# Finite Integral Formulas Involving Multivariable Aleph-Functions

Authors: Hagos Tadesse; D. L. Suthar; Minilik Ayalew
Journal: Journal of Applied Mathematics
 (2019)

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## Abstract

The integrals evaluated are the products of multivariable Aleph-functions with algebraic functions, Jacobi polynomials, Legendre functions, Bessel-Maitland functions, and general class of polynomials. The main results of our paper are quite general in nature and competent at yielding a very large number of integrals involving polynomials and various special functions occurring in the problem of mathematical analysis and mathematical physics.


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## Body

## 1. Introduction and Preliminaries

Throughout this paper, considerC,R,R+,Z0-, and N to be a set of complex numbers, positive real numbers, nonpositive integers, and positive integers, respectively. The multivariable Aleph (ℵ) function of several complex variables generalizes the multivariable I-function, recently studied by Sharma and Ahmad [1], which itself is a generalization of G- and H-functions of multiple variables as(1)ℵz1,z2,…,zr=ℵpi,qi,τi;R;pi1,qi1,τi1;R1,…,pir,qir,τir;Rr0,n:m1,n1,m2,n2,…,mr,nrz1⋮zr∣A:BC:D=12πωr∫L1⋯∫Lrψξ1,⋯,ξr∏i=1rϕiξiziξidξ1⋯dξr.where ω=-1,(2)A=aj;αj1,⋯,αjr1,n,τiaji;αji1,⋯,αjirn+1,piB=cj1,γj11,n1,τi1cji11,γji11n1+1,pi1;⋯;cjr,γjr1,nr,τircji1r,γji1rnr+1,pirC=…,τibji;βji1,⋯,βjirm+1,qiD=dj1,δj11,m1,τi1dji11,δji11m1+1,qi1;⋯;djr,δjr1,mr,τirdjirr,γjirrmr+1,qir(3)ψξ1,⋯,ξr=∏j=1nΓ1-aj+∑k=1rαjkξk∑i=1Rτi∏j=n+1piΓaji-∑k=1rαjikξk∏j=1qiΓ1-bji+∑k=1rβjikξk,(4)ϕkξk=∏j=1mkΓdjk-δjkξk∏j=1nkΓ1-cjk+γjkξk∑ik=1Rkτik∏j=mk+1qikΓ1-djikk+δjikkξk∏j=nk+1pikΓcjikk-γjikkξk,and aj(j=1,⋯,p);bj(j=1,⋯,q);cjk(j=1,⋯,nk),cjikk(j=nk+1,⋯,pik); djk(j=1,⋯,mk)djikk(j=mk+1,⋯,qik),(k=1,⋯,r;i=1,⋯,R; and i(k)=1,⋯,R(k) are complex numbers.(5)Uik=∑j=1nαjk+τi∑j=n+1piαjik+∑j=1nkγjk+τik∑j=nk+1pikγjikk-τi∑j=1qiβjikk-∑j=1mkδjk-τik∑j=mk+1qikδjikk≤0,τi(i=1,⋯,R),τik(ik=1,⋯,Rk) are positive real numbers.The integration pathLωγ∞γ∈R extends from γ-ω∞ to γ+ω∞ and the poles of Γdj(k)-δj(k)ξk,j=1,…,mk, do not coincide with the poles of Γ1-aj+∑i=1rαj(k)ξk,j=1,…,n and Γ1-cj(k)+γj(k)ξk,j=1,…,nk to the left of the contour Lk.The existence condition for multiple Mellin-Barnes type contours (1) can be given below:(6)arg⁡zk<12Aikπ where(7)Aik=∑j=1nαjk-τi∑j=n+1piαjik-τi∑j=1qiβjik+∑j=1nkγjk-τik∑j=nk+1pikγjikk+∑j=1mkδjk-τik∑j=mk+1qikδjikk>0,with =1,⋯,r; i=1,⋯,R; and i(k)=1,⋯,R(k).Remark 1.
By settingτi=τik=1, the multivariable Aleph-function reduces to multivariable I-function (see [1–3]]).Remark 2.
By settingτi=τik=1k∈1,…,r and R=R1=,…,=Rr=1, the multivariable Aleph-function reduces to multivariable H-function defined by Srivastava et al. [4].Remark 3.
When we setr=1, the multivariable Aleph-function reduces to Aleph-function of one variable defined by Su¨dland [5].For the definition of the H- function,ℵ-function, and its more generalization, the interested reader may refer to the papers [6–13].From Rainville [14], the integral representation of the gamma function Γx is defined as(8)Γx=∫0∞e-ttx-1dt,Rx>0.And also the beta integral is defined as follows:(9)Bx,y=∫01tx-11-ty-1dt=ΓxΓyΓx+y=By,x,Rx,Ry>0.Further, We will use the following notations in this paper:(10)U=m1,n1,m2,n2,…,mr,nr,V=pi1,qi1,τi1;R1,W=pir,qir,τir;Rr.


