# Relation of Some Known Functions in terms of Generalized MeijerG-Functions

Authors: Syed Ali Haider Shah; Shahid Mubeen; Gauhar Rahman; Jihad Younis
Journal: Journal of Mathematics
 (2021)

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## Abstract

The aim of this paper is to prove some identities in the form of generalized MeijerG-function. We prove the relation of some known functions such as exponential functions, sine and cosine functions, product of exponential and trigonometric functions, product of exponential and hyperbolic functions, binomial expansion, logarithmic function, and sine integral, with the generalized Meijer G-function. We also prove the product of modified Bessel function of first and second kind in the form of generalized Meijer G-function and solve an integral involving the product of modified Bessel functions.


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## Body

## 1. Introduction

The elementary functions such as trigonometric functions, exponential functions, logarithmic functions, hypergeometric functions, Bessel functions, Mittag-Leffler functions, and even binomial expansion are all special functions, which are widely applicable in many fields, especially in the field of sciences. The hypergeometric functions were firstly defined in series form by [1], also many properties were presented. Due to the wide range of applications, these hypergeometric functions have gained attention of the researchers. Almost all the elementary functions can be expressed in the form of hypergeometric functions.The exponential functions, sine and cosine functions, and binomial expansion in the form of hypergeometric functions are given in [2, 3] as, respectively,(1)eα=∑v=0∞αvv!=F00−;−;α,sinα=∑v=0∞−1vα2v+1v!3/2v22v=F10−;32;−α24,cosα=∑v=0∞−1vα2vv!1/2v22v=F10−;12;−α24,where ℜα>0 and(2)1−α−a=∑v=0∞avαvv!=F01a;−;α.The Bessel function is defined [4, 5] as(3)jıα=∑v=0∞−1vα2v+ıv!Γ1+v+ı22v+ı.Zhu et al. [6] estimated some weighted Simpson-like type integral inequalities, used them in some estimation type results to obtain the first-order differentiable functions, and applied them in some known special functions, modified Bessel function, q-digamma function, etc. Sarivastava et al. [7] worked on Mittag-Leffler type functions and estimated the Faber polynomial coefficient of biclose-to-convex functions connected with the Borel distribution of the Mittag-Leffler type. They considered the Fekete-Szegö type inequalities for biclose-to-convex function and also presented several results and related consequences. The researchers [8–10] also worked on inequalities involving special functions and proved various applications.The concept ofk-symbol was introduced by Diaz [11, 12]. The k-theory gave boost to the field of special functions. The researchers [13–15] started to work on this particular k-symbol and proved many properties and identities.Diaz defined Pochhammerk-symbol, gamma k-function, and beta