On Humbert matrix function of two complex variables under differential operator

Int. J.IndustrialMathematisVol. 2,No. 3(2010)167-179

On Humbert Matrix Funtion

1

(A;B;C;C 0

;z;w)

of Two Complex Variables under Dierential

Operator

S.Z. Rida a

,M.Abul-Dahab a

,M.A.Saleem b

,M.T.Mohamed

(a)DepartmentofMathematis,Faulty ofSiene,SouthValleyUniversity,Qena83523, Egypt.

(b)DepartmentofMathematis,Faulty ofSiene,SohagUniversity,Sohag82524, Egypt.

()Department ofMathematis andSiene,Faulty ofEduation(NewValley),AssiutUniversity,

Assiut71516,Egypt.

Reeived29February 2010;revised19September2010; aepted 22September2010.

|||||||||||||||||||||||||||||||-Abstrat

This paper deals with the study of the Humbert funtion with matrix arguments

1

(A;B;C ;C 0

;z;w).

The onvergent properties, an integral representation of

1

(A;B;C ;C 0

;z;w) and

on-tiguous funtion relations are presented. Some results are obtained from operating the

dierentialoperator D on Humbertmatrix funtion. Moreover a solutionof ertain

par-tialdierentialequation isgiven.

Keywords : Humbert's matrix funtions;Integralform; Contiguousfuntion relations;

Hypergeo-metrimatrixdierentialequation;Dierentialoperator.

||||||||||||||||||||||||||||||||{

1 Introdution

Humbert's funtions of salar oeÆients and variables appear in many elds suh as

statistial distribution theory, heat ow, astrophysis and related areas (see for instane

[4 , 6 , 19 ℄) and Srivastava and Karlsson [17 ℄. Humbert's funtions of real variables were

generalized to these funtions with matrix argument in [7℄ and [13 ℄, see also the books

of Mathai [11 , 12 ℄. Reently, Upadhyaya and Dhami have presented some properties of

theHumbert'sfuntions of matrixarguments in[20 , 21 ℄. Jdar and Cortsintroduedand

studied the hypergeometri matrix funtions in [9 , 10 ℄. Some properties of gamma and

betamatrixfuntions weregiven in[8 ℄.

Our mainpurpose in this paper is to obtain an extension of the hypergeometri matrix

funtion to funtions of more than one variable. Humbert's matrix funtions will be

introduedas funtions of two omplex variables with matrix oeÆients. The struture

of thispaperis asfollows:

Setion 2 is organized to establish the seven Humbert's matrix funtions and alulate

theirorresponding radiusof regularity. Insetion 3 some integral representationforthe

Humbert matrix funtion

1

(A;B;C ;C 0

;z;w) is given. The funtions ontiguous to the

Humbert matrix funtion

1

and its relations are given in setion 4. In Setion 5, a

solutionofertain partialdierentialequation isproposed.

AmatrixA inC NN

isapositivestablematrixifR e()>0forall2(A)where(A)

isthe setof all eigenvaluesof A andits two-norm denoted by

jjAjj=sup

x6=0 jjAxjj

2

jjxjj

2 ;

wherefora vetor y inC N

, jjyjj

2 =(y

T

y) 1

2

is Eulideannorm of y.

Let (A) and(A)be thereal numbers whih weredened in[9 ℄by

(A)=maxfR e(z) :z2(A)g; (A)=minfR e(z):z2(A)g: (1.1)

Iff(z)and g(z)are holomorphifuntionsof theomplexvariable zwhiharedenedin

anopenset oftheomplexplane andAisamatrixinC NN

suh that(A), then

from theproperties of thematrixfuntionalalulus(see[3 ℄), itfollowsthat

f(A)g(A)=g(A)f(A): (1.2)

Hene, ifB inC NN

isa matrixforwhih(B)and ifAB=BA,then

f(A)g(B)=g(B)f(A): (1.3)

The reiproal Gamma funtion denoted by 1

(z) = 1

(z)

is an entire funtion of the

omplexvariable z. Then theimage of 1

(z) atingon A denoted by 1

(A) is a

well-denedmatrix.

