ in theorem 3.1, then following relation holds
Jk,c,(γ,d);kδ,b,(αj,βj)m(z;p)=2B(γ,d−γ)∫π20(cosθ)2γ−1(sinθ)2d−2γ−1exp(−psin2θcos2θ)×J(αj)m,γ,c(βj)m,k,b,δ(zcos2kθ)dθ. | (3.5) |
Theorem 3.2. Let α,β,b,δ,γ,c∈C be such that ℜ(α)>max{0;ℜ(k)−1}; k>0, ℜ(β)>−1, ℜ(γ)>0, ℜ(δ)>0, then the following recurrence relation holds in the definition (1.17) for j=1 as
Jk,c,(γ,d);kδ,b,α,β(z;p)=(β+b+12)Jk,c,(γ,d);kδ,b,α,β+1(z;p)+αzddzJk,c,(γ,d);kδ,b,(α,β+1)(z;p). | (3.6) |
Proof. Consider the definition of (1.17) for j=1, and the right side of the Eq (3.6), we get
(β+b+12)Jk,c,(γ,d);kδ,b,(α,β+1)(z;p)+αzddzJk,c,(γ,d);kδ,b,(α,β+1)(z;p)=(β+b+12)∞∑n=0Bp(γ+kn,d−γ)B(γ,d−γ)cn(d)kn(−z)n(δ)nΓ(αn+β+1+1+b2)+αzddz∞∑n=0Bp(γ+kn,d−γ)B(γ,d−γ)cn(d)kn(−z)n(δ)nΓ(αn+β+1+1+b2)=∞∑n=0Bp(γ+kn,d−γ)B(γ,d−γ)cn(d)kn(δ)n×[(β+b+12)(−z)nΓ(αn+β+1+1+b2)+αzddz(−z)nΓ(αn+β+1+1+b2)]=∞∑n=0Bp(γ+kn,d−γ)B(γ,d−γ)cn(d)kn(−z)n(αn+β+1+b2)(δ)nΓ(αn+β+1+1+b2)=∞∑n=0Bp(γ+kn,d−γ)B(γ,d−γ)cn(d)kn(−z)n(δ)nΓ(αn+β+1+b2)=Jk,c,(γ,d);kδ,b,(α,β)(z;p) | (3.7) |
Theorem 3.3. For the E1GMBF we have the following higher derivative formula for δ=1, is given below
dndznJk,c,(γ,d);k1,b,(αj,βj)m(z;p)=(−c)n(d)k(d+k)k⋯(d+(n−1)k)kJk,c,(γ+kn,d+kn);k1,b,(αj,βj+αjn)m(z;p). | (3.8) |
where αj,βj,b,γ,c∈C (j=1,2⋯m) be such that ∑mj=1ℜ(αj)>max{0;ℜ(k)−1}; k>0, ℜ(βj)>−1, ℜ(γ)>0.
Proof. Differentiation with respect to z in Eq (1.17), we get
ddzJk,c,(γ,d);k1,b,(αj,βj)m(z;p)=∞∑n=1Bp(γ+kn,d−γ)B(γ,d−γ)cn(d)kn(−1)nnzn−1n!∏mj=1Γ(αjn+βj+1+b2)=∞∑n=1Bp(γ+k(n−1)+k,d−γ)B(γ,d−γ)(−c)n(d)k(n−1)+knzn−1n(n−1)!∏mj=1Γ(αjn+βj+1+b2) | (3.9) |
we can write the pochhammer symbols as
(d)k(n−1)+k=Γ(d+k(n−1)+k)Γ(d)=Γ(d+k(n−1)+k)Γ(d+k)Γ(d+k)Γ(d)=(d+k)(n−1)k(d)k. | (3.10) |
Now, using the Eq (3.10) in Eq (3.9), we have
ddzJk,c,(γ,d);k1,b,(αj,βj)m(z;p)=(−c)(d)k∞∑n=1Bp(γ+k+k(n−1),d−γ)(−c)n−1(d+k)k(n−1)zn−1B(γ,d−γ)(n−1)!∏mj=1Γ(αj(n−1)+αj+βj+1+b2)=(−c)(d)kJk,c,(γ+k,d+k);k1,b,(αj,βj+αj)m(z;p). | (3.11) |
Again differentiation with respect to z in Eq (3.9), we have
d2dz2Jk,c,(γ,d);k1,b,(αj,βj)m(z;p)=(−c)2(d)k(d+k)kJk,c,(γ+2k,d+2k);k1,b,(αj,βj+2αj)m(z;p), |
continue this technique up to n times, we obtain the desired result which state in the theorem 3.3.
