# An Expansion Theorem InvolvingH-Function of Several Complex Variables

Authors: Sébastien Gaboury; Richard Tremblay
Journal: International Journal of Analysis
 (2013)

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## Abstract

The aim of this present paper is to obtain a general expansion
theorem involvingH-functions of several complex variables. This is done by
making use of a Taylor-like expansion in terms of a rational function obtained
by means of fractional derivatives given recently by the authors. Special cases
are also computed.


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## Body

## 1. Introduction

In 1971, Osler obtained with the use of Cauchy integral formula for the fractional derivatives the following generalization for the Taylor's series [1]:
(1)∑k∈K‍a-1ω-γkf(θ-1(θ(z)ωk))=∑n=-∞∞‍Dz-ban+γ[f(z)θ′(z)[(z-z0)/θ(z)]an+γ+1]|z=z0θ(z)an+γΓ(an+γ+1),
where a is a positive real number, b≠z0,  ω=exp(2πi/a), α and γ are arbitrary complex numbers, f(z) is an analytic function in a simply connected region ℛ and θ(z)=(z-z0)q(z) with q(z) is a regular and univalent function without zero in ℛ, and K=0,1,…,[a], [a] being the largest integer not greater then a. If 0<a≤1 and θ(z)=(z-z0), then K=0 and the formula (1) reduces to
(2)f(z)=∑n=-∞∞‍[Dz0-ban+γf(z0)](z-z0)an+γ+1Γ(an+γ+1).
This last formula is usually called the Taylor-Riemann formula and has been studied in several papers [2–6]. But none considered a more general expansion of f(z) in terms of a power series of an arbitrary quadratic, cubic, or higher degrees functions. Recently, the authors [7] obtained the power series of an analytic function f(z) in terms of the rational expression ((z-z1)/(z-z2)) where z1 and z2 are two arbitrary points inside the region of analyticity ℛ of f(z). In particular, we obtain the following expansion:
(3)a-1f(z)(z-z1)ν(z-z2)μ(z1-z2)=∑n=-∞∞‍eiπa(n+1)sin((μ+an+γ)π)sin((μ-a+γ)π)Γ(1-ν+an+γ)×Dz-z2-ν+an+γ(z-z2)μ+an+γ-1f(z)|z=z1(z-z1z-z2)an+γ.
Several restrictions are imposed on the functions and parameters in (3). The following list is considered.  (i)
ν, μ, and γ are arbitrary complex numbers.(ii)
0<a≤1 is a real and n is the integral index of summation.(iii)
z1, z2 are fixed points in the z-plane and {z||θ(z)|=|θ((z1+z2)/2)|}, where θ(z)=(z-z1)(z-z2), defines a double-loop curve C=C1∪C2 on which the series (3) converges with z1≠z2.(iv)
z is on the loop C1 around the point z1 but z≠(z1+z2)/2 as shown in Figure 1.Figure 1