Furthermore, if

A+nI is invertible for all integern0; (1.4)

see[9 ℄. Thepohhammer symbolorshiftedfatorialdened by

(A)

n

=A(A+I):::(A+(n 1)I)

= (A+nI) 1

(A) ; n1;(A)

0 =I:

(1.5)

Jodar and Corteshave proved in[9℄that

(A)= lim

n !1

(n 1)![(A)

n ℄

1

n A

: (1.6)

The Shurdeompositionof A,wasgivenby[5 ℄ intheform:

ke tA

ke t(A)

r 1

X

(kAkr 1

2

t) k

k!

and

kn A

kn (A) r 1 X k=0 (kAkr 1 2 lnn) k k!

; n1: (1.7)

The hypergeometrimatrixfuntionisdened bythematrixpowerseriesintheform

2 F

1

(A;B;C;z)= 1 X n=0 (A) n (B) n [(C) n ℄ 1 n! z n : (1.8)

Ifn islargeenough, then

k(C+nI) 1

k 1

n kCk

; n>kCk; (1.9)

where C inC NN

suh thatC+nI isinvertibleforallintegersn0.

Let usdenote

(n) =k(C) 1

kk(C+I) 1

k:::k(C+(n 1)I) 1

k; n>0; (1.10)

2 Denition of Humbert's matrix funtions.

LetA;A 0

;B;B 0

;C andC 0

bematriesinC NN

suhthatC+nIandC 0

+nI areinvertible

forall integer n0.

We deneHumbert'smatrixfuntions asfollows:

1

(A;B;C;z;w) = P 1 m;n=0 (A) m+n (B)n[(C) m+n ℄ 1 m!n! z m w n ; 2 (A;A 0

;C;z;w)= P 1 m;n=0 (A)m(A 0 )n[(C) m+n ℄ 1 m!n! z m w n ; 3

(A;C;z;w)= P 1 m;n=0 (A)m[(C) m+n ℄ 1 m!n! z m w n ; 1

(A;B;C ;C 0

;z;w)= P 1 m;n=0 (A)m+n(B)m[(C)m℄ 1 [(C 0 )n℄ 1 m!n! z m w n ; 2

(A;C ;C 0

;z;w) = P 1 m;n=0 (A)m+n[(C)m℄ 1 [(C 0 )n℄ 1 m!n! z m w n ; 1 (A;A 0

;B;C;z;w) = P 1 m;n=0 (A) m (A 0 ) n (B) m [(C) m+n ℄ 1 m!n! z m w n ; 2

(A;B;C;z;w) = P 1 m;n=0 (A) m (B) m [(C) m+n ℄ 1 m!n! z m w n :

Now, let TheHumbert matrixfuntion

1

(A;B;C ;C 0

;z;w) of two omplexvariablesby

1

(A;B;C ;C 0

;z;w)) = P 1 m;n=0 (A) m+n (B) m [(C) m ℄ 1 (C 0 ) n ℄ 1 m!n! z m w n = P 1 m;n=0 U m;n (z;w); (2.11) where U m;n

We study thisfuntion byalulating its radiusof onvergene R . For thispurpose, we

reall therelationof[16℄and keep inmindthat

m;n

1. Hene

1

R

=lim

m+n!1 sup

kU

m;n k

m;n

1

m+n

=lim

m+n!1 sup

(A)

m+n

(B)m[(C)m℄ 1

[(C 0

)n℄ 1

m!n!m;n

1

m+n

=lim

m+n!1 sup

"

(m+n)

A

(A)m+n

(m+n 1)!

(m+n 1)!(m+n) A

m B

(B)m

(m 1)!

(m 1)!m B

m C

[(C)m℄ 1

(m 1)!

(m 1)!m C

: m

C 0

[(C 0

)n℄ 1

(n 1)!

(n 1)!n C

0

1

m!n!

m;n

#

1

m+n

1

R

=limsup

m+n!1 "

[ (A)℄ 1

[ (B)℄ 1

(C) (C 0

)

(m+n) A

m B C

n C

0

(m+n 1)!

m!(n 1)!

m;n

# 1

m+n

limsup

m+n!1 "

k(m+n) A

kkm B

kkm C

kkn C

0

k(m+n 1)!

m!(n 1)!