Theorem 3.4. Let αj,βj,d,γ,c,λ∈C (j=1,2⋯m), p≥0 be such that ∑mj=1ℜ(αj)>max{0;ℜ(k)−1}; k>0, ℜ(βj)>0, ℜ(d)>0, ℜ(γ)>0, ℜ(δ)>0 then the following relation holds as:
dndzn{zβ1⋯βm−1Jk,c,(γ,d);k1,−1,(αj,βj)m(λzα1⋯αm;p)}=Jk,c,(γ,d);k1,−1,(αj,βj−n)m(λzα1⋯αm;p)zn−β1⋯−βm+1. | (3.12) |
Proof. Replacing z by λzαj⋯αj, b=−1 and δ=1 in Eq (1.17), take its product zβ1⋯βj, and after taking differentiation with respect to z up to n times, we obtain our required result.
In this section, we establish some integral transforms (Euler, Mellin and Laplace transform) of E1GMBF in the form of theorems, and also discuss its sub cases.
Theorem 4.1. Euler transform of E1GMBF holds for αj,βj,b,δ,γ,c∈C (j=1,2⋯m) be such that ∑mj=1ℜ(αj)>max{0;ℜ(k)−1}; k>0, ℜ(βj)>−1, ℜ(γ)>0, ℜ(δ)>0.
B{Jk,c,(γ,d);kδ,−1,(αj,βj)m(λzαj;p);β1⋯βm,1}=Jk,c,(γ,d);kδ,−1,(αj,βj+1)m(λ;p). | (4.1) |
Proof. Apply the definition of Euler transform (1.9) in Eq (1.17), we get
B{Jk,c,(γ,d);kδ,−1,(αj,βj)m(λzαj;p);β1⋯βm,1}=∫10zβ1⋯βm−1(1−z)1−1∞∑n=0Bp(γ+kn,d−γ)B(γ,d−γ)×cn(d)kn(−1)n(λzαj)n(δ)n∏mj=1Γ(αjn+βj)dz. | (4.2) |
Interchanging the order of summations and integration in Eq (4.2), we get
B{Jk,c,(γ,d);kδ,−1,(αj,βj)m(λzαj;p);β1⋯βm,1}=∞∑n=0Bp(γ+kn,d−γ)B(γ,d−γ)cn(d)kn(−λ)n(δ)n∏mj=1Γ(αjn+βj)×∫10zβ1⋯βm+αjn−1(1−z)1−1dz. | (4.3) |
Using the Eq (1.6) and Eq (1.7) in Eq (4.3), then we obtain
B{Jk,c,(γ,d);kδ,−1,(αj,βj)m(λzαj;p);β1⋯βm,1}=∞∑n=0Bp(γ+kn,d−γ)cn(d)kn(−λ)nB(γ,d−γ)(δ)n∏mj=1Γ(αjn+βj+1)=Jk,c,(γ,d);kδ,−1,(αj,βj+1)m(λ;p). | (4.4) |
Theorem 4.2. The Mellin transform of E1GMBF is given by for αj,βj,b,δ,γ,c∈C (j=1,2⋯m) be such that ∑mj=1ℜ(αj)>max{0;ℜ(k)−1}; k>0, ℜ(βj)>−1, ℜ(γ)>0, ℜ(δ)>0. Then the following relation holds