Multiloops contour.The aim of this paper is to obtain a new expansion theorem involving theH-function of r complex variables z1,z2,…,zr defined by Srivastava and Panda [8–11]. We will define and represent it in the following form [12, page 251, equation (C.1)]: (4)H[z1,…,zr]=HP,Q:P′,Q′;…;P(r),Q(r)0,N:M′,N′;…;M(r),N(r)×[z1(aj;αj′,…,αj(r))1,P:(cj′,γj′)1,P′;…;(cj(r),γj(r))1,P(r)⋮zr(bj;βj′,…,βj(r))1,Q:(dj′,δj′)1,Q′;…;(dj(r),δj(r))1,Q(r)]=1(2πi)r∫L1‍⋯∫Lr‍ϕ1(ξ1)⋯ϕr(ξr)ψ(ξ1,…,ξr)×z1ξ1⋯zrξrdξ1⋯dξr,where i=-1,
(5)ϕk(ξk)=∏j=1M(k)‍Γ(dj(k)-δj(k)ξk)∏j=1N(k)‍Γ(1-cj(k)+γj(k)ξk)∏j=M(k)+1Q(k)‍Γ(1-dj(k)+δj(k)ξk)∏j=N(k)+1P(k)‍Γ(cj(k)-γj(k)ξk),
for all k∈{1,…,r} and
(6)ψ(ξ1,…,ξr)=∏j=1N‍Γ(1-aj+∑k=1r‍αj(k)ξk)∏j=N+1P‍Γ(aj-∑k=1r‍αj(k)ξk)∏j=1Q‍Γ(1-bj+∑k=1r‍βj(k)ξk).
Here, for convenience, (aj;αj′,…,αj(r))1,P abbreviates the p-member array
(7)((a1;α1′,…,α1(r)),…,(ap;αp′,…,αp(r))),
while (cj(k),γj(k))1,P(k) abbreviates the array of P(k) pairs of parameters:
(8)(c1(k),γ1(k)),…,(cP(k)(k),γP(k)(k))(k=1,…,r),
and so on. Suppose, as usual, that the parameters:
(9)aj,j=1,…,P;cj(k),j=1,…,P(k)bj,j=1,…,Q;dj(k),j=1,…,Q(k)(∀k∈{1,…,r})
are complex numbers and the associated coefficients
(10)αj(k),j=1,…,P;γj(k),j=1,…,P(k)βj(k),j=1,…,Q;δj(k),j=1,…,Q(k)(∀k∈{1,…,r})
are positive real numbers such that
(11)Λk∶=∑j=1P‍αj(k)-∑j=1Q‍βj(k)+∑j=1P(k)‍γj(k)-∑j=1Q(k)‍δj(k)≤0,(12)Ωk∶=-∑j=N+1P‍αj(k)-∑j=1Q‍βj(k)+∑j=1N(k)‍γj(k)-∑j=N(k)+1P(k)‍γj(k)+∑j=1M(k)‍δj(k)-∑j=M(k)+1Q(k)‍δj(k)>0,(∀k∈{1,…,r}),
where the integers N,P,Q,M(k),N(k),P(k), and Q(k) are constrained by the inequalities 0≤N≤P,Q≥0, 1≤M(k)≤Q(k), and 0≤N(k)≤P(k)(forallk∈{1,…,r}) and the equality in (11) holds true for suitably restricted values of the complex variables z1,…,zr.The multiple Mellin-Barnes contour integral [12, page 251, equation (C.1)] representing the multivariable H-function (4) converges absolutely, under the conditions (12), when
(13)|arg(zk)|<12Ωkπ,(∀k∈{1,…,r}),
the points zk=0(k=1,…,r) and various exceptional parameter values being tacitly excluded.  Furthermore, we have (cf. [9, page 131, equation (1.9)]):
(14)H[z1,…,zr]={O(|z1|ξ1…|zr|ξr),(max{|z1|,…,|zr|}→0),O(|z1|η1…|zr|ηr),(N=0;min{|z1|,…,|zr|}→∞),
where (k=1,…,r)(15)ξk=min{Re(dj(k))δj(k)},(j=1,…,M(k)),(16)ηk=max{Re(cj(k)-1)γj(k)},(j=1,…,N(k)),
provided that each of the inequalities in (11)–(13) holds true.Note that throughout this work, we will assume that the convergence and existence conditions corresponding appropriately to the ones detailed above are satisfied by each of the variousH-functions involved.


