4.1 Definition
[120.1.1] The H-function of order m,n,p,q∈N4 and with parameters
Ai∈R+(i=1,…,p), Bi∈R+(i=1,…,q),
ai∈C(i=1,…,p), and bi∈C(i=1,…,q)
is defined for z∈C,z≠0 by the contour integral
[55, 56, 57, 58, 59]
Hp,qm,n(z|a1,A1,…,ap,Apb1,B1,…,bq,Bq)=12πi∫Lη(s)z-sds | | (153) |
where the integrand is
ηs=∏i=1mΓbi+Bis∏i=1nΓ1-ai-Ais∏i=n+1pΓai+Ais∏i=m+1qΓ1-bi-Bis. | | (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
and empty products are interpreted as being unity.
[121.0.3] The parameters are restricted by the condition
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|<Cπ/2; C>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|<E-1 | | (159b) |
L | =L(-∞+iγ1,-∞+iγ2;Pa,Pb); D=0; |z|=E-1; C≥0; ReF<0 | | (159c) |
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 z≠0 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(n≥1) 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 πn∈Sn,πm∈Sm,πp-n∈Sp-n
and πq-m∈Sq-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 | =aπn1,Aπn1,…,aπnn,Aπnn | |
Pp-n | =aπp-nn+1,Aπp-nn+1,…,aπp-np,Aπp-np | |
Pm | =bπm1,Bπm1,…,bπmm,Bπmm | | (166) |
Pq-m | =bπq-mm+1,Bπq-mm+1,…,bπq-mq,Bπq-mq | |
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-1a1,A1)=Hp-1,q-1m,n-1(z|a2,A2,…,ap,Apb1,B1,…,bq-1,Bq-1) | | (167) |
[page 123, §0]
holds for n≥1 and q>m, and similarly
Hp,qm,n(z|a1,A1,a2,A2…,ap-1,Ap-1b1,B1b1,B1,b2,B2…,bq,Bq)=Hp-1,q-1m-1,n(z|a1,A1,…,ap-1,Ap-1b2,B2,…,bq,Bq) | | (168) |
for m≥1 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 n≥1 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,γA1,…,ap,γApb1,γB1,…,bq,γBq) | | (171) |
while for γ∈R
zγHp,qm,n(z|a1,A1,…,ap,Apb1,B1,…,bq,Bq)=Hp,qm,n(z|a1+γA1,A1,…,ap+γAp,Apb1+γB1,B1,…,bq+γBq,Bq) | | (172) |
[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≤mRe-biBi<s<min1≤i≤mRe1-aiAi | | (174) |
by the Mellin inversion theorem [60].
[124.2.2] Therefore
MHp,qm,nzs=ηs=∏i=1mΓbi+Bis∏i=1nΓ1-ai-Ais∏i=n+1pΓai+Ais∏i=m+1qΓ1-bi-Bis | | (175) |
whenever the inequality
max1≤i≤mRe-biBi<min1≤i≤mRe1-aiAi | | (176) |
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-uxHp,qm,n(x|a1,A1,…,ap,Apb1,B1,…,bq,Bq)dx | |
| =Hq,p+1n+1,m(u|1-b1-B1,B1,…,1-bq-Bq,Bq0,11-a1-A1,A1,…,1-ap-Ap,Ap) | |
| =1uHp+1,qm,n+1(1u|0,1a1,A1,…,ap,Apb1,B1,…,bq,Bq) | | (177) |
valid for Res>0, C>0, argz<12Cπ and
min1≤j≤mRebj/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+αHp,qm,ny | =1Γα∫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 min1≤j≤mRebj/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∞cik-1kk!Bizbi+k/Bi | | (180a) |
where |
cik=∏mj=1j≠iΓbj-bi+kBj/Bi∏nj=1Γ1-aj+bi+kAj/Bi∏qj=m+1Γ1-bj+bi+kBj/Bi∏pj=n+1Γaj-bi+kAj/Bi | | (180b) |
whenever D≥0, 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∞cik-1kk!Aiz-1-ai+k/Ai | | (181a) |
[page 126, §0]
where |
cik=∏nj=1j≠iΓ1-aj-1-ai+kAj/Ai∏mj=1Γbj+1-ai+kBj/Ai∏pj=n+1Γaj+1-ai+kAj/Ai∏qj=m+1Γ1-bj-1-ai+kBj/Ai | | (181b) |
whenever D≤0, L is as in (158) or (160)
and the poles in Pa are simple.