M{Jk,c,(γ,d);kδ,b,(αj,βj)m(z;p);s}=Γ(s)Γ(δ)Γ(d)Γ(d−γ+s)[Γ(γ)]2Γ(d−γ)3ψm+2[(γ,k)(γ+s,k)(1,1)(δ,1)(d+2s,k)(βj+1+b2,αj)|mj=1|−cz]. |
Proof. By applying the definition of the Mellin transform to the E1GMBF, we have
M{Jk,c,(γ,d);kδ,b,(αj,βj)m(z;p);s}=∫∞0ps−1Jk,c,(γ,d);kδ,b,(αj,βj)m(z;p)dp. | (4.5) |
Using theorem 3.1 in right side of Eq (4.5), we get
M{Jk,c,(γ,d);kδ,b,(αj,βj)m(z;p);s}=1B(γ,d−γ)∫∞0ps−1{∫10tγ−1(1−t)d−γ−1e−pt(1−t)J(αj)m,γ,c(βj)m,k,b,δ(tkz)dt}dp. | (4.6) |
Interchanging the order of integration in Eq (4.6), then we have
M{Jk,c,(γ,d);kδ,b,(αj,βj)m(z;p);s}=1B(γ,d−γ)∫10tγ−1(1−t)d−γ−1J(αj)m,γ,c(βj)m,k,b,δ(tkz){∫∞0ps−1e−pt(1−t)dp}dt. | (4.7) |
Now, putting pt(1−t)=u in Eq (4.7), and applying the mathematical formula of Eq (1.5), we get
M{Jk,c,(γ,d);kδ,b,(αj,βj)m(z;p);s}=Γ(s)B(γ,d−γ)∫10tγ+s−1(1−t)d−γ+s−1J(αj)m,γ,c(βj)m,k,b,δ(tkz)dt. | (4.8) |
Using Eq (1.1), and interchanging the order of integration and summation in Eq (4.8), we obtain
M{Jk,c,(γ,d);kδ,b,(αj,βj)m(z;p);s}=Γ(s)B(γ,d−γ)∞∑n=0(c)n(γ)kn(−z)n(δ)n∏mj=1Γ(αjn+βj+1+b2)∫10tγ+s+kn−1(1−t)d−γ+s−1dt. | (4.9) |
Using Eq (1.6) and Eq (1.7) in Eq (4.9), we get
M{Jk,c,(γ,d);kδ,b,(αj,βj)m(z;p);s}=Γ(s)B(γ,d−γ)∞∑n=0(c)n(γ)kn(−z)n(δ)n∏mj=1Γ(αjn+βj+1+b2)Γ(γ+s+kn)Γ(d−γ+s)Γ(2s+kn+d). | (4.10) |
After simplification in Eq (4.10), we get
M{Jk,c,(γ,d);kδ,b,(αj,βj)m(z;p);s}=Γ(s)Γ(δ)Γ(d)Γ(d−γ+s)[Γ(γ)]2Γ(d−γ)∞∑n=0Γ(γ+kn)(−cz)nΓ(δ+n)∏mj=1Γ(αjn+βj+1+b2)Γ(γ+s+kn)Γ(2s+kn+d)=Γ(s)Γ(δ)Γ(d)Γ(d−γ+s)[Γ(γ)]2Γ(d−γ)3ψm+2[(γ,k)(γ+s,k)(1,1)(δ,1)(d+2s,k)(βj+1+b2,αj)|mj=1|−cz]. |
Corollary 4.1. Let αj,βj,b,δ,γ,c∈C (j=1,2⋯m) be such that ∑mj=1ℜ(αj)>max{0;ℜ(k)−1}; k>0, ℜ(βj)>−1, ℜ(γ)>0, ℜ(δ)>0. Taking s=1 in theorem 4.2, then the following relation holds
