# Euler Type Integrals and Integrals in Terms of Extended Beta Function

Authors: Subuhi Khan; Mustafa Walid Al-Saad
Journal: Journal of Mathematics
 (2014)

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## Abstract

We derive the evaluations of certain integrals of Euler type involving generalized hypergeometric series. Further, we establish a theorem on extended beta function, which provides evaluation of certain integrals in terms of extended beta function and certain special polynomials. The possibility of extending some of the derived results to multivariable case is also investigated.


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## Body

## 1. Introduction

Euler generalized the factorial function from the domain of natural numbers to the gamma function(1)Γ(α)=∫0∞tα-1e-tdt(Re(α)>0),
defined over the right half of the complex plane. This led Legendre (in 1811) to decompose the gamma function into the incomplete gamma functions, γ(α,x) and Γ(α,x) [1],(2a)γ(α,x)=∫0x‍tα-1e-tdt(Re(α)>0),(2b)Γ(α,x)=∫x∞‍tα-1e-tdt,which are obtained from (1) by replacing the upper and lower limits by x, respectively. The closed-form solutions to a considerable number of problems in applied mathematics, astrophysics, nuclear physics, statistics, and engineering can be expressed in terms of incomplete gamma functions. These functions develop singularities at the negative integers. Chaudhry and Zubair [2] extended the domain of these functions to the entire complex plane by inserting a regularization factor exp⁡(-b/t) in the integrand of (1). For Re(b)>0, this factor clearly removes the singularity coming from the limit t=0. For b=0, this factor becomes unity, and thus we get the original gamma function. We note the following relation [3, page 20 (1.2)]:
(3)Γb(α)=∫0∞‍tα-1exp⁡(-t-bt)dt=2(b)α/2Kα(2b)(Re(α)>0),
where Kn(x) is the modified Bessel function of the second kind of order n (or Macdonald’s function) [1]. The relationships between the generalized gamma and Macdonald functions could not have been apparent in the original gamma function. These generalized gamma functions proved very useful in diverse engineering and physical problems; see, for example, [2, 3] and references therein.We note that Riemann’s zeta functionζ(x) defined by the series [1, page 85 (2.98)]
(4)ζ(x)=∑n=1∞1nx(x>1)
is useful in proving convergence or divergence of other series by means of comparison test. Zeta function is closely related to the logarithm of the gamma function and to the polygamma functions. The regularizer exp⁡(-b/t) also proved very useful in extending the domain of Riemann’s zeta function, thereby providing relationships that could not have been obtained with the original zeta function.In view of the fact that the regularization factor is useful in extending the domain of the gamma and zeta functions. The domain of other special functions could be usefully extended in a similar manner.In particular, Euler’s beta functionB(α,β) has a close relationship to gamma function, that is,
(5)B(α,β)=B(β,α)=Γ(α)Γ(β)Γ(α+β),
and can be usefully extendable. We first recall the basic Euler integral which defines the beta function as follows:
(6)B(α,β)=∫01uα-1(1-u)β-1du(Re(α),Re(β)>0).Keeping in view the fact that for theα-β symmetry to be preserved there must be symmetry of the integrand in u and 1-u, Chaudhry et al. [3] introduced an extension of Euler’s beta function B(α,β) in the following form:
(7)B(α,β;b)=∫01‍tα-1(1-t)β-1exp⁡(-bt(1-t))dt555555555555555555555555555555(Re(b)>0),
which for b=0 reduces to the original beta function.The transformationst=(u+1)/2 and t=(u-a)/(c-a) in (7) yield the following integral representations for the extended beta function (EBF) B(α,β;b) [3, page 22 (2.9), (2.11)]:
(8)B(α,β;b)=21-α-β×∫-11‍(1+u)α-1(1-u)β-1exp⁡⁡(-4b(1-u2))du55555555555555555555555555555555(Re(b)>0),(9)B(α,β;b)=(c-a)1-α-β×∫ac‍(u-a)α-1(c-u)β-1exp⁡⁡(-b(c-a)2(u-a)(c-u))du5555555555555555555555555555555555555(Re(b)>0),