## 2. Pochhammer Contour Integral Representation for Fractional Derivative

The use of contour of integration in the complex plane provides a very powerful tool in both classical and fractional calculus. The most familiar representation for fractional derivative of orderα of zpf(z) is the Riemann-Liouville integral [13–15], that is,
(17)Dzαzpf(z)=1Γ(-α)∫0z‍f(ξ)ξp(ξ-z)-α-1dξ,
which is valid for Re(α)<0, Re(p)>1 and where the integration is done along a straight line from 0 to z in the ξ-plane. By integrating by part m times, we obtain
(18)Dzαzpf(z)=dmdzmDzα-mzpf(z).
This allows to modify the restriction Re(α)<0 to Re(α)<m [15]. Another used representation for the fractional derivative is the one based on the Cauchy integral formula widely used by Osler [4, 16–18]. These two representations have been used in many interesting research papers. It appears that the less restrictive representation of fractional derivative according to parameters is the Pochhammer's contour definition introduced in [19, 20].Definition 1.
Letf(z) be analytic in a simply connected region ℛ. Let g(z) be regular and univalent on ℛ and let g-1(0) be an interior point of ℛ then if α is not a negative integer, p is not an integer, and z is in ℛ-{g-1(0)}, we define the fractional derivative of order α of g(z)pf(z) with respect to g(z) by
(19)Dg(z)αg(z)pf(z)=e-iπpΓ(1+α)4πsin(πp)×∫C(z+,g-1(0)+,z-,g-1(0)-;F(a),F(a))‍f(ξ)g(ξ)pg′(ξ)(g(ξ)-g(z))α+1dξ.
For noninteger α and p, the functions g(ξ)p and (g(ξ)-g(z))-α-1 in the integrand have two branch lines which begin, respectively, at ξ=z and ξ=g-1(0), and both pass through the point ξ=a without crossing the Pochhammer contour P(a)={C1∪C2∪C3∪C4} at any other point as shown in Figure 2. F(a) denotes the principal value of the integrand in (19) at the beginning and ending point of the Pochhammer contour P(a) which is closed on Riemann surface of the multiple-valued function F(ξ).Figure 2
Pochhammer's contour.Remark 2.
In Definition1, the function f(z) must be analytic at ξ=g-1(0). However it is interesting to note here that we could also allow f(z) to have an essential singularity at ξ=g-1(0), and (19) would still be valid.Remark 3.
The Pochhammer contour never crosses the singularities atξ=g-1(0) and ξ=z in (19), then we know that the integral is analytic for all p and for all α and for z in ℛ-{g-1(0)}. Indeed, the only possible singularities of Dg(z)αg(z)pf(z) are α=-1,-2,… and p=0,±1,±2,… which can directly be identified from the coefficient of the integral (19). However, integrating by parts N times the integral in  (19) by two different ways, we can show that α=-1,-2,…, and p=0,1,2,… are removable singularities (see [19]).In 1985, Srivastava and Goyal [21, page 644, equation (17)] obtained the following fractional derivative formula, by using the well-known Riemann-Liouville's definition of the fractional derivative, for the multivariable H-function (4): (20)Dxα{xβ(x+ξ)λH[z1xρ1(x+ξ)σ1,…,zrxρr(x+ξ)σr]}=ξλxβ-α∑m=0∞‍(x/ξ)mm!×HP+2,Q+2:P′,Q′;…;P(r),Q(r)0,N+2:M′,N′;…;M(r),N(r)×[z1xρ1ξσ1(-λ;σ1,…,σr),(-β-m;ρ1,…,ρr),⋮zrxρrξσr(m-λ;σ1,…,σr),(α-β-m;ρ1,…,ρr),(aj;αj′,…,αj(r))1,P:(cj′,γj′)1,P′;…;(cj(r),γj(r))1,P(r)(bj;βj′,…,βj(r))1,Q:(dj′,δj′)1,Q′;…;(dj(r),δj(r))1,Q(r)]provided (in addition to the usual convergence and existence conditions) that min{ρi,σi}>0,i=1,…,r,|arg(x/ξ)|<π, Re(μ)<0 and
(21)Re(β)+∑i=1r‍ρiξi>-1,
where ξ1,…,ξr are given by (15).Remark 4.
Adopting the representation based on the Pochhammer contour for fractional derivative and in view of Remark3, the restrictions Re(β)+∑i=1r‍ρiξi>-1 can be modified to Re(β)+∑i=1r‍ρiξi not a negative integer and Re(μ)<0 to μ not a negative integer.Lettingσi↓0,foralli∈{1,…,r} and λ=0 in (20) yields the following useful particular case:
(22)Dxα{xβH[z1xρ1,…,zrxρr]}=xβ-αHP+1,Q+1:P′,Q′;…;P(r),Q(r)0,N+1:M′,N′;…;M(r),N(r)×[z1xρ1(-β;ρ1,…,ρr),(aj;αj′,…,αj(r))1,P:⋮zrxρr(α-β;ρ1,…,ρr),(bj;βj′,…,βj(r))1,Q:(cj′,γj′)1,P′;…;(cj(r),γj(r))1,P(r)(dj′,δj′)1,Q′;…;(dj(r),δj(r))1,Q(r)].