# 1

m+n

Forpositive numbersand positive integersn, we an write

m=n (2.12)

Usingtherelation(1:8) and theStirling formula,we get

1

R

lim

n(+1)!1 sup

"

n(+1)

(A) P

N 1

k=0 (kAkN

1

2

ln((+1)n) k

)

k!

n

(B)

P

N 1

k=0 (kBkN

1

2

lnn) k

k!

n

(C) P

N 1

k=0 (kCkN

1

2

lnn) k

k!

n

(C 0

)

P

N 1

k=0 (kC

0

kN 1

2

lnn) k

k!

p

2fn(+1) 1gf

n(+1) 1

e g

n(+1) 1

p

2n( n

e )

n p

2(n 1) ( (n 1)

e )

(n 1) #

1

n(+1)

:

So,

P

N 1

k=0 (kAkN

1

2

ln(n(+1)) k

k!

(Nln(n(+1)) N 1

P

N 1

k=0 (kAk)

k

k!

and

1

R

lim

n(+1)!1 sup

"

n

(A)+(B) (C) (C 0

)

(Nln((+1)n) N 1

e kAk

(Nln(n) N 1

e kBk

(Nln(n) N 1

e kCk

(Nln(n) N 1

e kC

0

k

# 1

n(+1)

:

Therefore

lim

n(+1)!1 sup

h p

2fn(+1) 1g f

n(+1) 1

e g

n(+1) 1

p

2n( n

e )

n p

2(n 1)( (n 1)

e )

(n 1) i 1

n(+1)

=1;

i.e. theradius ofonvergene ofthe Humbertmatrixfuntion

1

isone.

By analogous way, one an prove that theother Humbert's matrix funtions are regular

inthehypersphere S

R

; R=1.

3 An integral representation.

We beginthissetionbyonsideringthebinomialfuntionofamatrixexponent. Ifuand

b are omplexnumbers withjuj <1, then (1 u) b

=exp(blog(1 u)); where Log is the

prinipal branh of the logarithm funtion, (see[18℄). The Taylor expansionof (1 u) a

aboutu=0 isgiven by

(1 u) a

= X

n0 (a)

n

n! u

n

; juj<1; a2C : (3.13)

Now, we onsider the funtionof the omplexvariable a denedby (3:13). Letf

n (a) be

thefuntiondened by

f

n (a)=

(a)

n

n! u

n

=

a(a+1):::(a+n 1)

n!

u n

; a2C : (3.14)

For a xed omplex number u with juj<1, it is lear that f

n

is a holomorphifuntion

of variable a dened inthe omplex plane. Given a losed bounded disD

R

=fa 2C :

jajR g,one gets

jf

n (a)j

(jaj)

n

n! juj

n

(R )

n

n! juj

n

; n0;jajR :

Sine P

n0 (R)n

n! juj

n

< +1;by theWeierstrass theorem forthe onvergene of

holo-morphi funtion(see[18℄)it followsthat

g(a)= X

n0 (a)

n

n! u

n

=(1 u) a

is holomorphiin C. By appliation of the holomorphifuntionalalulus ([3℄) forany

matrixA inC NN

, theimage ofg bythisfuntionalalulusatingonA yields

(1 u) A

= X

(A)

n

n! u

n

where(A)

n

is given by(1:6):

Let B and C be matriesin C NN

suh that

BC=CB: (3.16)

where

C ;B andC B arepositivestable: (3.17)

By (3:13) and (3:17), one gets

(B)

m [(C)

m ℄

1

= (B+mI) 1

(B)[ (C+mI) 1

(C) ℄ 1

= 1

(B) 1

(C B) (C B) (B+mI) 1

(C+mI) (C):

(3.18)

By lemma2 in[8 ℄ and (3:16) and (3:17), we ndthat

R

1

0 t

B+(m 1)I

(1 t) C B I

dt =B(B+mI;C B)

= (C B) (B+mI) 1

(C+mI) :

(3.19)

Also,by(3:18) and (3:19), we get

(B)

m [(C)

m ℄

1

= 1

(B) 1

(C B)

R

1

0 t

B+(m 1)I

(1 t) C B I

dt

(C) (3.20)

Hene,formally one an write

1

(A;B;C ;C 0

;z;w) = P

1

m;n=0 (A)

m+n (B)

m [(C)

m ℄

1

[(C 0

)

n ℄

1

m!n!

z m

w n

= P

1

m;n=0 (A)

m+n 1

(B) 1

(C B) (C)

(C 0

)

n m!n!