respectively.The EBFB(α,β;b) is extremely useful in the sense that most of the properties of the beta function carry over naturally and simply for it. This extension is also important due to the fact that this function is related to other special functions for particular values of the variables.We note the following connections [3]:
(10)B(α,-α;b)=2e-2bKα(2b)(Re(b)>0),(11)B(α,α;b)=π2-αb(α-1)/2e-2bW-α/2,α/2(4b)=π21-2αe-4bΨ(12,1-α;4b)(Re(b)>0),
where Wm,n(z) denotes the Whittaker function [1] and Ψ(a,c;z) denotes the second form of solutions of Kummer’s equation [4].Also, we note that(12)B(0,0;b)=2e-2bK0(2b)(Re(b)>0),B(12,-12;b)=πbe-4b(Re(b)>0),B(12,12;b)=πErfc⁡(2b)(Re(b)>0),
where Erfc⁡(z)=(π/2)erfc(z) denotes the complementary error function [4].Explicit evaluations of some integrals of Euler type(13)∫01uα-1(1-u)β-1f(u)du
for some particular functions f, specially in the symmetric case α=β, are derived in [5]. These evaluations are related to various reduction formulae for hypergeometric functions represented by such integrals. These formulae generalize the evaluations of some symmetric Euler integrals implied by the following result due to Pitman [6].If a standard Brownian bridge is sampled at time0, time 1 and at n independent random times with uniform distribution on [0,1], then the broken line approximation to the bridge obtained from these n+2 values has a total variation whose mean square is n(n+1)/(2n+1).Motivated and inspired by the work of Ismail and Pitman [5] and Chaudhry et al. [3], in this paper, we derive the evaluations of certain Euler type integrals and some integrals in terms of EBF B(α,β;b). In Section 2, we obtain the evaluations of certain integrals of the following type:
(14)Id,α,β,γ,a,e[ϕ(t),ψ(t)]=1B(α,β)∫ae‍(t-a)α-1(e-t)β-1(ϕ(t))γ×Ftr[(fr);dψ(t)(gt);]dt,
where Frt denotes the generalized hypergeometric series [4], for particular functions ϕ(t) and ψ(t). In Section 3, we establish a theorem on EBF B(α,β;b) and apply it to obtain evaluations of certain integrals in terms of EBF B(α,β;b). Finally, we give some concluding remarks in Section 4.


## 2. Euler Type Integrals

We derive the evaluations of certain integrals of the following type:(15)Id,α,β,γ,a,e[ϕ(t),ψ(t)]=1B(α,β)∫ae‍(t-a)α-1(e-t)β-1(ϕ(t))γ×Ftr[(fr);dψ(t)(gt);]dt.We evaluate the integrals of type (15) for particular functions ϕ(t) and ψ(t) by considering the following cases.Case 1.
Takingϕ(t)=(1-x1t)-α1(1-x2t)-α2, ψ(t)=1/t(1-t), a=0, and e=γ=1 in (15) and using [5, page 962 (7)]
(16)∫01‍tα-1(1-t)β-1(1-x1t)-α1(1-x2t)-α2dt=B(α,β)F1[α,α1,α2;α+β;x1,x2](Re(α),Re(β)>0;max⁡{|x1|,|x2|}<1),
where F1 denotes the Appell function [4], in the r.h.s. after expanding Frt[(fr);dψ(t)(gt);] and simplifying, we find the following integral:(17)Id,α,β,1,0,1[(1-x1t)-α1(1-x2t)-α2,1t(1-t)]=∑m1,m2=0∞(α)m1+m2(α1)m1(α2)m2x1m1x2m2(α+β)m1+m2m1!m2!×Fr+2t+2[(fr),12(1-α-β-m1-m2),1-12(α+β+m1+m2);4d(gt),1-(α+m1+m2),1-β;](Re(α),Re(β)>0;max⁡⁡{|x1|,|x2|}<1).Case 2.
Takingϕ(t)=Fqp[(ap);ct(e-t)(bq);], ψ(t)=1/t(e-t), a=0, and γ=1 in (15) and using [7, page 303 (1)]
(18)∫0e‍tα-1(e-t)β-1Fpq[(ap);ct(e-t)(bq);]dt=(e)α+β-1B(α,β)×Fq+2p+2[(ap),α,β;ce24(bq),α+β2,α+β+12;](e,Re(α),Re(β)>0;|arg(4-ce2)|<π),
we find the following integral:
(19)Id,α,β,1,0,e[Fpq[(ap);ct(e-t)(bq);],1t(e-t)]=(e)α+β-1∑m=0∞‍(ap)m(α)m(β)m(ce2/4)m(bq)m((α+β)/2)m((α+β+1)/2)mm!×Fr+2t+2[(fr),12(1-α-β)-m,1-12(α+β)-m;4de2(gt),1-α-m,1-β-m;](e,Re(α),Re(β)>0;|arg(4-ce2)|<π).Case 3.
Takingϕ(t)=Fqp[(ap);ct(e-t)(bq);], ψ(t)=1/(t2(e-t)2), a=0, and γ=1 and proceeding as in Case 2, we find the following integral:(20)Id,α,β,1,0,e[Fpq[(ap);ct(e-t)(bq);],1t2(e-t)2]=(e)α+β-1∑m=0∞‍(ap)m(α)m(β)m(ce2/4)m(bq)m((α+β)/2)m((α+β+1)/2)mm!×Fr+6t+8[(fr),j-α-β2,12(1-α-β)-m,1-12(α+β)-m;4de4(gt),j-α-β4,1-α-m2,2-α-m2,1-β-m2,2-β-m2;](j=1,2,3,4)(e,Re(α),Re(β)>0;|arg(4-ce2)|<π).Case 4.