∫∞0Jk,c,(γ,d);kδ,b,(αj,βj)m(z;p)dp=(d−γ)Γ(δ)Γ(γ)3ψm+2[(γ,k)(γ+1,k)(1,1)(δ,1)(d+2,k)(βj+1+b2,αj)|mj=1|−cz]. | (4.11) |
Corollary 4.2. Let αj,βj,b,δ,γ,c∈C (j=1,2⋯m) be such that ∑mj=1ℜ(αj)>max{0;ℜ(k)−1}; k>0, ℜ(βj)>−1, ℜ(γ)>0, ℜ(δ)>0. Applying the inverse Mellin transform on left and right side of Eq (1.17), we gain the important complex integral representation as follows:
M−1{Jk,c,(γ,d);kδ,b,(αj,βj)m(z;p);s}=12πiΓ(γ)Γ(d−γ)∫λ+i∞λ−i∞Γ(s)Γ(δ)Γ(d−γ+s)×3ψm+2[(γ,k)(γ+s,k)(1,1)(δ,1)(d+2s,k)(βj+1+b2,αj)|mj=1|−cz]p−sds. | (4.12) |
Theorem 4.3. The Laplace transform of E1GMBF is given as for αj,βj,b,δ,γ,c∈C (j=1,2⋯m) be such that ∑mj=1ℜ(αj)>max{0;ℜ(k)−1}; k>0, ℜ(βj)>−1, ℜ(γ)>0, ℜ(δ)>0.
Ł(Jk,c,(γ,d);k1,b,(αj,βj)m(z;p))=1sJk,c,(γ,d);k1,b,(αj,βj)m(1s;p). | (4.13) |
Proof. Using the definition of Laplace transform (1.8) in Eq (1.17), we have
Ł(Jk,c,(γ,d);k1,b,(αj,βj)m(z;p))=∫∞0e−st∞∑n=0Bp(γ+kn,d−γ)B(γ,d−γ)cn(d)kn(−t)nn!∏mj=1Γ(αjn+βj+1+b2)dt=∞∑n=0Bp(γ+kn,d−γ)B(γ,d−γ)cn(d)kn(−1)nn!∏mj=1Γ(αjn+βj+1+b2)∫∞0e−sttndt=∞∑n=0Bp(γ+kn,d−γ)B(γ,d−γ)cn(d)kn(−1)nn!∏mj=1Γ(αjn+βj+1+b2)n!sn+1=1s∞∑n=0Bp(γ+kn,d−γ)B(γ,d−γ)cn(d)kn(−s)n∏mj=1Γ(αjn+βj+1+b2)=1sJk,c,(γ,d);k1,b,(αj,βj)m(1s;p). | (4.14) |
In this section, the authors represent the E1GMBF in terms of Laguerre polynomial, and Whittaker function in the form of theorems.
Theorem 5.1. Let αj,βj,b,d,δ,γ,c∈C (j=1,2⋯m), p≥0 be such that ∑mj=1ℜ(αj)>max{0;ℜ(k)−1}; k>0, ℜ(βj)>0, ℜ(d)>0, ℜ(γ)>0, ℜ(δ)>0, then the E1GMBF holds
e2pJk,c,(γ,d);kδ,b,(αj,βj)m(z;p)=Γ(δ)∑∞a,b=0Lb(p)La(p)Γ(b+d−γ+1)Γ(γ)B(γ,d−γ)×3ψm+2[(γ,k)(a+γ+1,k)(1,1)(δ,1)(a+b+d+2,k)(βj+1+b2,αj)|mj=1|−cz]. | (5.1) |