R

1

0 t

B+(m 1)I

(1 t) C B I

dt

z m

w n

:

Sine (A)

m+n =(A)

n

(A+n)

m

, forjzj<1,bytherelation(3:15), we get

1

(A;B;C ;C 0

;z;w)= 1

X

m;n=0 (A)

m+n (B)

m [(C)

m ℄

1

[(C 0

)

n ℄

1

m!n!

z m

w n

= Z

1

0 t

B I

(1 t) C B I

(1 tz) A

1 1

(A; ; ;C 0

; w

1 tz dt

1

(B) 1

(C B) (C):

We have proved thefollowingtheorem:

Theorem 3.1. Let A, B and C be matries in C NN

suh that CB =BC where C, B,

C-B are positive stable. Then for jzj<1 we have

1

(A;B;C ;C 0

;z;w)=

Z

1

0 t

B I

(1 t) C B I

(1 tz) A

1 1

(A; ; ;C 0

; w

1 tz dt

1

(B) 1

(C B) (C):

4 The ontiguous funtion relations.

In thissetionsome reurrenerelationsare arried outon theHorn matrixfuntion. In

this onnetion the following ontiguous funtions relations follow diretly by inreasing

ordereasingone inoriginal relation,weuse thenotations

1 =

1

(A;B;C ;C 0

;z;w);

1

(A+)=

1

(A+I;B;C ;C 0

;z;w);

1

(A )=

1

(A I;B;C ;C 0

;z;w);

(4.22)

together with notations of

1 (B+),

1 (B ),

1 (C+),

1 (C ),

1 (C

0

+),

1 (C

0

), for

theother funtionsontiguous to

1

. Now, we maywritethefuntions ontiguousto

1

intheform

1

(A+)= 1

X

m;n=0

(A+I)

m+n (B)

m [(C)

m ℄

1

[(C 0

)

n ℄

1

m!n!

z m

w n

=A 1

1

X

m;n=0

(A+(m+n)I)(A)

m+n (B)

m [(C)

m ℄

1

[(C 0

)

n ℄

1

m!n!

z m

w n

=A 1

1

X

m;n=0

(A+(m+n)I)U

m;n (z;w);

(4.23)

and

1

(A )= 1

X

m;n=0

(A I)

m+n (B)

m [(C)

m ℄

1

[(C 0

)

n ℄

1

m!n!

z m

w n

= 1

X

m;n=0

(A+(m+n 1)I) 1

(A I)(A)

m+n (B)

m [(C)

m ℄

1

[(C 0

)

n ℄

1

m!n!

z m

w n

= 1

X

m;n=0

(A+(m+n 1)I) 1

(A I)U

m;n (z;w):

(4.24)

Similarly

1

(B+)=B 1

1

X

m;n=0

(B+mI)U

m;n (z;w);

1

(B )= 1

X

m;n=0

(B+(m 1)I) 1

(B I)U

m;n (z;w);

1

(C+)= 1

X

m;n=0

(C+mI) 1

CU

m;n (z;w);

1

(C )= 1

X

m;n=0

(C I) 1

(C+(m 1)I)U

m;n (z;w);

1 (C

0

+)= 1

X

(C 0

+nI) 1

C 0

U

1 (C

0

)= 1

X

m;n=0 (C

0

I) 1

(C 0

+(n 1)I)U

m;n (z;w):

Forany integer k1,we deduethat

1

(A+kI)= Q

k

r=1

(A+(r 1)I) 1

P

1

m;n=0 Q

k

r=1

(A+(m+n+(r 1)I)U

m;n (z;w);

(4.25)

1

(A kI)= k

Y

r=1

(A rI) 1

X

m;n=0 k

Y

r=1

(A+(m+n r)I) 1

U

m;n (z;w);

(4.26)

1

(B+kI)= k

Y

r=1

(B+(r 1)I) 1

1

X

m;n=0 k

Y

r=1

(B+(m+(r 1)I))U

m;n

(z;w); (4.27)

1

(B kI)= k

Y

r=1

(B rI) 1

X

m;n=0 k

Y

r=1

(B+(m r)I) 1

U

m;n (z;w);

(4.28)

1

(C+kI)= k

Y

r=1

(C+(r 1)I) 1

X

m;n=0 k

Y

r=1

(C+(m+(r 1)I)) 1

U

m;n

(z;w); (4.29)

1

(C kI)= k

Y

r=1

(C rI) 1

1

X

m;n=0 k

Y

r=1

(C+(m r)I)U

m;n (z;w);