Takingϕ(t)=K(ct(e-t)), ψ(t)=1/t(e-t), a=0, and γ=1 in (15), where K(z) is the complete elliptic integral of the first kind [4, page 35 (32)], defined by
(21)K(z)=∫0π/2‍dθ(1-z2sin2θ)
and using [7, page 265 (4)]
(22)∫0e‍tα-1(e-t)β-1K(ct(e-t))dt=π2(e)α+β-1B(α,β)F34[12,12,α,β;c2e241,α+β2,α+β+12;](Re(α),Re(β)>0;|arg(4-c2e2)|<π),
we find the following integral:
(23)Id,α,β,1,0,e[K(ct(e-t)),1t(e-t)]=π2(e)α+β-1∑m=0∞‍(1/2)m(1/2)m(α)m(β)m(c2e2/4)m(1)m((α+β)/2)m((α+β+1)/2)mm!×Fr+2t+2[(fr),12(1-α-β)-m,1-12(α+β)-m;4de2(gt),1-α-m,1-β-m;](Re(α),Re(β)>0;|arg(4-c2e2)|<π).Case 5.
Takingϕ(t)=E(ct(e-t)), ψ(t)=1/t(e-t), a=0, and γ=1 in (15), where E(z) is the complete elliptic integral of the second kind [4, page 35 (33)], defined by
(24)E(z)=∫0π/2‍(1-z2sin2θ)dθ
and using [7, page 280 (3)]
(25)∫0e‍tα-1(e-t)β-1E(ct(e-t))dt=π2(e)α+β-1B(α,β)F34[-12,12,α,β;c2e241,α+β2,α+β+12;](e,Re(α),Re(β)>0;|arg(4-c2e2)|<π),
we find the following integral:
(26)Id,α,β,1,0,e[E(ct(e-t)),1t(e-t)]=π2(e)α+β-1∑m=0∞‍(-1/2)m(1/2)m(α)m(β)m(c2e2/4)m(1)m((α+β)/2)m((α+β+1)/2)mm!×Fr+2t+2[(fr),12(1-α-β)-m,1-12(α+β)-m;4de2(gt),1-α-m,1-β-m;](e,Re(α),Re(β)>0;|arg(4-c2e2)|<π).Case 6.
Takingϕ(t)=arcsin(ct(e-t)), ψ(t)=1/t(e-t), a=0, and γ=2 in (15) and using [7, page 173 (151)]
(27)∫0e‍tα-1(e-t)β-1arcsin2(ct(e-t))dt=c2(e)α+β+1B(α+1,β+1)×F45[1,1,1,α+1,β+1;c2e2432,2,α+β+22,α+β+32;](e>0;Re(α),Re(β)>-1;|arg(4-c2e2)|<π),
we find the following integral:
(28)Id,α,β,2,0,e[arcsin(ct(e-t)),1t(e-t)]=αβc2(e)α+β+1(α+β)(α+β+1)×∑m=0∞‍(1)m(1)m(1)m(α+1)m(β+1)m(c2e2/4)m(3/2)m(2)m((α+β+2)/2)m((α+β+3)/2)mm!×Fr+2t+2[(fr),-12(1+α+β)-m,-12(α+β)-m;4de2(gt),-α-m,-β-m;](e>0;Re(α),Re(β)>-1;|arg(4-c2e2)|<π).Case 7.
Takingϕ(t)=exp⁡(c2t(e-t))erf⁡(ct(e-t)), ψ(t)=1/t(e-t), a=0, and γ=1 in (15) and using [7, page 185 (1)]
(29)∫0e‍tα-1(e-t)β-1exp⁡(c2t(e-t))erf⁡(ct(e-t))dt=2πc(e)α+βB(α+12,β+12)×F33[1,α+12,β+12;c2e2432,α+β+12,α+β+22;](e>0;Re(α),Re(β)>-12),
where erf⁡(z)=(2/π)Erf(z) denotes the error function [4], we find the following integral:
(30)Id,α,β,1,0,e[exp⁡(c2t(e-t))erf⁡(ct(e-t)),1t(e-t)]=2πc(e)α+βB(α+(1/2),β+(1/2))B(α,β)×∑m=0∞‍(1)m(α+(1/2))m(β+(1/2))m(c2e2/4)m(3/2)m((α+β+1)/2)m((α+β+2)/2)mm!×Fr+2t+2[(fr),12(1-α-β)-m,-12(α+β)-m;4de2(gt),12-α-m,12-β-m;](e>0;Re(α),Re(β)>-12).
Now, we establish a theorem and apply it to obtain the evaluations of certain integrals in terms of extended beta function.