k-function, for k>0, respectively, as(4)lm,k=ll+kl+2k⋯l+m−1k,m≥1,k>0,Γkl=limm⟶∞m!kmmkl/k−1lm,k,k>0,l∈ℂ\kZ−,βkl,ω=1k∫01xl/k−11−xω/k−1dx,ℜl>0,ℜω>0.The integral form of gammak-function is given as(5)Γkl=∫0∞xl/k−1e−xk/kdx,ℜl>0,k>0.Mubeen and Rehman [16] gave the k-form of some elementary functions.The MeijerG-functions which are considered to be the general functions are the particular special functions which have gained the attention of many researchers [17–24]. Many special functions can be obtained by considering the specific variation of parameters of Meijer G-function. Most of the special functions are G-functions or can be expressed in the form of product of G-functions with elementary functions [25–29].MeijerG-function is defined [30] as(6)Gp,qm,nepfq|z=12πi∮LΓf1,m−sΓ1−e1,n+sΓen+1,p−sΓ1−fm+1,q+szsds,where Γ⋅ is well known gamma function and 0≤m≤q,0≤n≤p,z≠0.The definitions of MeijerG-function in the form of hypergeometric function are given [31], respectively, as(7)Gp,qm,nepfq|z=∑i=1m∏h=1mΓfh−fi∗∏h=1nΓ1−eh+fizfi∏h=m+1qΓ1−fh+fi∏h=n+1pΓeh−fi×Fq−1p1−ep+fi1−fq+fi∗|−1p−m−nz,for p<q or p=q, z<1 and(8)Gp,qm,nepfq|z=∑i=1m∏h=1nΓei−eh∗∏h=1mΓ1−ei+fhzei−1∏h=n+1pΓ1−ei+eh∏h=m+1qΓei−fh×Fp−1q1−ei+fq1−ei+ep∗|−1q−m−nz−1,for p>q or p=q and z>1.The relation of MeijerG-functions with elementary functions as exponential, logarithmic, cosine, and Bessel functions is given by Santosh [32] as(9)et=G1,00,1−0|−t,ln1+t=G2,21,21,11,0|t,cost=πG0,21,0−0,12|t24,jvt=G0,21,0−v2,−v2|t24,−π2<argt<π2,Mt=12G2,21,212,120,0|−t,where jvt and Mt are the Bessel and modified Bessel functions, respectively.Roach [33] gave the relation of hypergeometric functions with Meijer G-function as(10)Fe;f;z=ΓfΓeG1−e;;0;1−f;log−z,and proved some results related to Meijer G-function. Dyda et al. [34] worked on fractional Laplace operators and Meijer G-functions and discussed some transformation properties of Meijer G-function given as(11)tdGp,qm,nef|t=Gp,qm,ne+df+d|t,Gp,qm,nef|t−1=Gp,qm,n1−e1−f|t,Gp,qm,nd,e′f′,d|t=Gp−1,q−1m,n−1e′f′|t,Gp,qm,ne′,dd,f′|t=Gp−1,q−1m−1,ne′f′|t,where e′=e1,e2,…,ep−1 and f′=f1,f2,…,fp−1. Pishkoo and Darus [35] gave series representations of three basic univalent G-functions by using Mellin-Barnes type contour integral representation.In our current findings, we first define the generalized form of MeijerG-function in the integral and hypergeometric forms and obtain some known special functions such as the Bessel function, exponential function, sine function, cosine function, sine and cosine hyperbolic functions, product of exponential and trigonometric functions, product of exponential and hyperbolic functions, binomial expansions, and logarithmic function by using the definitions of generalized Meijer G-functions for different choice of parameters. We define the generalized form of modified Bessel functions of the first and second kind and prove results for the product of modified Bessel functions in the form of generalized Meijer G-function.