## 2. Integrals Involving Multivariable Aleph-Function with Algebraic Function

In this section, we evaluate integrals, the product of multivariable Aleph-functions with various algebraic functions.(11)I1=∫01x-ρ1-xρ-σ-1ℵpi,qi,τi;R;V,…,W0,n:Uz1xω1,…,zrxωrdx=12πωr∫L1⋯∫Lrψξ1,…,ξr∏i=1rϕiξiziξi×∫01x1-ρ+∑i=1rωiξi-11-xρ-σ-1dxdξ1…dξr=12πωr∫L1⋯∫Lrψξ1,…,ξr∏i=1rϕiξiziξiB1-ρ+∑i=1rωiξi,ρ-σdξ1⋯dξr=Γρ-σ12πωr∫L1⋯∫Lrψξ1,…,ξr∏i=1rϕiξiziξiΓ1-ρ+∑i=1rωiξiΓ1-σ+∑i=1rωiξidξ1⋯dξr=Γρ-σℵpi+1,qi+1,τi;R;V,…,W0,n+1:Uz1⋮zr∣ρ;ω1,…,ωr,A:Bσ;ω1,…,ωr,C:D.(12)I2=∫01xρ-11-xσ-1ℵpi,qi,τi;R;V,…,W0,n:Uz1xω1,…,zrxωrdx=12πωr∫L1⋯∫Lrψξ1,…,ξr∏i=1rϕiξiziξi∫01xρ+∑i=1rωiξi-11-xσ-1dxdξ1…dξr=12πωr∫L1⋯∫Lrψξ1,…,ξr∏i=1rϕiξiziξiBρ+∑i=1rωiξi,σdξ1⋯dξr=Γσ12πωr∫L1⋯∫Lrψξ1,…,ξr∏i=1rϕiξiziξiΓρ+∑i=1rξiΓρ+σ+∑i=1rξidξ1⋯dξr=Γσℵpi+1,qi+1,τi;R;V,…,W0,n+1:Uz1⋮zr∣1-ρ;ω1,…,ωr,A:B1-ρ-σ;ω1,…,ωr,C:D.(13)I3=∫1∞x-ρx-1σ-1ℵpi,qi,τi;R;V,…,W0,n:Uz1xω1,…,zrxωrdx=12πωr∫L1⋯∫Lrψξ1,…,ξr∏i=1rϕiξiziξi∫1∞x-ρ+∑i=1rωiξix-1σ-1dxdξ1…dξrPutting x=t+1, we have dx=dt, and we use the following relation.