## 3. Integrals in Terms of Extended Beta Function

Consider an(r+1)-variable generating function G(x1,x2,…,xr;t) which possesses a formal (not necessarily convergent for t≠0) power series expansion in t such that [4, page 80 (8)]
(31)G(x1,x2,…,xr;t)=∑n=0∞‍cngn(x1,x2,…,xr)tn,
where each member of the generated set {gn(x1,x2,…,xr)}n=0∞ is independent of t and the coefficient set {cn}n=0∞ may contain the parameters of the set {gn(x1,x2,…,xr)}n=0∞ but is independent of x1,x2,…,xr and t.Theorem 1.
Let the generating functionG(x1,x2,…,xr;t) defined by (31) be such that G(x1,x2,…,xr;t(u-a)ρ(c-u)σ) remains uniformly convergent for u∈(a,c), ρ,σ≥0, and ρ+σ>0. Then
(32)∫ac‍(u-a)λ-1(c-u)μ-λ-1×G(x1,x2,…,xr;t(u-a)ρ(c-u)σ)×exp⁡(-b(c-a)2(u-a)(c-u))du=∑n=0∞‍(c-a)μ+(ρ+σ)n-1cngn(x1,x2,…,xr)×B(λ+ρn,μ-λ+σn;b)tn(Re(μ)>Re(λ)>0;Re(b)>0),
where B(x,y;b) is the extended beta function defined by (7).Proof.
Applying the definition ofG(x1,x2,…,xr;t) given in (31) in the l.h.s. of (32), we get
(33)∑n=0∞‍cngn(x1,x2,…,xr)tn×∫ac‍(u-a)λ+ρn-1(c-u)μ-λ+σn-1×exp⁡(-b(c-a)2(u-a)(c-u))du,
which by using (9) yields the r.h.s. of (32).Corollary 2.
With definition (31) and notations as in Theorem 1, one has
(34)∫01‍uλ-1(1-u)μ-λ-1G(x1,x2,…,xr;tuρ(1-u)σ)×exp⁡(-bu(1-u))du=∑n=0∞‍cngn(x1,x2,…,xr)B(λ+ρn,μ-λ+σn;b)tn(Re(μ)>Re(λ)>0;Re(b)>0).Proof.
Takinga=0 and c=1 in (32), we get (34).Corollary 3.
With definition (31) and notations as in Theorem 1, one has
(35)∫-11‍(1+u)λ-1(1-u)μ-λ-1×G(x1,x2,…,xr;t(1+u)ρ(1-u)σ)×exp⁡(-4b(1-u2))du=∑n=0∞‍(2)μ+(ρ+σ)n-1cngn(x1,x2,…,xr)×B(λ+ρn,μ-λ+σn;b)tn(Re(μ)>Re(λ)>0;Re(b)>0).Proof.
Takinga=-1 and c=1 in (32), we get (35).Corollary 4.
With definition (31) and notations as in Theorem 1, one has
(36)∫ac‍(u-a)λ-1(c-u)-λ-1G(x1,x2,…,xr;t(c-uu-a)σ)×exp⁡⁡(-b(c-a)2(u-a)(c-u))du=2(c-a)e2b∑n=0∞‍cngn(x1,x2,…,xr)Kλ-σn(2b)tn(Re(λ)>0;Re(b)>0).Proof.
Takingμ=0 and ρ=-σ in (32) and making use of relation (10), we get (36).
Now, we apply Theorem1 to derive the evaluations of certain Euler type integrals in terms of EBF B(α,β;b). We consider the following cases.
Case
1. Consider the generating function of the multivariable Hermite polynomials Hn(1,2,…,r)(x1,x2,…,xr) [8, page 602 (20,21)]
(37)exp⁡(x1t+x2t2+⋯+xrtr)=∑n=0∞‍Hn(1,2,…,r)(x1,x2,…,xr)tnn!.
Applying Theorem1 to generating function (37), we find the following integral:
(38)∫ac‍(u-a)λ-1(c-u)μ-λ-1×exp⁡(-b(c-a)2(u-a)(c-u)+∑j=1r‍xj(t(u-a)ρ(c-u)σ)jb(c-a)2(u-a)(c-u))du=∑n=0∞‍(c-a)μ+(ρ+σ)n-1Hn(1,2,…,r)(x1,x2,…,xr)×B(λ+ρn,μ-λ+σn;b)tnn!(Re(μ)>Re(λ)>0;Re(b)>0;ρ,σ≥0;ρ+σ>0).
Takingr=3 in (38), we get
(39)∫ac‍(u-a)λ-1(c-u)μ-λ-1×exp⁡(-b(c-a)2(u-a)(c-u)+∑j=13‍xj(t(u-a)ρ(c-u)σ)jb(c-a)2(u-a)(c-u))du=∑n=0∞‍(c-a)μ+(ρ+σ)n-1Hn(x1,x2,x3)×B(λ+ρn,μ-λ+σn;b)tnn!(Re(μ)>Re(λ)>0;Re(b)>0;ρ,σ≥0;ρ+σ>0),
where Hn(x1,x2,x3) are the 3-variable Hermite polynomials (3VHP) [9].