## 2. Generalized MeijerG-Functions and Their Relation with Some Known Functions

In this section, we define the generalized form of (7) and (8) in the form of hypergeometric k-function and (6) in integral form for k>0. The relations of some known functions with generalized Meijer G-function are also considered.Definition 1.
Forp<q or p=q and z<1, the generalized Meijer G-function can be defined as(12)Gk,p,qm,nepfq|z=∑i=1m∏h=1mΓkfh−fi∗∏h=1nΓkk−eh+fizfi/k∏h=m+1qΓkk−fh+fi∏h=n+1pΓkeh−fi×Fq−1,kpk−ep+fik−fq+fi∗|−1p−m−nz.Definition 2.
Forp>q or p=q and z>1, the generalized Meijer G-function can be defined as(13)Gk,p,qm,nepfq|z=∑i=1m∏h=1nΓkei−eh∗∏h=1mΓkk−ei+fhzei/k−1∏h=n+1pΓkk−ei+eh∏h=m+1qΓkei−fh×Fp−1,kqk−ei+fqk−ei+ep∗|−1q−m−nz−1.Definition 3.
The integral form of generalized MeijerG-function for k>0 is given as(14)Gk,p,qm,nepfq|z=12πi∮LΓkf1,m−sΓkk−e1,n+sΓken+1,p−sΓkk−fm+1,q+szs/kds,where Γk⋅ is the well-known gamma k-function and 0≤m≤q,0≤n≤p,z≠0.
Now, we prove some known special functions by considering different choices of parameters of generalized MeijerG-functions defined in (12) and (13), for k>0. Different identities are proved as follows.Proposition 1.
Letf1=α+♭/2, f2=−α+♭/2, z=x2/4, p=0,q=2,m=1,n=0, then (12) gives the generalized Meijer G-function in the form of Bessel function as(15)2♭/kGk,0,21,0−α+♭/2,−α+♭/2|x24=x♭/kjα,kx.Proof.
By choosingf1=α+♭/2, f2=−α+♭/2, z=x2/4, p=0,q=2,m=1,n=0 in (12), we have(16)Gk,0,21,0−α+♭/2,−α+♭/2|x24=x2/4α+♭/2kΓkk+α−♭/2+α+♭/2F1,k0−k+α|−x24=x♭/k2♭/kjα,kx,which gives the desired result.Proposition 2.
Lete1=0, e2=k/2, f1=0, f2=k/2, z=x,p=2,q=2,m=1,n=1, then (12) gives(17)πkGk,2,21,10,k20,k2|x=1+kx−1.Proof.
By choosinge1=0, e2=k/2, f1=0, f2=k/2, z=x,p=2,q=2,m=1,n=1 in (12), we have(18)Gk,2,21,10,k20,k2|x=ΓkkΓkk−k/2Γkk/2F1,k2k,k2k2|−x=1Γkk/2Γkk/2∑n=0∞kn,k−1nxnn!.
SinceΓkk/2=π/k, so(19)Gk,2,21,10,k20,k2|x=kπ1+kx−1.Proposition 3.
Lete1=k−ρ, f1=0, z=x,p=1,q=1,m=1,n=1, then (12) gives the relation of Meijer G-function with binomial series as(20)1ΓkρGk,1,11,1k−ρ0|x=1+kx−ρ/k.Proof.
By choosinge1=k−ρ, f1=0, z=x,p=1,q=1,m=1,n=1 in (12), we have(21)Gk,1,11,1k−ρ0|x=ΓkρF0,k1ρ−|−x=Γkρ∑n=0∞ρn,k−1nxnn!=Γkρ1+kx−ρ/k.Proposition 4.
Lete1=β, f1=0, z=x,p=1,q=1,m=1,n=0 in (12), then the generalized Meijer G-function can be expressed in series form as(22)Gk,1,11,0β0|x=1Γkβ∑n=0∞k−βn,kxnn!.Proof.
By choosinge1=β, f1=0, z=x,p=1,q=1,m=1,n=0 in (12), we have(23)Gk,1,11,0β0|x=1Γkβ1F0,kk−β−|x=1Γkβ∑n=0∞k−βn,kxnn!.Corollary 1.
By taking same choices as in the above case in (13), we can obtain(24)ΓkβGk,1,11,0β0|x=∑n=0∞k−βn,kxnn!.Proposition 5.