(14)ΓαΓβΓα+β=∫0∞xα-1x+1-α+βdx=∫0∞xβ-1x+1-α+βdx(15)I3=12πωr∫L1⋯∫Lrψξ1,…,ξr∏i=1rϕiξiziξi∫0∞tσ-11+t-ρ-∑i=1rωiξidtdξ1…dξr=Γσ12πωr∫L1⋯∫Lrψξ1,…,ξr∏i=1rϕiξiziξiΓρ-σ-∑i=1rωiξiΓρ-∑i=1rωiξidξ1⋯dξr=Γσℵpi+1,qi+1,τi;R;V,…,W0,n+1:Uz1⋮zr∣ρ;ω1,…,ωr,A:Bρ-σ;ω1,…,ωr,C:D.(16)I4=∫0∞xρ-1x+β-σℵpi,qi,τi;R;V,…,W0,n:Uz1xω1,…,zrxωrdx=12πωr∫L1⋯∫Lrψξ1,…,ξr∏i=1rϕiξiziξi∫0∞xρ+∑i=1rωiξi-1x+β-σdxdξ1…dξrSetting x=βt implies dx=βdt, and then we obtain the following.(17)=βρ-σ2πωr∫L1⋯∫Lrψξ1,…,ξr∏i=1rϕiξiβωiziξi∫0∞tρ+∑i=1rωiξi-11+t-σdtdξ1…dξr By applying (14), we have the following.(18)=βρ-σΓσ12πωr∫L1⋯∫Lrψξ1,…,ξr∏i=1rϕiξiβωiziξi×Γρ+∑i=1rωiξiΓσ-ρ-∑i=1rωiξidξ1…dξr(19)=βρ-σΓσℵpi+1,qi+1,τi;R;V,…,W0,n+1:Uβω1z1⋮βωrzr∣1-ρ;ω1,…,ωr,A:Bσ-ρ;ω1,…,ωr,C:D.(20)I5=∫-111-xρ1+xσℵpi,qi,τi;R;V,…,W0,n:Uz11-xμ1,…,zr1-xμrdx=12πωr∫L1⋯∫Lrψξ1,…,ξr∏i=1rϕiξiziξi∫-111-xρ+∑i=1rμiξi1+xσdxdξ1…dξrNow, we use the following formula ([14], p.261).(21)∫-111-xn+α1+xn+βdx=22n+α+β+1B1+α+n,1+β+nHence, we arrive at(22)=2ρ+σ+1Γ1+σ12πωr∫L1⋯∫Lrψξ1,…,ξr∏i=1rϕiξi2μiziξi×Γ1+ρ+∑i=1rμiξiΓ2+ρ+σ+μ∑i=1rμiξidξ1…dξr(23)=2ρ+σ+1Γ1+σℵpi+1,qi+1,τi;R;V,…,W0,n+1:U2μ1z1⋮2μrzr∣-ρ;μ1,…,μr,A:B-1-σ-ρ;μ1,…,μr,C:D.