Again, takingr=2 in (38), we get
(40)∫ac‍(u-a)λ-1(c-u)μ-λ-1×exp⁡(-b(c-a)2(u-a)(c-u)+∑j=12‍xj(t(u-a)ρ(c-u)σ)jb(c-a)2(u-a)(c-u))du=∑n=0∞‍(c-a)μ+(ρ+σ)n-1Hn(x1,x2)×B(λ+ρn,μ-λ+σn;b)tnn!(Re(μ)>Re(λ)>0;Re(b)>0;ρ,σ≥0;ρ+σ>0),
where Hn(x1,x2) are the 2-variable Hermite-Kampé de-Fériet polynomials (2VHKdFP) [10].
Next, replacingx1 by x, xr by y and taking x2=x3=⋯=xr-1=0 in (38) and using the relation Hn(1,2,…,r)(x,0,0,…,0,y)=gnr(x,y), we get
(41)∫ac‍(u-a)λ-1(c-u)μ-λ-1×exp⁡⁡(-b(c-a)2(u-a)(c-u)+xt(u-a)ρ(c-u)σ+y(t(u-a)ρ(c-u)σ)rb(c-a)2(u-a)(c-u))du=∑n=0∞‍(c-a)μ+(ρ+σ)n-1gnr(x,y)×B(λ+ρn,μ-λ+σn;b)tnn!(Re(μ)>Re(λ)>0;Re(b)>0;ρ,σ≥0;ρ+σ>0),
where gnr(x,y) are the Gould-Hopper generalized Hermite polynomials (GHGHP) [11].
Further, replacingx1 by νx and taking x2=x3=⋯=xr-1=0, xr=-1 in (38) and using the relation Hn(1,2,…,r)(νx,0,0,…,0,-1)=Hn,r,ν(x), we get
(42)∫ac‍(u-a)λ-1(c-u)μ-λ-1×exp⁡(-b(c-a)2(u-a)(c-u)+νxt(u-a)ρ(c-u)σ-(t(u-a)ρ(c-u)σ)rb(c-a)2(u-a)(c-u))du=∑n=0∞‍(c-a)μ+(ρ+σ)n-1Hn,r,ν(x)×B(λ+ρn,μ-λ+σn;b)tnn!(Re(μ)>Re(λ)>0;Re(b)>0;ρ,σ≥0;ρ+σ>0),
where Hn,r,ν(x) are the generalized Hermite polynomials [12].
Furthermore, takingr=ν=2 in (42) and using the relation Hn,2,2(x)=Hn(x), we get
(43)∫ac‍(u-a)λ-1(c-u)μ-λ-1×exp⁡⁡(-b(c-a)2(u-a)(c-u)+2xt(u-a)ρ(c-u)σ-(t(u-a)ρ(c-u)σ)2b(c-a)2(u-a)(c-u))du=∑n=0∞‍(c-a)μ+(ρ+σ)n-1Hn(x)×B(λ+ρn,μ-λ+σn;b)tnn!(Re(μ)>Re(λ)>0;Re(b)>0;ρ,σ≥0;ρ+σ>0),
where Hn(x) are the ordinary Hermite polynomials [1].
Case
2. Consider the generating function of the 2-variable Laguerre polynomials Ln(x,y) [13]
(44)1(1-yt)exp⁡(-xt1-yt)=∑n=0∞‍Ln(x,y)tn(|yt|<1),
which can also be expressed as
(45)exp⁡(yt)C0(xt)=∑n=0∞‍Ln(x,y)tnn!,
where C0(x) denotes the 0th-order Tricomi function [4].
Applying Theorem1 to generating functions (44) and (45), we find the following integrals:
(46)∫ac‍(u-a)λ-1(c-u)μ-λ-11-yt(u-a)ρ(c-u)σ×exp⁡(-b(c-a)2(u-a)(c-u)-xt(u-a)ρ(c-u)σ1-yt(u-a)ρ(c-u)σ)du=∑n=0∞‍(c-a)μ+(ρ+σ)n-1Ln(x,y)×B(λ+ρn,μ-λ+σn;b)tn(Re(μ)>Re(λ)>0;Re(b)>0;|yt(u-a)ρ(c-u)σ|<1;ρ,σ≥0;ρ+σ>0),(47)∫ac‍(u-a)λ-1(c-u)μ-λ-1×exp⁡(-b(c-a)2(u-a)(c-u)+yt(u-a)ρ(c-u)σ)×C0(xt(u-a)ρ(c-u)σ)du=∑n=0∞‍(c-a)μ+(ρ+σ)n-1Ln(x,y)×B(λ+ρn,μ-λ+σn;b)tnn!(Re(μ)>Re(λ)>0;Re(b)>0;ρ,σ≥0;ρ+σ>0),
respectively.