Proof. We being recalling the valuable identity [25] which is
e−pt(1−t)=e−2p∞∑a,b=0Lb(p)La(p)ta+1(1−t)b+1,(0<t<1). | (5.2) |
Applying Eq (5.2) in theorem 3.1, we get
Jk,c,(γ,d);kδ,b,(αj,βj)m(z;p)=1B(γ,d−γ)∫10tγ−1(1−t)d−γ−1e−2p∞∑a,b=0Lb(p)La(p)ta+1(1−t)b+1J(αj)m,γ,c(βj)m,k,b,δ(tkz)dt=1B(γ,d−γ)∫10tγ−1(1−t)d−γ−1e−2p∞∑a,b=0Lb(p)La(p)ta+1(1−t)b+1×∞∑n=0(c)n(γ)kn(−tkz)n(δ)n∏mj=1Γ(αjn+βj+1+b2)dt. | (5.3) |
Interchanging the order of integration and summations in Eq (5.3), we obtain
Jk,c,(γ,d);kδ,b,(αj,βj)m(z;p)=e−2pB(γ,d−γ)∞∑a,b,n=0Lb(p)La(p)(γ)kn(−cz)n(δ)n∏mj=1Γ(αjn+βj+1+b2)∫10ta+kn+γ(1−t)b+d−γdt. | (5.4) |
Using Eq (1.6) and Eq (1.7) in Eq (5.4), then we have
e2pJk,c,(γ,d);kδ,b,(αj,βj)m(z;p)=1B(γ,d−γ)∞∑a,b,n=0Lb(p)La(p)(γ)kn(−cz)n(δ)n∏mj=1Γ(αjn+βj+1+b2)Γ(a+kn+γ+1)Γ(b+d−γ+1)Γ(a+b+d+kn+2)=Γ(δ)∑∞a,b=0Lb(p)La(p)Γ(b+d−γ+1)Γ(γ)B(γ,d−γ)×3ψm+2[(γ,k)(a+γ+1,k)(1,1)(δ,1)(a+b+d+2,k)(βj+1+b2,αj)|mj=1|−cz]. | (5.5) |
Theorem 5.2. For the E1GMBF with αj,βj,b,δ,γ,c∈C (j=1,2⋯m) be such that ∑mj=1ℜ(αj)>max{0;ℜ(k)−1}; k>0, ℜ(βj)>−1, ℜ(γ)>0, ℜ(δ)>0, we have
e3p2Jk,c,(γ,d);kδ,b,(αj,βj)m(z;p)=e−pB(γ,d−γ)∞∑a,n=0La(p)(δ)kn(−cz)n(δ)n∏mj=1Γ(αjn+βj+b+12)×Γ(d−γ+1)pγ+kn2W−1+γ−2d−kn2,γ+kn2. | (5.6) |
Proof. Allowing for the following equality e−pt(1−t)=e(−p1−t)e(−pt) and via generating function related to the Laguerre polynomial [25], we obtain
e−pt(1−t)=e−pe−pt(1−t)∞∑a=0La(p)tn. | (5.7) |
Using Eq (5.7) in Eq (1.17), we have
Jk,c,(γ,d);kδ,b,(αj,βj)m(z;p)=1B(γ,d−γ)∫10tγ−1(1−t)d−γ−1e−pe−pt(1−t)∞∑a=0La(p)tnJ(αj)m,γ,c(βj)m,k,b,δ(tkz)dt=e−pB(γ,d−γ)∞∑a,n=0La(p)(δ)kn(−cz)n(δ)n∏mj=1Γ(αjn+βj+b+12)∫10tγ+kn−1(1−t)d−γe−ptdt. | (5.8) |