## 3. Integrals Involving Multivariable Aleph-Function with Jacobi Polynomials

The Jacobi polynomial ([15], 4.21.2)) with parameters α,β∈R is defined by(24)pnα,βx=1+αnn!F12-n,1+α+β+n;1+α;1-x2n≥1,where F12. is the classical hypergeometric function. By substituting α=β=0, the Jacobi polynomial (24) reduces to Lagrange polynomial ( [14], p. 157) as(25)pnα,β1=1+αnn!.In this section, we derive integral formulas involving multivariable Aleph-functions multiplied by Jacobi polynomials.(26)I6=∫-11xλ1-xα1+xμPnα,βxℵpi,qi,τi;R;V,…,W0,n:Uz11+xl1,…,zr1+xlrdx=12πωr∫L1⋯∫Lrψξ1,…,ξr∏i=1rϕiξiziξi×∫-11xλ1-xα1+xμ+∑i=1rliξiPnα,βxdxdξ1…dξrNext, we use the following formula:(27)∫-11xλ1-xα1+xμpnα,βxdx=-1n2α+μ+1Γμ+1Γn+α+1Γμ+β+1n!Γμ+β+n+1Γμ+α+n+2×F32-λ,μ+β+1,μ+1μ+β+n+1,μ+α+n+2;1,where α>-1 and β>-1. Also, F32 is the special case of generalized hypergeometric series.Then, we have the following.(28)=12πωr∫L1⋯∫Lrψξ1,…,ξr∏i=1rϕiξiziξi-1n2α+μ+∑i=1rliξi+1n!×Γμ+∑i=1rliξi+1Γn+α+1Γμ+∑i=1rliξi+β+1Γμ+∑i=1rliξi+β+n+1Γμ+∑i=1rliξi+α+n+2×F32-λ,μ+∑i=1rliξi+β+1,μ+∑i=1rliξi+1μ+∑i=1rliξi+β+n+1,μ+∑i=1rliξi+α+n+2;1dξ1…dξrUsing the definition of hypergeometric function and some simplifications, the above expression becomes(29)I6=-1n2α+μ+1Γα+n+1n!∑k=0∞-λk1kk!12πωr∫L1⋯∫Lrψξ1,…,ξr∏i=1rϕiξi×2liziξiΓμ+∑i=1rliξi+β+k+1Γμ+∑i=1rliξi+k+1Γμ+∑i=1rliξi+β+n+k+1Γμ+∑i=1rliξi+α+n+k+2dξ1…dξr=-1n2α+μ+1Γα+n+1n!∑k=0∞-λk1kk!ℵpi+2,qi+2,τi;R;V,…,W0,n+2:U2l1z1⋮2lrzr∣-μ-β-k;l1,…,lr,-μ-k;l1,…,lr,A:B-μ-β-n-k;l1,…,lr,-1-μ-α-n-k;l1,…,lr,C:D.Thus, α>-1, β>-1, Rλ>-1, and argz<1/2πΩ.(30)
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rFinally, we arrive at(32)I7=2δ+v+1Γ1+ρ+mΓ1+μ+nΓv+1m!n!×∑k=0∞-mk-nk1+ρ+σ+mk1+μ+v+nkΓ1+ρ+kΓ1+μ+kk!2×ℵpi+1,qi+1,τi;R;V,…,W0,n+1:U2l1z1⋮2lrzr∣-δ-2k;l1,…,lr,A:B-1-δ-2k-v;l1,…,lr,C:D.Thus, δ>0, Rv>-1, and argz<1/2πΩ.(33)I8=∫-111-xρ1+xσPnμ,vx×ℵpi,qi,τi;R;V,…,W0,n:Uz11-xl11+xh1,…,zr1-xlr1+xhrdx=Γ1+μ+nn!∑k=0∞-nk1+μ+v+nkΓ1+μ+k2kk!12πωr∫L1⋯∫Lrψξ1,…,ξr∏i=1rϕiξiziξi×∫-111-x1+ρ+∑i=1rliξi+k-11+x1+σ+∑i=1rhiξi-1dxdξ1…dξrUsing (21), we have the following.(34)I8=2ρ+σ+1Γ1+μ+nn!∑k=0∞-nk1+μ+v+nkΓ1+μ+kk!×12πωr∫L1⋯∫Lrψξ1,…,ξr∏i=1rϕiξi2li+hiziξi×Γ1+ρ+k+∑i=1rliξiΓ1+σ+∑i=1rhiξiΓ2+ρ+σ+k+∑i=1rli+hiξidξ1…dξrFinally, rewriting the above equation by virtue of (1), we arrive at(35)I8=2ρ+σ+1Γ1+μ+nn!∑k=0∞-nk1+μ+v+nkΓ1+μ+kk!×ℵpi+2,qi+1,τi;R;V,…,W0,n+2:U2l1+h1z1⋮2lr+hrzr∣-ρ-k;l1,…,lr,-σ;h1,…,hr,A:B-1-ρ-σ-k,l1+h1,…,lr+hr,C:D.Thus, Rv>-1, Rμ>-1, and argz<1/2πΩ.