Takingy=1 in (46) and (47) and using the relation Ln(x,1)=Ln(x), we get
(48)∫ac‍(u-a)λ-1(c-u)μ-λ-11-t(u-a)ρ(c-u)σ×exp⁡⁡(-b(c-a)2(u-a)(c-u)-xt(u-a)ρ(c-u)σ1-t(u-a)ρ(c-u)σ)du=∑n=0∞‍(c-a)μ+(ρ+σ)n-1Ln(x)×B(λ+ρn,μ-λ+σn;b)tn(Re(μ)>Re(λ)>0;Re(b)>0;|t(u-a)ρ(c-u)σ|<1;ρ,σ≥0;ρ+σ>0),(49)∫ac‍(u-a)λ-1(c-u)μ-λ-1×exp⁡⁡(-b(c-a)2(u-a)(c-u)+t(u-a)ρ(c-u)σ)×C0(xt(u-a)ρ(c-u)σ)du=∑n=0∞‍(c-a)μ+(ρ+σ)n-1Ln(x)×B(λ+ρn,μ-λ+σn;b)tnn!(Re(μ)>Re(λ)>0;Re(b)>0;ρ,σ≥0;ρ+σ>0),
respectively, where Ln(x) are the ordinary Laguerre polynomials [1].
Again, takingy=0 in (46) and (47) and using the relation Ln(x,0)=(-x)n/n!, we get
(50)∫ac‍(u-a)λ-1(c-u)μ-λ-1×exp⁡⁡(-b(c-a)2(u-a)(c-u)-xt(u-a)ρ(c-u)σ)du=∑n=0∞‍(c-a)μ+(ρ+σ)n-1×B(λ+ρn,μ-λ+σn;b)(-xt)nn!(Re(μ)>Re(λ)>0;Re(b)>0;ρ,σ≥0;ρ+σ>0),(51)∫ac‍(u-a)λ-1(c-u)μ-λ-1×exp⁡⁡(-b(c-a)2(u-a)(c-u))C0(xt(u-a)ρ(c-u)σ)du=∑n=0∞‍(c-a)μ+(ρ+σ)n-1×B(λ+ρn,μ-λ+σn;b)(-xt)n(n!)2(Re(μ)>Re(λ)>0;Re(b)>0;ρ,σ≥0;ρ+σ>0),
respectively.
Next, takingx=0 in (46) and (47) and using the relation Ln(0,y)=yn, we get
(52)∫ac‍(u-a)λ-1(c-u)μ-λ-11-yt(u-a)ρ(c-u)σexp⁡(-b(c-a)2(u-a)(c-u))du=∑n=0∞‍(c-a)μ+(ρ+σ)n-1×B(λ+ρn,μ-λ+σn;b)(yt)n(Re(μ)>Re(λ)>0;Re(b)>0;ρ,σ≥0;ρ+σ>0),(53)∫ac‍(u-a)λ-1(c-u)μ-λ-1×exp⁡⁡(-b(c-a)2(u-a)(c-u)+yt(u-a)ρ(c-u)σ)=∑n=0∞‍(c-a)μ+(ρ+σ)n-1×B(λ+ρn,μ-λ+σn;b)(yt)nn!(Re(μ)>Re(λ)>0;Re(b)>0;ρ,σ≥0;ρ+σ>0),
respectively. Replacing y by -x in (53), it reduces to (50).
Case
3. Consider the generating function of the Hermite-Appell polynomials AHn(x1,x2,x3) [14, page 759 (2.3)]
(54)A(t)exp⁡(x1t+x2t2+x3t3)=∑n=0∞‍AnH(x1,x2,x3)tnn!.
Applying Theorem1 to generating function (54), we find the following integral:
(55)∫ac‍(u-a)λ-1(c-u)μ-λ-1A(t(u-a)ρ(c-u)σ)×exp⁡⁡(-b(c-a)2(u-a)(c-u)+∑j=13‍xj(t(u-a)ρ(c-u)σ)j)du=∑n=0∞‍(c-a)μ+(ρ+σ)n-1HAn(x1,x2,x3)×B(λ+ρn,μ-λ+σn;b)tnn!(Re(μ)>Re(λ)>0;Re(b)>0;ρ,σ≥0;ρ+σ>0).
Takingx2=x3=0 and replacing x1 by x in (55) and using the relation AHn(x,0,0)=An(x) [14, page 760 (2.6)], we get
(56)∫ac‍(u-a)λ-1(c-u)μ-λ-1A(t(u-a)ρ(c-u)σ)×exp⁡⁡(-b(c-a)2(u-a)(c-u)+xt(u-a)ρ(c-u)σ)du=∑n=0∞‍(c-a)μ+(ρ+σ)n-1An(x)×B(λ+ρn,μ-λ+σn;b)tnn!(Re(μ)>Re(λ)>0;Re(b)>0;ρ,σ≥0;ρ+σ>0),
where An(x) are the Appell polynomials [4, 15].