## 3. Expansion Theorem and Special Cases

In this section, we establish the new expansion theorem for theH-functions of several complex variables. Two special cases (presumably new) are also computed to demonstrate the importance of this new expansion theorem to the theory of special functions of mathematical physics.Theorem 5.
Assume that the convergence and existence conditions corresponding appropriately to the ones detailed in Section1 are satisfied for each H-functions involved. Assume also that the list of restrictions of (3) is satisfied, then the following expansion(23)HP,Q:P′,Q′;…;P(r),Q(r)0,N:M′,N′;…;M(r),N(r)×[y1xρ1(aj;αj′,…,αj(r))1,P:⋮yrxρr(bj;βj′,…,βj(r))1,Q:(cj′,γj′)1,P′;…;(cj(r),γj(r))1,P(r)(dj′,δj′)1,Q′;…;(dj(r),δj(r))1,Q(r)]=(az1)xμ(x-z1)ν×∑n=-∞∞‍eiπa(n+1)sin((μ+an+γ)π)sin((μ-a+γ)π)Γ(1-ν+an+γ)×(x-z1x)an+γ×z1μ+ν-1HP+1,Q+1:P′,Q′;…;P(r),Q(r)0,N+1:M′,N′;…;M(r),N(r)×[y1z1ρ1(-μ-an-γ+1;ρ1,…,ρr),⋮yrz1ρr(1-μ-ν;ρ1,…,ρr),(aj;αj′,…,αj(r))1,P:(cj′,γj′)1,P′;…;(cj(r),γj(r))1,P(r)(bj;βj′,…,βj(r))1,Q:(dj′,δj′)1,Q′;…;(dj(r),δj(r))1,Q(r)]holds true.Proof.
By settingf(x)=H[y1xρ1,…,yrxρr], z2=0 in (3) and with the help of (22), we find first that(24)Dx-ν+an+γxμ+an+γ-1H[y1xρ1,…,yrxρr]|x=z1=z1μ+ν-1HP+1,Q+1:P′,Q′;…;P(r),Q(r)0,N+1:M′,N′;…;M(r),N(r)×[y1z1ρ1(-μ-an-γ+1;ρ1,…,ρr),⋮yrz1ρr(1-μ-ν;ρ1,…,ρr),(aj;αj′,…,αj(r))1,P:(cj′,γj′)1,P′;…;(cj(r),γj(r))1,P(r)(bj;βj′,…,βj(r))1,Q:(dj′,δj′)1,Q′;…;(dj(r),δj(r))1,Q(r)].Substituting the last result into (3) yields the desired result.Example 6.
ForN=P=Q=0 and ρ1=ρ2=⋯=ρr=1 the multivariable H-function in the L.H.S. of (23) breaks up into product of r   H-functions and there holds the following result [22]:
(25)H0,0:P′,Q′;…;P(r),Q(r)0,0:M′,N′;…;M(r),N(r)×[y1x-:(cj′,γj′)1,P′;…;(cj(r),γj(r))1,P(r)⋮yrx-:(dj′,δj′)1,Q′;…;(dj(r),δj(r))1,Q(r)]=∏i=1rHP(i),Q(i)M(i),N(i)[(cj(i),γj(i))1,P(i)yix(dj(i),δj(i))1,Q(i)].
Now setting P(i)=Q(i)=M(i)=N(i)=γj(i)=δj(i)=1, dj(i)=0, i=1,…,r, substituting cj(i) by 1-cj(i), i=1,…,r into (25), replacing x by -x, and making use of the following special case [23]
(26)H1,11,1[(1-ν,1)z(0,1)]=Γ(ν)(1+z)-ν,|z|<1,
gives
(27)H0,0:1,1;…;1,10,0:1,1;…;1,1×[-y1x-:(1-c1′,1);…;(1-c1(r),1)⋮-yrx-:(0,1);…;(0,1)]=∏i=1rΓ(c1(i))(1-yix)-c1(i).
Moreover, we also know from [21, page 649, equation (3.6)] that
(28)Dxλ-μxλ-1∏i=1r(1-xzi)-αi=Γ(λ)Γ(μ)xμ-1FD(r)[λ,α1,…,αr;μ;xz1,…,xzr],