## 4. Integrals Involving Multivariable Aleph-Function with Legendre Function

The solution of Legendre differential equation in the form of Gauss hypergeometric type is as follows:(36)pvμx=1Γ1-μx+1x-11/2μF21-v,v+1;1-μ;1-x2,1-x<2.Here pvμx is known as the Legendre function of the first kind [16].Next, we derive the integrals with Legendre function.(37)I9=∫01xσ-11-x2μ/2pvμxℵpi,qi,τi;R;V,…,W0,n:Uz1xρ1,…,zrxρrdx=12πωr∫L1⋯∫Lrψξ1,…,ξr∏i=1rϕiξiziξi×∫01xσ+∑i=1rρiξi-11-x2μ/2Pvμxdxdξ1…dξrNext, we use the following formula ([16], Sec. 3.12) for μ∈N,Rσ>0(38)∫01xσ-11-x2μ/2pvμxdx=-1μ2-σ-μπ1/2ΓσΓ1+μ+vΓ1-μ+vΓ1/2+σ/2+μ/2-v/2Γ1+σ/2+μ/2+v/2.Now, applying (38), we obtain(39)I9=2-σ-μ-1μπ1/2Γ1+μ+vΓ1-μ+v12πωr∫L1⋯∫Lrψξ1,…,ξr∏i=1rϕiξi2-ρiziξi×Γσ+∑i=1rρiξiΓ1/2+σ/2+μ/2-v/2+∑i=1rρiξi/2Γ1+σ/2+μ/2+v/2+∑i=1rρiξi/2dξ1…dξr(40)I9=2-σ-μ-1μπ1/2Γ1+μ+vΓ1-μ+vℵpi+1,qi+2,τi;R;V,…,W0,n+1:U2-ρ1z1⋮2-ρrzr∣1-σ;ρ1,…,ρr,A:B12-μ2+v2-σ2;ρ12,…,ρr2,-μ2-v2-σ2;ρ12,…,ρr2,C:D.Thus, argz<1/2πΩ, σ>0, and μ∈N∪0.(41)I10=∫01xσ-11-x2-μ/2pvμxℵpi,qi,τi;R;V,…,W0,n:Uz1xρ1,…,zrxρrdx=12πωr∫L1⋯∫Lrψξ1,…,ξr∏i=1rϕiξiziξi×∫01xσ+∑i=1rρiξi-11-x2-μ/2Pvμxdxdξ1…dξrNext we use the following formula ([16], Sec. 3.12) for μ∈N,Rσ>0(42)∫01xσ-11-x2-μ/2pvμxdx=2-σ+μπ1/2ΓσΓ1/2+σ/2-μ/2-v/2Γ1+σ/2-μ/2-v/2.Using (42), we obtain(43)I10=2-σ+μπ1/212πωr∫L1⋯∫Lrψξ1,…,ξr∏i=1rϕiξi2-ρiziξi×Γσ+∑i=1rρiξiΓ1/2+σ/2-μ/2-v/2+∑i=1rρiξi/2Γ1+σ/2-μ/2-v/2+∑i=1rρiξi/2dξ1…dξr(44)I10=2-σ+μπ1/2ℵpi+1,qi+2,τi;R;V,…,W0,n+1:U2-ρ1z1⋮2-ρrzr∣1-σ;ρ1,…,ρr,A:B12+μ2+v2-σ2;ρ12,…,ρr2,μ2+v2-σ2;ρ12,…,ρr2,C:D.Thus, argz<1/2πΩ, Rσ>0, and Rμ∈N∪0.


## 5. Integrals Involving Multivariable Aleph-Function and Bessel-Maitland Function

The Bessel-Maitland function is defined by Kiryakova [17] as follows:(45)Jvμx=ϕμ,v+1:x=∑k=0∞1Γμk+v+1-xkk!,μ>-1,x∈C.(46)I11=∫0∞xρJvμxℵpi,qi,τi;R;V,…,W0,n:Uz1xσ1,…,zrxσrdxdx=12πωr∫L1⋯∫Lrψξ1,…,ξr∏i=1rϕiξiziξi∫0∞xρ+∑i=1rσiξiJvμxdxdξ1…dξrNow, using the following formula from [18](47)∫0∞xρJvμxdx=Γρ+1Γ1+v-μ-μρRρ>-1,0<μ<1,and, further, applying (47), we obtain(48)I14=12πωr∫L1⋯∫Lrψξ1,…,ξr∏i=1rϕiξiziξiΓ1+ρ+∑i=1rσiξiΓ1+v-μ-μρ-μ∑i=1rσiξidξ1…dξr(49)I14=ℵpi+2,qi,τi;R;V,…,W0,n+1:Uz1⋮zr∣-ρ;σ1,…,σr,1+v-μ-μρ;μσ1,…,μσr,A:BC:D.Thus, argz<1/2πΩ,σ-μσ>0,σ>0,0<μ<1, and Rρ+1>0.