Case
4. Consider the generating function of the Laguerre-Appell polynomials ALn(x,y) [16]
(57)A(t)[1(1-yt)exp⁡(-xt1-yt)]=∑n=0∞‍AnL(x,y)tn(|yt|<1),
which can also be expressed as follows:
(58)A(t)exp⁡(yt)C0(xt)=∑n=0∞‍AnL(x,y)tnn!.
Applying Theorem1 to generating functions (57) and (58), we find the following integrals:
(59)∫ac‍(u-a)λ-1(c-u)μ-λ-1A(t(u-a)ρ(c-u)σ)1-yt(u-a)ρ(c-u)σ×exp⁡⁡(-b(c-a)2(u-a)(c-u)-xt(u-a)ρ(c-u)σ1-yt(u-a)ρ(c-u)σ)du=∑n=0∞‍(c-a)μ+(ρ+σ)n-1LAn(x,y)×B(λ+ρn,μ-λ+σn;b)tn(Re(μ)>Re(λ)>0;Re(b)>0;|yt(u-a)ρ(c-u)σ|<1;ρ,σ≥0;ρ+σ>0),(60)∫ac‍(u-a)λ-1(c-u)μ-λ-1A(t(u-a)ρ(c-u)σ)×exp⁡⁡(-b(c-a)2(u-a)(c-u)+yt(u-a)ρ(c-u)σ)×C0(xt(u-a)ρ(c-u)σ)du=∑n=0∞‍(c-a)μ+(ρ+σ)n-1LAn(x,y)×B(λ+ρn,μ-λ+σn;b)tnn!(Re(μ)>Re(λ)>0;Re(b)>0;ρ,σ≥0;ρ+σ>0),
respectively.
Takingy=0 and replacing x by -x in (59) and (60) and using the relation ALn(x,0)=An(-x)/n! [16, page 9 (2.8)], we get (56) and
(61)∫ac‍(u-a)λ-1(c-u)μ-λ-1A(t(u-a)ρ(c-u)σ)×exp⁡⁡(-b(c-a)2(u-a)(c-u))C0(-xt(u-a)ρ(c-u)σ)du=∑n=0∞‍(c-a)μ+(ρ+σ)n-1An(x)×B(λ+ρn,μ-λ+σn;b)tn(n!)2(Re(μ)>Re(λ)>0;Re(b)>0;ρ,σ≥0;ρ+σ>0),
respectively.
Case
5. Consider the generating function of the Hermite-Sheffer polynomials sHn(x1,x2,x3) [17]
(62)A(t)exp⁡(x1H(t)+x2H2(t)+x3H3(t))=∑n=0∞‍snH(x1,x2,x3)tnn!.
Applying Theorem1 to generating function (62), we find the following integral:
(63)∫ac‍(u-a)λ-1(c-u)μ-λ-1A(t(u-a)ρ(c-u)σ)×exp⁡⁡(-b(c-a)2(u-a)(c-u)+∑j=13‍xjHj(t(u-a)ρ(c-u)σ))du=∑n=0∞‍(c-a)μ+(ρ+σ)n-1Hsn(x1,x2,x3)×B(λ+ρn,μ-λ+σn;b)tnn!(Re(μ)>Re(λ)>0;Re(b)>0;ρ,σ≥0;ρ+σ>0).
Takingx2=x3=0 and replacing x1 by x in (63) and using the relation sHn(x,0,0)=sn(x) [17, page 16 (2.3)], we get
(64)∫ac‍(u-a)λ-1(c-u)μ-λ-1A(t(u-a)ρ(c-u)σ)×exp⁡⁡(-b(c-a)2(u-a)(c-u)+xH(t(u-a)ρ(c-u)σ))du=∑n=0∞‍(c-a)μ+(ρ+σ)n-1sn(x)×B(λ+ρn,μ-λ+σn;b)tnn!(Re(μ)>Re(λ)>0;Re(b)>0;ρ,σ≥0;ρ+σ>0),
where sn(x) are the Sheffer polynomials [15, 18, 19].


## 4. Concluding Remarks

In this paper, we have derived the evaluations of certain Euler type integrals. We have also established a theorem and applied it to obtain evaluations of some integrals in terms of EBFB(α,β;b). We remark that these results can be extended to multivariable case.To give an example, we consider the following integral representation [5, page 965 (20)] of Lauricella’s multiple hypergeometric series FD(n) [4]:
(65)∫01‍tα-1(1-t)β-1∏i=1n‍(1-xit)-αidt=B(α,β)FD(n)[α,α1,α2,…,αn;α+β;x1,x2,…,xn](Re(α),Re(β)>0;max⁡{|x1|,|x2|,…,|xn|}<1).Now, takingϕ(t)=∏i=1n(1-xit)-αi,ψ(t)=1/t(1-t), a=0, and e=γ=1 in (15), using integral (65) in the r.h.s. after expanding Frt[(fr);dψ(t)(gt);] in series and simplifying, we get(66)Id,α,β,1,0,1[∏i=1n‍(1-xit)-αi,1t(1-t)]=∑m1,m2,…,mn=0∞‍(α)m1+m2+⋯+mn(α1)m1(α2)m2⋯(αn)mn(α+β)m1+m2+⋯+mnx1m1m1!x2m2m2!⋯xnmnmn!×Fr+2t+2[(fr),12(1-α-β-∑i=1n‍mi),1-12(α+β+∑i=1n‍mi);4d(gt),1-(α+∑i=1n‍mi),1-β;](Re(α),Re(β)>0;max⁡{|x1|,|x2|,…,|xn|}<1).We note that, forα3=α4=⋯=0, (66) reduces to (17).Further, we extend Theorem1 as follows.Theorem 5.