Now, integral representation of Whittaker function is defined [26] as follows
∫10tμ−1(1−t)ν−1e−ptdt=Γ(ν)pμ−12e−p2W1−μ−2ν2,μ2(p). | (5.9) |
Using Eq (5.9) in Eq (5.8), then we have
Jk,c,(γ,d);kδ,b,(αj,βj)m(z;p)=e−pB(γ,d−γ)∞∑a,n=0La(p)(δ)kn(−cz)n(δ)n∏mj=1Γ(αjn+βj+b+12)×Γ(d−γ+1)pγ+kn2e−p2W−1+γ−2d−kn2,γ+kn2. | (5.10) |
Theorem 5.3. Let αj,βj,b,d,δ,γ,σ,η,c∈C (j=1,2⋯m), p≥0 be such that ∑mj=1ℜ(αj)>max{0;ℜ(k)−1}; k>0, ℜ(βj)>0, ℜ(d)>0, ℜ(γ)>0, ℜ(δ)>0 the E1GMBF holds
(Iα,ˊα,β,ˊβ,λ0+Jc,b,δ(γ,d);k[(αj,βj)m;(t−σ+η;p)])(x)=∞∑n=0Bp(γ+kn,d−γ)(−c)nxσn−ηn+α−λ+ˊαΓ[(d+kn)(δ)(λ−ˊα−σn+ηn−α−β+1)(γ)(d−γ)(δ+n)(λ−ˊα−σn+ηn−β+1)×(1−ˊα−σn+ηn+ˊβ)(1−σn+ηn)(1−σn+ηn+ˊβ)(λ−ˊα−σn+ηn−α+1)(αjn+βj+1+b2)|mj=1]. |
Proof. Consider the composition of generalized fractional integral operator having Appell function as its kernel with the E1GMBF,
(Iα,ˊα,β,ˊβ,λ0+Jc,b,δ(γ,d);k[(αj,βj)m;(t−σ+η;p)])(x)=x−αΓ(λ)∫x0(x−t)λ−1t−ˊαF3(α,ˊα,β,ˊβ;λ;1−tx;1−xt)∞∑n=0Bp(γ+kn,d−γ)B(γ,d−γ)×cn(d)kn(−1)nt−σn+ηn(δ)n∏mj=1Γ(αjn+βj+1+b2)dt=x−α+λ−1Γ(λ)Jk,c,(γ,d);kδ,b,(αj,βj)m(1;p)|∞n=0∫x0(1−tx)λ−1t−ˊα−σn+ηn∞∑m,s=0(α)m(ˊα)s(β)m(ˊβ)sλm+sm!s!×(1−tx)m(1−xt)sdt=x−α+λ−1Γ(λ)Jk,c,(γ,d);kδ,b,(αj,βj)m(1;p)|∞n=0∞∑m,s=0(α)m(ˊα)s(β)m(ˊβ)sλm+sm!s!∫x0(1−tx)m+λ−1×(1−xt)st−ˊα−σn+ηndt. | (5.11) |
Putting these values tx=τ ⇒ dτ=xdt, t=x⇒τ=1 and t=0⇒τ=0 in Eq (5.11), then we have
(Iα,ˊα,β,ˊβ,λ0+Jc,b,δ(γ,d);k[(αj,βj)m;(t−σ+η;p)])(x)=x−α+λ−1Γ(λ)Jk,c,(γ,d);kδ,b,(αj,βj)m(1;p)|∞n=0∞∑m,s=0(α)m(ˊα)s(β)m(ˊβ)sλm+sm!s!∫10(1−τ)m+λ−1×(1−1τ)s(xτ)−ˊα−σn+ηnxdτ=x−α−ˊα+λΓ(λ)Jk,c,(γ,d);kδ,b,(αj,βj)m(xηxσ;p)|∞n=0∞∑m,s=0(α)m(ˊα)s(β)m(ˊβ)s(−1)sλm+sm!s!∫10(1−τ)s+m+λ−1×τ−ˊα−σn+ηn−sdτ. | (5.12) |
Using Eqs (1.6) and (1.7) in Eq (5.12), we obtain