(4.30)

1 (C

0

+kI)= k

Y

r=1 (C

0

+(r 1)I) 1

X

m;n=0 k

Y

r=1 (C

0

+(m+(r 1)I)) 1

U

m;n

(z;w) (4.31)

and

1 (C

0

kI)= k

Y

r=1 (C

0

rI) 1

1

X

m;n=0 k

Y

r=1 (C

0

+(m r)I)U

m;n (z;w):

(4.32)

Bysimilararguments, we angetsome examplesof ontiguousfuntionrelationsdiretly

asfollows:

1

(A+;B+)=A 1

B 1

1

X

m;n=0

(A+(m+n)I)(B+mI)U

m;n (z;w)

1

(A ;B )=(A I)(B I) 1

X

m;n=0

[(A+(m+n 1)I)℄ 1

(B+(m 1)I) 1

U (z;w);

1

(B+;C+)= 1

X

m;n=0 CB

1

(B+mI)(C+(m+n)I) 1

U

m;n (z;w);

1

(B ;C )= 1

X

m;n=0

(B I)(B+(m 1)I) 1

(C I) 1

(C+(m+n 1)I)U

m;n (z;w);

1

(B++;C 0

+)= 1

X

m;n=0 C

0

B 1

(B+I) 1

(B+mI)(B+(m+1)I)

(C 0

+(m+n)I) 1

U

m;n (z;w);

1

(B++;(C+I)+)= 1

X

m;n=0

(C+I)B 1

(B+I) 1

(B+mI)(B+(m+1)I)

(C+(m+n+1)I) 1

U

m;n (z;w):

(4.34)

ConsiderthedierentialoperatorD, as given in[14 , 15 , 16 ℄,takesthe form

D =

d

1 +d

2

; m;n1;

1; otherwise,

(4.35)

whered

1 =z

z and d

2 =w

w

: Thisoperatorhasthepartiularlypleasantproperty

Dz m

w n

=(m+n)z m

w n

:

Now, thefollowingontiguous funtionrelationsfortheHumbert matrixfuntionan be

dedued

(DI+A)

1 =

X

m;n=0

(A+(m+n)I)(A)

m+n (B)

m [(C)

m+n ℄

1

m!n!

z m

w n

= X

m;n=0

(A+(m+n)I)U

m;n

(z;w)=A

1 (A+):

(4.36)

By thesame way, weget

(d

1

I+B)

1 =B

1 (B+);

(d

1 I+C)

1

=(C I)

1

(C )+

1

(d

2 I+C

0

)

1 =(C

0

I)

1 (C

0

)+

1 :

(4.37)

From(4.33) and(4.34), it followsat one that

(A (C+C 0

)+2I)

1 =A

1

(A+) (C I)

1

(C ) (C 0

I)

1 (C

0

);

(A B)

1

=A

1

(A+) B

1

(B+) d

2 1 (B);

(B C+I)

1 =B

1

(B+) (C I)

1 (C );

(A (B+C 0

I))

1 =A

1

(A+) B

1

(B+) (C 0

I)

1 (C

0

):

theoperatingwith D on theHumbert matrixfuntionyields

D

1 =

1

X

m;n=1

(m+n)(A)

m+n (B)

m [(C)

m ℄

1

[(C 0

)

n ℄

1

m!n!

z m

w n

= 1

X

m1;n=0 m(A)

m+n (B)

m [(C)

m ℄

1

[(C 0

)

n ℄

1

m!n!

z m

w n

+ 1

X

m=0;n=1 n(A)

m+n (B)

m [(C)

m ℄

1

[(C 0

)

n ℄

1

m!n!

z m

w n

=z 1

X

m;n=0

(A+(m+n)I)(B+mI)[(C+mI)℄ 1

U

m;n (z;w)

=w 1

X

m;n=0

(A+(m+n)I)[(C 0

+nI)℄ 1

U

m;n (z;w):

(4.39)

Alternatively, equation (4:36) an be writtenintheform

D

1

=zABC 1

1

(A+;B+;C+;C 0

;z;w)

+wAC 0 1

1

(A+;B;C ;C 0

+;z;w):

(4.40)

Now,letusoperatewithD ontheseriesdening

1

(A ),wethusobtainfromtherelation

(4.24) that

1

(A )= 1

X

m;n=0

(A I)[A+(m+n 1)I℄ 1

U

m;n (z;w)

= 1

X

m;n=0

I (m+n)[A+(m+n 1)I℄ 1

!