Let the conditions forG(x1,x2,…,xr;t) defined by (31) be the same as in Theorem 1, and then one has
(67)∫ac‍(u-a)λ-1(c-u)μ-λ-1×∏i=1s‍(1-yi(u-a))-αi×G(x1,x2,…,xr;t(u-a)ρ(c-u)σ)×exp⁡⁡(-b(c-a)2(u-a)(c-u))du=∑n,m1,m2,…,ms=0∞‍(c-a)μ+(ρ+σ)n+m1+m2+⋯+ms-1×cngn(x1,x2,…,xr)×B(λ+ρn+∑i=1s‍mi,μ-λ+σn;b)∏i=1s‍(αi)miyimimi!tn(Re(μ)>Re(λ)>0;Re(b)>0;ρ,σ≥0;ρ+σ>0).
The proof of this theorem is similar to that of Theorem1.
The following are the consequences of Theorem5.Corollary 6.
Let the conditions forG(x1,x2,…,xr;t) defined by (31) be the same as in Theorem 1, and then for b=0 and s=2, one has
(68)∫ac‍(u-a)λ-1(c-u)μ-λ-1×∏i=12‍(1-yi(u-a))-αi×G(x1,x2,…,xr;t(u-a)ρ(c-u)σ)du=∑n=0∞‍(c-a)μ+(ρ+σ)n-1cngn(x1,x2,…,xr)×B(λ+ρn,μ-λ+σn)×F1[λ+ρn,α1,α2;μ+(ρ+σ)n;(c-a)y1,(c-a)y2]tn(Re(μ)>Re(λ)>0;ρ,σ≥0;ρ+σ>0).Corollary 7.
Let the conditions forG(x1,x2,…,xr;t) defined by (31) be the same as in Theorem 1, and then for b=0 and ρ=σ=1, one has
(69)∫ac‍(u-a)λ-1(c-u)μ-λ-1×∏i=1s‍(1-yi(u-a))-αi×G(x1,x2,…,xr;t(u-a)(c-u))du=B(λ,μ-λ)∑n=0∞‍(c-a)μ+2n-1cngn(x1,x2,…,xr)×(λ)n(μ-λ)n(μ)2n×FD(s)[λ+n,α1,α2,…,αs;μ+2n;(c-a)y1,(c-a)y2,…,(c-a)ys]tn(Re(μ)>Re(λ)>0).Example 8.
Applying Corollary7 to generating function (37) of the multivariable Hermite polynomials Hn(1,2,…,r)(x1,x2,…,xr), we get
(70)∫ac‍(u-a)λ-1(c-u)μ-λ-1×∏i=1s‍(1-yi(u-a))-αi×exp⁡⁡(∑j=1r‍xj(t(u-a)(c-u))j)du=B(λ,μ-λ)∑n=0∞‍(c-a)μ+2n-1Hn(1,2,…,r)(x1,x2,…,xr)×(λ)n(μ-λ)n(μ)2n×FD(s)[λ+n,α1,α2,…,αs;μ+2n;(c-a)y1,(c-a)y2,…,(c-a)ys]tnn!(Re(μ)>Re(λ)>0).Remark 9.
Takinga=0 and c=1 in (67), we get an extension of integral (34).Remark 10.
Takinga=-1 and c=1 in (67), we get an extension of integral (35).Clearly, it appears that, by applying Theorem5 and Corollaries 6 and 7 to the generating functions defined by (37), (44), (45), (54), (57), (58), and (62) and also to some other generating functions, a number of interesting integrals can be obtained.Further, it is remarked that the operational methods provide useful tools to estimate specific theorems in various fields of analysis. This approach requires the validation of any of its consequences by a more rigorous procedure. This is indeed the case of the Ramanujan master theorem [20, 21]. An operational method, already employed to formulate a generalization of the Ramanujan master theorem, is applied for the evaluation of certain integrals by Babusci et al. [22].The technique adopted in [22] provides a very flexible and powerful tool yielding new results encompassing different aspects of the theory of special functions. The possibility of using the method outlined in [22], for the integrals evaluated in this article, is a further research problem.



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