(Iα,ˊα,β,ˊβ,λ0+Jc,b,δ(γ,d);k[(αj,βj)m;(t−σ+η;p)])(x)−x−α+λ−ˊαJk,c,(γ,d);kδ,b,(αj,βj)m(xηxσ;p)|∞n=0=∞∑m,s=0(α)m(ˊα)s(β)m(ˊβ)s(−1)sλm+sm!s!Γ(s+m+λ)Γ(−ˊα−σn+ηn−s+1)Γ(λ)Γ(m+λ−ˊα−σn+ηn+1)=Γ(−ˊα−σn+ηn+1)Γ(λ−ˊα−σn+ηn+1)∞∑m=0(α)m(β)m(λ−ˊα−σn+ηn+1)mm!∞∑s=0(ˊα)s(ˊβ)s(ˊα+σn−ηn)ss!=Γ(−ˊα−σn+ηn+1)Γ(λ−ˊα−σn+ηn−α−β+1)Γ(ˊα+σn−ηn)Γ(σn−ηn−ˊβ)Γ(λ−ˊα−σn+ηn−α+1)Γ(λ−ˊα−σn+ηn−β+1)Γ(σn−ηn)Γ(ˊα+σn−ηn−ˊβ)=Γ(λ−ˊα−σn+ηn−α−β+1)Γ(1−ˊα−σn+ηn+ˊβ)Γ(1−σn+ηn)Γ(λ−ˊα−σn+ηn−α+1)Γ(λ−ˊα−σn+ηn−β+1)Γ(1−σn+ηn+ˊβ). | (5.13) |
we have the required result
(Iα,ˊα,β,ˊβ,λ0+Jc,b,δ(γ,d);k[(αj,βj)m;(t−σ+η;p)])(x)=∞∑n=0Bp(γ+kn,d−γ)(−c)nxσn−ηn+α−λ+ˊαΓ[(d+kn)(δ)(λ−ˊα−σn+ηn−α−β+1)(γ)(d−γ)(δ+n)(λ−ˊα−σn+ηn−β+1)×(1−ˊα−σn+ηn+ˊβ)(1−σn+ηn)(1−σn+ηn+ˊβ)(λ−ˊα−σn+ηn−α+1)(αjn+βj+1+b2)|mj=1]. |
Theorem 5.4. Let αj,βj,b,d,δ,γ,σ,η,c∈C (j=1,2⋯m), p≥0 be such that ∑mj=1ℜ(αj)>max{0;ℜ(k)−1}; k>0, ℜ(βj)>0, ℜ(d)>0, ℜ(γ)>0, ℜ(δ)>0, then the E1GMBF holds true:
(Iα,ˊα,β,ˊβ,λ−Jc,b,δ(γ,d);k[(αj,βj)m;(tσ−dη+b;p)])(x)=∞∑n=0Bp(γ+kn,d−γ)(−c)nxσn−ηn+α−λ+ˊαΓ[(d+kn)(δ)((d−σ)n(η+b)n−β)((d−σ)n(η+b)n+α−λ+ˊβ)(γ)(d−γ)(δ+n)((d−σ)n(η+b)n)((d−σ)n(η+b)n+α−β)×((d−σ)n(η+b)n+α−λ+ˊα)((d−σ)n(η+b)n+α−λ+ˊα+ˊβ)(αjn+βj+1+b2)|mj=1]. |
Proof. Consider the composition of right side generalized fractional integral operator with the E1GMBF \newpage
(Iα,ˊα,β,ˊβ,λ−Jc,b,δ(γ,d);k[(αj,βj)m;(tσ−dη+b;p)])(x)=x−ˊαΓ(λ)∫∞x(t−x)λ−1t−αF3(α,ˊα,β,ˊβ;λ;1−xt;1−tx)∞∑n=0Bp(γ+kn,d−γ)B(γ,d−γ)×cn(d)kn(−1)ntσn−dnηn+bn(δ)n∏mj=1Γ(αjn+βj+1+b2)dt=x−ˊαΓ(λ)Jk,c,(γ,d);kδ,b,(αj,βj)m(1;p)|∞n=0∫∞x(1−xt)λ−1t(σ−d)n(η+b)n−α+λ−1∞∑m,s=0(α)m(ˊα)s(β)m(ˊβ)sλm+sm!s!×(1−xt)m(1−tx)sdt=x−ˊαΓ(λ)Jk,c,(γ,d);kδ,b,(αj,βj)m(1;p)|∞n=0∞∑m,s=0(α)m(ˊα)s(β)m(ˊβ)sλm+sm!s!∫∞x(1−xt)λ+m−1(1−tx)s×t(σ−d)n(η+b)n−α+λ−1dt. | (5.14) |