U

m;n (z;w)

=

1

(A I) 1

D

1 (A );

i.e. the

1

(A;B;C;z;w) is asolutionof thepartialdierentialequation

D

1

(A ) (A I)

1

+(A I)

1

(A )=0: (4.41)

Now, forA=

2 0

0 2

, we have

1

(2I;B;C ;C 0

;z;w)= X

m;n=0 (2I)

m+n (B)

m [(C)

m ℄

1

[(C 0

)

n ℄

1

m!n!

z m

w n

= X

m;n=0

((m+n+1)I)(I)

m+n (B)

m [(C)

m ℄

1

[(C 0

)

n ℄

1

m!n!

z m

w n

= X

m;n=0

((m+n)I)(I)

m+n (B)

m [(C)

m ℄

1

[(C 0

)

n ℄

1

m!n!

z m

w n

+ X

m;n=0 (I)

m+n (B)

m [(C)

m ℄

1

[(C 0

)

n ℄

1

m!n!

z m

w n

=D (I;B;C ;C 0

;z;w)+ (I;B;C ;C 0

;z;w)

It turnsoutthat

D

1

(I;B;C ;C 0

;z;w)=

1

(2I;B;C ;C 0

;z;w)

1

(I;B;C ;C 0

;z;w): (4.42)

Ontheother hand, we an use

1

(C+) to get

1

(C+)= 1

X

m;n=0

(C+mI) 1

CU

m;n (z;w)

=

1 C

1

d

1 1 (C+);

d

1 1

(C+)=C

1 C

1

(C+): (4.43)

From (4:42), we obtain

d

1 1

(C+)+d

2 1

(C+)=C

1 C

1

(C+)+d

2 1 (C+);

i.e. Operatewith D on thefuntion

1

(C+),weobtain

D

1

(C+)=C

1 C

1

(C+)+d

2 1

(C+); (4.44)

and

D

1 (C

0

+)=C 0

1 C

0

1 (C

0

+)+d

1 1 (C

0

+): (4.45)

5 The Humbert matrix: dierential equation.

The operators D, d

1 and d

2

whih have been already used in the derivation of the

on-tiguousfuntion relations,are helpfulinderivingdierentialequation satisedby(2:11).

h

d

1 (d

1

I+C I)+d

2 (d

2 I +C

0

I) i

1

= 1

X

m=1;n=0

m(C +(m 1)I)(A)

m+n (B)

m [(C)

m ℄

1

[(C 0

)

n ℄

1

m!n!

z m

w n

+ 1

X

m=0;n=1 m(C

0

+(n 1)I)(A)

m+n (B)

m [(C)

m ℄

1

[(C 0

)

n ℄

1

m!n!

z m

w n

= 1

X

m=1;n=0 (A)

m+n (B)

m [(C)

m 1 ℄

1

[(C 0

)

n ℄

1

(m 1)!n!

z m

w n

+ 1

X

(A)

m+n (B)

m [(C)

m ℄

1

[(C 0

)

n 1 ℄

1

m!(n 1)!

z m

w n

A shiftofindexyields

d

1 (d

1

I +C I)+d

2 (d

2 I+C

0

I)

1

= 1

X

m=1;n=0 (A)

m+1+n (B)

m+1 [(C)

m+1 ℄

1

[(C 0

)

n ℄

1

m!n!

z m+1

w n

+ 1

X

m=0;n=1 (A)

m+n+1 (B)

m [(C)

m ℄

1

[(C 0

)

n ℄

1

m!n!

z m

w n+1

=z 1

X

m;n=0

((A+(m+n)I)((B+mI))U

m;n (z;w)

+w 1

X

m;n=1

((A+(m+n)I)U

m;n (z;w)

=z(DI+A)(d

1

I+B)

1

+w(DI+A)

1 :

ItiseasytoseethattheHumbertmatrixfuntion

1

(A;B;C ;C 0

;z;w)shouldbeasolution

of apartialdierentialequation givenby

"

z(DI+A)(d

1

I+B)+w(DI+A)

d

1 (d

1

I+C I)+d

2 (d

2 I+C

0

I)

#

1

(A;B;C ;C 0

;z;w)=0:

(5.46)

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