Letf1=k/2, f2=0, z=x,p=0,q=2,m=1,n=0, then (12) gives the relation of generalized Meijer G-function with sine function as(25)πGk,0,21,0−k2,0|x=sin2xk.Proof.
By choosingf1=k/2, f2=0, z=x,p=0,q=2,m=1,n=0 in (12), we have(26)Gk,0,21,0−k2,0|x=x1/2Γk3k/2F1,k0−3k2|−x=2x1/2kπ∑n=0∞−xn3k/2n,kn!=1πsin2xk.Proposition 6.
Letf1=0, f2=k/2, z=x,p=0,q=2,m=1,n=0, then (12) gives the relation of generalized Meijer G-function with cosine function as(27)πGk,0,21,0−0,k2|x=kcos2xk.Proof.
By choosingf1=0, f2=k/2, z=x,p=0,q=2,m=1,n=0 in (12), we have(28)Gk,0,21,0−0,k2|x=1Γkk/20F1,k−k2|−x=kπ∑n=0∞−xnk/2n,kn!=kπcos2xk.Proposition 7.
Lete1=k, f1=0, z=xk,p=1,q=0,m=0,n=1 in (12), then we have(29)Gk,1,00,1k0|xk=e−xk.Proof.
By choosinge1=k, f1=0, z=xk,p=1,q=0,m=0,n=1 in (12), we have(30)Gk,1,00,1k0|xk=F0,k0−−|−xk=∑n=0∞−xknn!=e−xk.Corollary 2.
By taking the same choice of parameters as in the previous proposition andz=x, then (13) gives(31)Gk,1,00,1k0|x=e−1/x.Proof.
By choosinge1=k, f1=0, z=x,p=1,q=0,m=0,n=1 in (13), we obtain(32)Gk,1,00,1k0|x=F0,k0−−|−x−1=∑n=0∞−1nxnn!=e−1x.Proposition 8.
Lete1=k, f1=k,f2=0, z=x,p=1,q=2,m=1,n=1, then (12) gives the relation of generalized Meijer G-function with the product of exponential and sine hyperbolic function as(33)Gk,1,21,1kk,0|x=2ke−x/2sinhx2.Proof.
By choosinge1=k, f1=k,f2=0, z=x,p=1,q=2,m=1,n=1 in (12), we have(34)Gk,1,21,1kk,0|x=xΓk2kF1,k1k2k|−x=xk∑n=0∞kn,k−xn2kn,kn!=2ke−x/2sinhx2.Corollary 3.
Fore1=0, e2=k, f1=0, z=x,p=2,q=1,m=1,n=1, (13) gives(35)Gk,2,11,10,k0|x=2ke−1/2xsinh12x.Proof.
By choosinge1=0, e2=k, f1=0, z=x,p=2,q=1,m=1,n=1 in (13), we can obtain(36)Gk,2,11,10,k0|x=x−1Γk2kF1,k1k2k|−x−1=1k∑n=0∞kn,k1/xn+1−1n2kn,kn!=2ke−1/2xsinh12x.Proposition 9.
Lete1=k/2, e2=3k/2, z=x,p=2,q=0,m=0,n=1 in (13), then(37)πGk,2,00,1k2,3k2−|x=sin2kx−1/2.Proof.
By choosinge1=k/2, e2=3k/2, z=x,p=2,q=0,m=0,n=1 in (13), we obtain(38)Gk,2,00,1k2,3k2−|x=x−1/2Γk3k/2F1,k0−3k2|−x−1=1Γk3k/2∑n=0∞−1n3k/2n,kxn+1/2n!=1πsin2kx−1/2.Proposition 10.
Lete1=0, e2=k, e3=k/2, f1=0, f2=0, f3=k/2, z=x,p=3,q=3,m=2,n=1, then (13) gives the relation of generalized Meijer G-function with logarithmic function as(39)kπGk,3,32,10,k,k20,0,k2|x=ln1−kx.Proof.
By choosinge1=0, e2=k, e3=k/2, f1=0, f2=0, f3=k/2, z=x,p=3,q=3,m=2,n=1 in (13), we have(40)Gk,3,32,10,k,k20,0,k2|x=x−1Γk−k/2Γk3k/2Γk2kF2,k3k,k,3k22k,3k2|1x=−2π∑n=0∞kn,k2kn,kxn+1n!=1kπln1−kx.Proposition 11.
Lete1=k/2, e2=k, e3=k, f1=0z=x,p=3,q=1,m=1,n=1 in (13), then we have(41)Gk,3,11,1k2,k,k0|x=2kπSi2kx,where Si is the sine integral.Proof.
By choosinge1=k/2, e2=k, e3=k, f1=0z=x,p=3,q=1,m=1,n=1 in (13), we have(42)Gk,3,11,1k2,k,k0|x=x−1/2Γkk/2Γk3k/2Γk3k/2F2,k1k23k2,3k2|−1x=4kkπ∑n=0∞k/2n,k−1n3k/2n,k3k/2n,kxn+1/2n!=2kπSi2kx.