Putting these values xt=u ⇒ −xu2du=dt, t=x⇒u=1 and t=∞⇒u=0 in Eq (5.14), then we have
(Iα,ˊα,β,ˊβ,λ−Jc,b,δ(γ,d);k[(αj,βj)m;(tσ−dη+b;p)])(x)−x−ˊαΓ(λ)Jk,c,(γ,d);kδ,b,(αj,βj)m(1;p)|∞n=0=∞∑m,s=0(α)m(ˊα)s(β)m(ˊβ)sλm+sm!s!∫01(1−u)λ+m−1(1−1u)s(xu)(σ−d)n(η+b)n−α+λ−1(−xu2)du=∞∑m,s=0(α)m(ˊα)s(β)m(ˊβ)s(−1)sλm+sm!s!x(σ−d)n(η+b)n−α+λ∫10(1−u)λ+m+s−1u(d−σ)n(η+b)n+α−λ−s−1du. | (5.15) |
Using Eqs (1.6) and (1.7) in Eq (5.15), we have
(Iα,ˊα,β,ˊβ,λ−Jc,b,δ(γ,d);k[(αj,βj)m;(tσ−dη+b;p)])(x)−x−ˊα+λ−αJk,c,(γ,d);kδ,b,(αj,βj)m(xσ−dη+b;p)|∞n=0=∞∑m,s=0(α)m(ˊα)s(β)m(ˊβ)s(−1)sλm+sm!s!Γ(λ+m+s)Γ((d−σ)n(η+b)n+α−λ−s)Γ(λ)Γ((d−σ)n(η+b)n+α+m)=Γ((d−σ)n(η+b)n+α−λ)Γ((d−σ)n(η+b)n+α)∞∑m=0(α)m(β)m((d−σ)n(η+b)n+α)mm!∞∑s=o(ˊα)s(ˊβ)s(1−(d−σ)n(η+b)n−α+λ)ss!=Γ((d−σ)n(η+b)n+α−λ)Γ((d−σ)n(η+b)n−β)Γ((d−σ)n(η+b)n)Γ((d−σ)n(η+b)n+α−β)Γ(1−(d−σ)n(η+b)n−α+λ)Γ(1−(d−σ)n(η+b)n−α+λ−ˊα−ˊβ)Γ(1−(d−σ)n(η+b)n−α+λ−ˊα)Γ(1−(d−σ)n(η+b)n−α+λ−ˊβ)=Γ((d−σ)n(η+b)n−β)Γ((d−σ)n(η+b)n+α−λ+ˊβ)Γ((d−σ)n(η+b)n+α−λ+ˊα)Γ((d−σ)n(η+b)n)Γ((d−σ)n(η+b)n+α−β)Γ((d−σ)n(η+b)n+α−λ+ˊα+ˊβ). |
We have a desired result
(Iα,ˊα,β,ˊβ,λ−Jc,b,δ(γ,d);k[(αj,βj)m;(tσ−dη+b;p)])(x)=∞∑n=0Bp(γ+kn,d−γ)(−c)nxσn−ηn+α−λ+ˊαΓ[(d+kn)(δ)((d−σ)n(η+b)n−β)((d−σ)n(η+b)n+α−λ+ˊβ)(γ)(d−γ)(δ+n)((d−σ)n(η+b)n)((d−σ)n(η+b)n+α−β)×((d−σ)n(η+b)n+α−λ+ˊα)((d−σ)n(η+b)n+α−λ+ˊα+ˊβ)(αjn+βj+1+b2)|mj=1]. |
In this research, we described extension of extended generalized multi-index Bessel function (E1GMBF) 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 E1GMBF and obtained results in terms of Wright functions.
The authors declare that they have no competing interests.
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