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# 4 H-Functions

## 4.1 Definition

[120.1.1] The H-function of order m,n,p,qN4 and with parameters AiR+(i=1,,p), BiR+(i=1,,q), aiC(i=1,,p), and biC(i=1,,q) is defined for zC,z0 by the contour integral [55, 56, 57, 58, 59]

 Hp,qm,n(z|a1,A1,…,ap,Apb1,B1,…,bq,Bq)=12⁢π⁢i∫Lη(s)z-sds (153)

where the integrand is

 η⁢s=∏i=1mΓ⁢bi+Bi⁢s⁢∏i=1nΓ⁢1-ai-Ai⁢s∏i=n+1pΓ⁢ai+Ai⁢s⁢∏i=m+1qΓ⁢1-bi-Bi⁢s. (154)

[page 121, §0]    [121.0.1] In (153) z-s=exp-slogz-iargz and argz is not necessarily the principal value. [121.0.2] The integers m,n,p,q must satisfy

 0≤m≤q,0≤n≤p, (155)

and empty products are interpreted as being unity. [121.0.3] The parameters are restricted by the condition

 Pa∩Pb=∅ (156)

where

 Pa ={poles of Γ(1-ai-Ais)}={1-ai+kAi∈C:i=1,…,n;k∈N0} Pb ={poles of Γ(bi+Bis)}={-bi-kBi∈C:i=1,…,m;k∈N0} (157)

are the poles of the numerator in (154). [121.0.4] The integral converges if one of the following conditions holds [59]

 L =L(c-i∞,c+i∞;Pa,Pb);    |argz|0 (158a) L =L(c-i∞,c+i∞;Pa,Pb);    |argz|=Cπ/2;    C≥0;    cD<-ReF (158b)
 L =L(-∞+iγ1,-∞+iγ2;Pa,Pb);    D>0;    0<|z|<∞ (159a) L =L(-∞+iγ1,-∞+iγ2;Pa,Pb);    D=0;    0<|z|
 L =L(∞+iγ1,∞+iγ2;Pa,Pb);    D<0;    0<|z|<∞ (160a) L =L(∞+iγ1,∞+iγ2;Pa,Pb);    D=0;    |z|>E-1 (160b) L =L(∞+iγ1,∞+iγ2;Pa,Pb);    D=0;    |z|=E-1; C≥0; ReF<0 (160c)

where γ1<γ2. [121.0.5] Here Lz1,z2;G1,G2 denotes a contour in the complex plane starting at z1 and ending at z2 and separating the points in G1 from those

[page 122, §0]    in G2, and the notation

 C =∑i=1nAi-∑i=n+1pAi+∑i=1mBi-∑i=m+1qBi (161) D =∑i=1qBi-∑i=1pAi (162) E =∏i=1pAiAi⁢∏i=1qBi-Bi (163) F =∑i=1qbi-∑i=1paj+p-q/2+1 (164)

was employed. [122.0.1] The H-functions are analytic for z0 and multivalued (single valued on the Riemann surface of logz).

## 4.2 Basic Properties

[122.1.1] From the definition of the H-functions follow some basic properties. [122.1.2] Let Sn(n1) denote the symmetric group of n elements, and let πn denote a permutation in Sn. [122.1.3] Then the product structure of (154) implies that for all πnSn,πmSm,πp-nSp-n and πq-mSq-m

 Hp,qm,n(z|a1,A1,…,ap,Apb1,B1,…,bq,Bq)=Hp,qm,n(z|Pn,Pp-nPm,Pq-m) (165)

where the parameter permutations

 Pn =aπn⁢1,Aπn⁢1,…,aπn⁢n,Aπn⁢n Pp-n =aπp-n⁢n+1,Aπp-n⁢n+1,…,aπp-n⁢p,Aπp-n⁢p Pm =bπm⁢1,Bπm⁢1,…,bπm⁢m,Bπm⁢m (166) Pq-m =bπq-m⁢m+1,Bπq-m⁢m+1,…,bπq-m⁢q,Bπq-m⁢q

have to be inserted on the right hand side. [122.1.4] If any of n,m,p-n or q-m vanishes the corresponding permutation is absent.

[122.2.1] The order reduction formula

 Hp,qm,n(z|a1,A1,a2,A2⁢…,ap,Apb1,B1,b2,B2⁢…,bq-1,Bq-1⁢a1,A1)=Hp-1,q-1m,n-1(z|a2,A2,…,ap,Apb1,B1,…,bq-1,Bq-1) (167)

[page 123, §0]    holds for n1 and q>m, and similarly

 Hp,qm,n(z|a1,A1,a2,A2⁢…,ap-1,Ap-1⁢b1,B1b1,B1,b2,B2⁢…,bq,Bq)=Hp-1,q-1m-1,n(z|a1,A1,…,ap-1,Ap-1b2,B2,…,bq,Bq) (168)

for m1 and p>n. [123.0.1] The formula

 Hp,qm,n(z|a,0,a2,A2⁢…,ap,Apb1,B1,…,bq,Bq)=Γ(1-a)Hp-1,qm,n-1(z|a2,A2,…,ap,Apb1,B1,…,bq,Bq) (169)

holds for n1 and Re1-a>0. [123.0.2] Analogous formulae are readily found if a parameter pair a,0 or b,0 appears in one of the other groups.