where FD(r) denotes the Lauricella's hypergeometric function of r variables [24, page 33, equation (4)]. Thus, we have
(29)Dx-ν+an+γxμ+an+γ-1∏i=1r(1-yix)-c1(i)|x=z1=Γ(μ+an+γ)Γ(μ+ν)z1μ+ν-1FD(r)×[μ+an+γ,c1(1),…,c1(r);μ+ν;y1z1,…,yrz1]
with max{|y1|,…,|yr|}<1. Combining (27) and (29) with Theorem 5 yields the following new expansion involving the Lauricella's hypergeometric function of r variables(30)∏i=1r(1-yix)-c1(i)=axμ(x-z1)ν×∑n=-∞∞‍eiπa(n+1)sin((μ+an+γ)π)Γ(μ+an+γ)sin((μ-a+γ)π)Γ(1-ν+an+γ)×z1μ+νΓ(μ+ν)FD(r)[μ+an+γ,c1(1),…,c1(r);μ+ν;y1z1,…,yrz1c1(r)](x-z1x)an+γwith max{|y1z1|,…,|yrz1|}<1.Example 7.
Settingρ1=⋯=ρr=1 and αj(i)=βj(i)=γj(i)=δj(i)=1, i=1,…,r in (23) and with the help of the result given by Srivastava and Panda [10] connecting the generalized Lauricella function and the multivariable H-function: (31)HP,Q:P′,Q′+1;…;P(r),Q(r)+10,P:1,P′;…;1,P(r)×[xy1(aj;αj′,…,αj(r))1,P:⋮xyr(bj;βj′,…,βj(r))1,Q:(cj′,γj′)1,P′;⋯;(cj(r),γj(r))1,P(r)(0,1),(dj′,δj′)1,Q′;⋯;(0,1),(dj(r),δj(r))1,Q(r)]=[∏j=1P‍Γ(1-aj)][∏j=1P′‍Γ(1-cj′)]⋯[∏j=1P(r)‍Γ(1-cj(r))][∏j=1Q‍Γ(1-bj)][∏j=1Q′‍Γ(1-dj′)]⋯[∏j=1Q(r)‍Γ(1-dj(r))]×FQ:Q′;…;Q(r)P:P′;…;P(r)×[(1-aj;αj′,…,αj(r))1,P:(1-cj′,γj′)1,P′;…;(1-bj;βj′,…,βj(r))1,Q:(1-dj′,δj′)1,Q′;…;(1-cj(r),γj(r))1,P(r)-xy1,…,-xyr(1-dj(r),δj(r))1,Q(r)],we obtain the following expansion involving an extension of the Kampé de Fériet series [24, page 38, equation (24)] (32)FQ:Q′;…;Q(r)P:P′;…;P(r)[(1-aP):(1-cP′′);…;(1-cP(r)(r));xy1,…,xyr(1-bQ):(1-dQ′′);…;(1-dQ(r)(r));]=az1μ+νxμ(x-z1)ν∑n=-∞∞‍eiπa(n+1)sin((μ+an+γ)π)Γ(μ+an+γ)sin((μ-a+γ)π)Γ(1-ν+an+γ)Γ(μ+ν)×(x-z1x)an+γFQ+1:Q′;…;Q(r)P+1:P′;…;P(r)[μ+an+γ,(1-aP):(1-cP′′);…;μ+ν,(1-bQ):(1-dQ′′);…;(1-cP(r)(r));z1y1,…,z1yr(1-dQ(r)(r));].If  we  put r=1,  P=1, Q=0, and z1=1 in (32), we have(33)1F0[1-a1;xy1-;]=axμ(x-1)ν×∑n=-∞∞‍eiπa(n+1)sin((μ+an+γ)π)Γ(μ+an+γ)sin((μ-a+γ)π)Γ(1-ν+an+γ)Γ(μ+ν)×(x-1x)an+γ×F21[μ+an+γ,1-a1;y1μ+ν;].Making the following changes
(34)y1↦1-y12,a1↦1+γ
and using the hypergeometric representation for the Jacobi functions [25, page 35]
(35)Pμ(θ,γ)(z)=Γ(1+θ+μ)Γ(1+μ)Γ(1+θ)F21[μ+γ+θ+1,-μ;1-z21+θ;]
yields
(36)1F0[-γ;x(1-y1)2-;]=[1-(x(1-y1)2)]γ=axμ(x-1)ν×∑n=-∞∞‍eiπa(n+1)sin((μ+an+γ)π)Γ(μ+an+γ)sin((μ-a+γ)π)Γ(1-ν+an+γ)Γ(μ+ν)×(x-1x)an+γ×Γ(1+γ)Γ(μ+ν)Γ(μ+ν+γ)Pγ(μ+ν-1,an-ν)(y1).



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