## 6. Integrals Involving Multivariable Aleph-Function and General Class of Polynomials

The general class of polynomialsSn1,…,nrm1,…,mr(x) defined by Srivastava [19] is as follows(50)Sn1,…,nrm1,…,mrx=∑li=0n1/m1⋯∑lr=0nr/mr∏i=1r-nimilili!Ani,lixli.Here n1,n2,…,nr∈N0; m1,m2,…,mr∈N; and the coefficients Ani,lini,li≥0 are any constant numbers. By suitable restriction of the coefficient Ani,li the general class of polynomials has various special cases. These include the Jacobi polynomials, the Laguerre polynomials, the Bessel polynomials, the Hermite polynomials, the Gould-Hopper polynomials, and the Brafman polynomials ([20], p. 158-161).Next, we establish the following integral.(51)I12=∫-111-xρ-11+xσ-1Sn1,…,nr′m1,…,mr′y1-xμ1+xv×ℵpi,qi,τi;R;V,…,W0,n:Uz11-xh11+xk1,…,zr1-xhr1+xkrdx=∑l1=0n1/m1⋯∑lr′nr′/mr′∏i=1r′-njmjljlj!Anj,ljylj12πωr∫L1⋯∫Lrψξ1⋯ξr∏i=1rϕiξiziξi×∫-111-xρ+∑i=1rhiξi+μ∑j=1r′lj-11+xσ+∑i=1rkiξi+v∑j=1r′lj-1dxdξ1…dξrBy applying (21), we have(52)I12=2ρ+σ-1∑l1=0n1/m1⋯∑lr′nr′/mr′∏i=1r′-njmjljlj!Anjlj2μ+vylj×12πωr∫L1⋯∫Lrψξ1⋯ξr∏i=1rϕiξi2hi+kiziξi×Γρ+∑i=1rhiξi+μ∑j=1r′ljΓσ+∑i=1rkiξi+v∑j=1r′ljΓρ+σ+∑i=1rhi+kiξi+μ+v∑j=1r′ljdξ1…dξr(53)I12=2ρ+σ-1∑l1=0n1/m1⋯∑lr′nr′/mr′∏i=1r′-njmjljlj!Anjlj2μ+vyljℵpi+2,qi,τi;R;V;…;W0,n+1:U2h1+k1z1⋮2hr+krzr∣1-ρ-μ∑j=1r′lj;h1,…,hr,1-σ-v∑j=1r′lj;k1,…,kr,A:B1-ρ-σ-μ+v∑j=1r′lj;h1+k1,…,hr+kr,C:D.These converge under the following conditions:(1)
argz<1/2πΩ,(2)
ρ≥1,σ≥1,μ≥0,v≥0,h≥0,k≥0 (h and k are not both zero simultaneously),(3)
Rρ+hmin⁡Rbj/Bj>0 and Rσ+kmin⁡Rbj/Bj>0.


## 7. Special Cases

(i) By setting δ by η-1 and ρ=σ=μ=v=0, (32) transforms to the following integral:(54)I13=2η∑k=0∞-mk-nk1+mk1+nkΓ1+kΓ1+kk!2×ℵpi+1,qi+1,τi;R;V;…;W0,n+1:U2lz1⋮2lzr∣1-η-2k;l1,…,lr,A:B-η-2k;l1,…,lr,C:D.Thus, argz<1/2πΩ.(ii) When we substitute ρ by ρ-1 and σ by σ-1 and put μ=v=0, integral (35) transforms to the following:(55)I14=2ρ+σ-1∑k=0∞-nk1+nkΓ1+kk!×ℵpi+2,qi+1,τi;R;V;…;W0,n+2:U2l1+h1z1⋮2lr+hrzr∣1-ρ-k;l1,…,lr,1-σ;h1,…,hr,A:B1-ρ-σ-k;l1+h1,…,lr+hr,C:D.Thus, argz<1/2πΩ.


## 8. Concluding Remarks

In the present paper, we evaluated new integrals involving the multivariable Aleph-function with certain special functions. Certain special cases of integrals that follow Remarks1–3 have been investigated in the literature by a number of authors [21–24]) with different arguments. Therefore, the results presented in this paper are easily converted in terms of a similar type of new interesting integrals with different arguments after some suitable parameter substitutions. In this sequel, one can obtain integral representation of more generalized special function, which has many applications in physics and engineering science.



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