[123.1.1] A change of variables in (153) shows

 Hp,qm,n(z|a1,A1,…,ap,Apb1,B1,…,bq,Bq)=Hq,pn,m(1z|1-b1,B1,…,1-bq,Bq1-a1,A1,…,1-ap,Ap) (170)

which allows to transform an H-function with D>0 and argz to one with D<0 and arg1/z. [123.1.2] For γ>0

 1γHp,qm,n(z|a1,A1,…,ap,Apb1,B1,…,bq,Bq)=Hp,qm,n(zγ|a1,γ⁢A1,…,ap,γ⁢Apb1,γ⁢B1,…,bq,γ⁢Bq) (171)

while for γR

 zγHp,qm,n(z|a1,A1,…,ap,Apb1,B1,…,bq,Bq)=Hp,qm,n(z|a1+γ⁢A1,A1,…,ap+γ⁢Ap,Apb1+γ⁢B1,B1,…,bq+γ⁢Bq,Bq) (172)

[page 124, §0]    holds.

[124.1.1] For m=0 with conditions (159) the integrand is analytic and thus

 Hp,q0,n(z|a1,A1,…,ap,Apb1,B1,…,bq,Bq)=0. (173)

## 4.3 Integral Transformations

[124.2.1] The definition of an H-function in eq. (153) becomes an inverse Mellin transform if L is chosen parallel to the imaginary axis inside the strip

 max1≤i≤m⁡Re⁢-biBi

by the Mellin inversion theorem [60]. [124.2.2] Therefore

 M⁢Hp,qm,n⁢z⁢s=η⁢s=∏i=1mΓ⁢bi+Bi⁢s⁢∏i=1nΓ⁢1-ai-Ai⁢s∏i=n+1pΓ⁢ai+Ai⁢s⁢∏i=m+1qΓ⁢1-bi-Bi⁢s (175)

whenever the inequality

 max1≤i≤m⁡Re⁢-biBi

is fulfilled.

[124.3.1] The Laplace transform of an H-function is obtained from eq. (175) by using eq. (78). [124.3.2] One finds

 L{Hp,qm,n(z)}(u)=∫0∞e-u⁢xHp,qm,n(x|a1,A1,…,ap,Apb1,B1,…,bq,Bq)dx =Hq,p+1n+1,m(u|1-b1-B1,B1,…,1-bq-Bq,Bq0,1⁢1-a1-A1,A1,…,1-ap-Ap,Ap) =1uHp+1,qm,n+1(1u|0,1⁢a1,A1,…,ap,Apb1,B1,…,bq,Bq) (177)

valid for Res>0, C>0, argz<12Cπ and min1jmRebj/Bj>-1.

[page 125, §1]
[125.1.1] The definite integral found in [59, 2.25.2.2.]

 ∫0yxβ-1(y-x)γ-1Hp,qm,n(Cxδ(y-x)η|a1,A1,…,ap,Apb1,B1,…,bq,Bq)dx =yβ+γ-1Hp+2,q+1m,n+2(Cyδ+η|1-β,δ,1-γ,η,a1,A1,…,ap,Apb1,B1,…,bq,Bq,1-β-γ,δ+η) (178)

contains as a special case the fractional Riemann-Liouville integral [58, (2.7.13)]

 I0+α⁢Hp,qm,n⁢y =1Γ⁢α∫0y(y-x)α-1Hp,qm,n(x|a1,A1,…,ap,Apb1,B1,…,bq,Bq)dx =yαHp+1,q+1m,n+1(y|0,1,a1,A1,…,ap,Apb1,B1,…,bq,Bq,-α,1) (179)

valid if min1jmRebj/Bj>0. [125.1.2] The fractional Riemann-Liouville derivative is obtained from this formula by analytic continuation to α<0.

## 4.4 Series Expansions

[125.2.1] The H-functions may be represented as the series [56, 57, 58, 59]

 Hp,qm,n(z|a1,A1,…,ap,Apb1,B1,…,bq,Bq)=∑i=1m∑k=0∞ci⁢k-1kk!⁢Bizbi+k/Bi (180a) where ci⁢k=∏mj=1j≠iΓ⁢bj-bi+k⁢Bj/Bi⁢∏nj=1Γ⁢1-aj+bi+k⁢Aj/Bi∏qj=m+1Γ⁢1-bj+bi+k⁢Bj/Bi⁢∏pj=n+1Γ⁢aj-bi+k⁢Aj/Bi (180b)

whenever D0, L is as in (158) or (159) and the poles in Pb are simple. [125.2.2] Similarly

 Hp,qm,n(z|a1,A1,…,ap,Apb1,B1,…,bq,Bq)=∑i=1n∑k=0∞ci⁢k-1kk!⁢Aiz-1-ai+k/Ai (181a) [page 126, §0]    where ci⁢k=∏nj=1j≠iΓ⁢1-aj-1-ai+k⁢Aj/Ai⁢∏mj=1Γ⁢bj+1-ai+k⁢Bj/Ai∏pj=n+1Γ⁢aj+1-ai+k⁢Aj/Ai⁢∏qj=m+1Γ⁢1-bj-1-ai+k⁢Bj/Ai (181b)

whenever D0, L is as in (158) or (160) and the poles in Pa are simple.