## 3. Product of Modified Bessel Functions in terms of Generalized MeijerG-Function and Their Relations

In this section, we first define the modified Bessel function of the first and second kind in the form of generalized MeijerG-function and then prove the relation of the product of modified Bessel functions with the generalized Meijer G-function by taking the specific choice of parameter of the Bessel functions. At the end, we solve an integral involving the product of modified Bessel functions in the integrand.For complex argumentx, the modified Bessel functions of the first and second kind are defined, respectively, as(43)lv,kx=ι−vGk,0,21,0−vk2,−vk2|−x24,(44)Mv,kx=12Gk,0,22,0−vk2,−vk2|x24,where v is an integer.Theorem 1.
Letv=3 in (43) and (44), then the product of modified Bessel functions in terms of generalized Meijer G-function is given as(45)l3,kxM3,kx=k2πGk,1,32,1k23k,0,−3k|x2.Proof.
We first consider the left hand side as(46)l3,kx=ι−3Gk,0,21,0−3k2,−3k2|−x24=−1ιΓk4k∑j=0∞−12j+3/2x2j+34kj,kj!22j+3=18Γk4k∑j=0∞x2j+34kj,kj!22j,(47)M3,kx=12Gk,0,22,0−3k2,−3k2|x24=12Γk−3k∑j=0∞x2j+34kj,kn!22j+3+Γk3k∑j=0∞x2j−3−2kj,kj!22j−3.
So, from (46) and (47), we have(48)l3,kxM3,kx=18Γk4k∑j=0∞x2j+34kj,kj!22j12Γk−3k∑j=0∞x2j+34kj,kj!22j+3+Γk3k∑j=0∞x2j−3−2kj,kj!22j−3.
By working on (48), we obtain(49)l3,kxM3,kx=Γk−3k128Γk4k∑j=0∞7k/2j,kx2j+64kj,k7kj,kj!+Γk3k2Γk4k∑j=0∞k/2j,kx2j−2kj,k4kj,kj!.
Now, by considering the right hand side and applying (12), we have(50)Gk,1,32,1k23k,0,−3k|x2=Γk−3kΓk7k/2Γk7k∑j=0∞7k/2j,kx2j+64kj,k7kj,kj!+Γk3kΓkk/2Γk4k∑j=0∞k/2j,kx2j−2kj,k4kj,kj!.
From (49) and (50), we obtain the required result.Theorem 2.
Letu be the variable of integration, k>0, and uz be the argument of the modified Bessel functions of the first and second kind, then we have(51)∫0∞euk/kGk,1,32,1k23k,0,−3k|uzdu=Gk,2,32,2k2,03k,0,−3k|z.Proof.
From Theorem 3.1, we can write(52)l3,kuzM3,kuz=k2πGk,1,32,1k23k,0,−3k|uz=Γk−3kΓk7k/2Γk7k∑j=0∞7k/2j,kuzj+34kj,k7kj,kj!+Γk3kΓkk/2Γk4k∑j=0∞k/2j,kuzj−2kj,k4kj,kj!.
Now, multiplying byeuk/k and taking integral, we have(53)∫0∞euk/kl3,kuzM3,kuzdu=Γk−3kΓk7k/2Γk7k∑j=0∞7k/2j,kzj+34kj,k7kj,kj!∫0∞uj+3eukkdu+Γk3kΓkk/2Γk4k∑j=0∞k/2j,kzj−2kj,k4kj,kj!∫0∞ujeuk/kdu=Γk−3kΓk7k/2Γk4kΓk7k∑j=0∞7k/2j,k4kj,kzj+34kj,k7kj,kj!+Γk3kΓkk/2ΓkkΓk4k∑j=0∞k/2j,kkj,kzj−2kj,k4kj,kj!∫0∞euk/kGk,1,32,1k23k,0,−3k|uzdu=Γk−3kΓk7k/2Γk4kΓk7k∑j=0∞7k/2j,k4kj,kzj+34kj,k7kj,kj!+Γk3kΓkk/2ΓkkΓk4k∑j=0∞k/2j,kkj,kzj−2kj,k4kj,kj!.
Also, by using (12) on the right hand side, we have(54)Gk,2,32,2k2,03k,0,−3k|z=Γk−3kΓk7k/2Γk4kΓk7k∑j=0∞7k/2j,k4kj,kzj+34kj,k7kj,kj!+Γk3kΓkk/2ΓkkΓk4k∑j=0∞k/2j,kkj,kzj−2kj,k4kj,kj!.
Hence, from (53) and (54), we obtained the desired result.


## 4. Conclusions

In this research work, we gave some functions as sine, cosine, exponential, logarithmic functions, etc., in terms of the generalized form of MeijerG-function. We proved the theorems, which gave the relation of the product of Bessel functions of the first and second kind with the generalized Meijer G-function, and solved the integral involving modified Bessel functions as an integrand, respectively.



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