GENERALIZED LAURICELLA HYPERGEOMETRIC FUNCTIONS

VOL. 15 NO. 4 (1992) 653-658

INTEGRAL REPRESENTATIONS OF

GENERALIZED LAURICELLA HYPERGEOMETRIC FUNCTIONS

VU

KIM TUAN

InstituteofMathematics P.O.Box 631,

B

H5

Hanoi,Vietnam

AND

R.G. BUSCHMAN

Departmentof Mathematics Box 3036,UniversityStation

Laramie,WY82071,USA (Received August I0, 1990 and in revised form March 24, 1992)

ABSTRACT. The generalized hypergeometricfunctionwasintroduced bySrivastava andDaoust. Inthepresent paper a new integral representation isderived. Similarlynew integral representations ofLauricellaand Appell functionareobtained.

KEYWORDS. Lauricella, Appell, integralrepresentations, Mellin transformation.

1991 AMS SUBJECT CLASSIFICATION CODES. 33A35, 44A30 1. INTRODUCTION.

The generalizedLauricella functionofseveralcomplexvariableswasintroducedbySrivastava andDaoust

[7,8].

The only integral representationwhich seemsto be known for this functionisinterms ofa Mellin-Barnesinte- gral

[6,8].

^FromafundamentalresultaboutMellintransformsinn-dimensions,^weobtainanewrepresentation for the generalized Lauricella function under suitable restrictionsonthe parameters. Inasimilarmanner newintegral representations areobtained forthe Lauricella functions

F( ^"), F(

^")and

F( n),

andconsequently for theAppell func- tionsF1, F2, and

F4.

^Fromthesederivations it isclearthat themethod doesnotprovide representations for

F(s

ⁿ⁾

and

F3.

2. THE FUNDAMENTALTHEOREM.

Althoughthefollowing theoremisquitesimple, neverthelessithasbasic importance.

THEOREM. Let

f(z)z

^’-1E

L(0, oo)

and

M. (m={x,,

.-.,

x.})l(s, .--,s.) /(max{x,, ,x.}) z,

s

+---+sf.(

+’"+s)

81 ^$n

He

Mn ^psen

^then-dmensional Melhn integral

tmsfoation [].

PROOF. Infactwehave

()

(2)

i=1$152 Sn

f(z’)’’+ +’)"-ldz*

⁸¹

$1

+ 8n+

^Snf*($1

""""" " ^sn)

Thus theformula

(2)

^isproved.

Theonly related rultswhichwehave foundintheliteraturearethosefor the twdimensional Laplacetrans- formationin thetabiofVoelker and Doetsch

[;

^{p. 15,}

(30),(32)]

^andinthe work of

ernov [4;

^p.

145].

Inan analogousmannertoourthrem, thee results eilycanbederivedand extendedtohigher dimensions.

3.

THE

INTEGRAL REPSENTATION OFTHEGENERALIZEDLAURICELLA FUNCTION.

Thegeneralized hypergeometricfunctionofSrivtavaandDust

[6]

is definedby

’" ^b; d ^c. . ^{,,...,, (3)}

H=,(.,)-,+.+-., H,,, ^(i).,

where forbeolute convergenceit sucientthat

l+9+q-p-pk0;

:

1,9,...,n.

Thefunction in

(3)

specicoftheH-functionsof

sever

wriablwhich were defined in

[3,].

The fund- mentltheoremlestothefollowing^)ultwhichinvolvtheG-functionof Meijer

[8].

Let

(

^{),. ,)}

) / ^{+ ni= ’ ( +}

P

j=l /=1

/’() - ⁾ s) ^>

^0.

Therefore

(2)

becomes

I’[=11’(a

P

+ s +... + s,) (s; +... + ^s,)

_{Resl,... Res,}

_>

_0.

H,":, ^{Gamma(b, +}

^s,

+ + s,)

^{s, ...s,}

(4)

rl.=1(aj).,,+

p

+m.(n)m,+..+,,.(n+ mx +-.-+m,,)

! " E

^..))0

^H=(,),,+

^+,.

(1"1

4"

1re,4-

^+,.()’1’i

+ 1) (ran + 1)

The representation for the generalized Lauricella functioncan now be obtainedasfollows. LetPi

_<

qi, P

1,2,...,n,andeither p qandRe=1

a ^<

^Re

_

^j-1

^b

^orp

^>

^q.Then

Thuswehave proved

F^pp’’

_’ , ₍ ^%;c-; _{b," d_ a.} ^c-’zt,.., _x.)= ⁽⁵⁾

F (hi).- .F

(b)

(

,0 ,1

’

Cnt"" G],p+, ^max{t,,.., tn}

. ^{r(.) r(..)} ...

i=l

Formula

(5)

^{is valid}when

Rea/ >

^n;j 1,2,...,p;pi

<_

qi,i=1,2,.--,n’p

>_

q,l+q+qi-p-pi

>_

0; 1,2,...,n.

p

(The

restrictionRe

E ^{bi + >}

^Re

E

^aiisneeded whenp q.)

=I j=l

4. INTEGRALREPRESENTATIONSOF LAURICELLAHYPERGEOMETRIC FUNCTIONS.

Derivations similar tothosein the previous sectionleadto representationsfor 3 ofthe Lauricella functions.

(Sincein

(5)

^p

_>

qwedonotget

F(Bn).)

^Weintroducethe operator

rt +zi

(a)

Usingtheformula

(2)

^weget

oo oo

^exp(-

max{zz,---,z.}) ^I

z,"

^"-’+’- mx

^...dz.

i=I

r(a+ml+.-.+mn)

(+ ,)...( +,),

Consequently,fortheLauricella function

FA

^(")wehave

Rea>

1.

Consequently

F(A

ⁿ⁾

(a;bt,"

,bn"cl, ",cn’z,"

,zn)

ml, ,am’-0

E r-c ^{t/’Z+m’-x}

^exp

^(-

^max

{tl,-.", in})

mz, ,m,⁼⁰ i=1

=

ix

(c,), m! dtz"’dtn

r(.) (tl "tn) ’z-1

_exp

_(--

_max

_{{tl, ",tn))}

(7)

(b)

From

(2)

^wehave

j01 ^(I

^max

^{{zz, ,} z.})’-" I z, "’+m

^,-Idz]."dzn

r(a+m +...+ m.)r(b- a+ 1)

+

^Reb

>

^Rea

>

r(b+m, +...+m.)(". +m,)...(". +m.)

(8)

656 V.K. TUAN

Hence

Thereforefor the Lauricella function

F

^")weget

F

^")(a;bt,--. ,b,,"c;

,

,z,,)

... ^(C)m+

-1

(1

max

{tt,...,/,})c-,

ra)r-

a

+ ¹⁾

_,=

fi ( ) (b,),,(x,t,)"’

Z ^a_l+m,

t=lm,=0 ¹

dt dt,

F(o")(a;b,--.,b.’c;z,...,x.)

r(a)r(-+x) v, t’-t(1-max(t,,...,tn}) (l-x,t,)-b’dt,’"dtn,

t=l t=l

+

Rec

>

Rea

>

(c)

Using the formula

(2)

again,^wehave

(max {,---, n}) rr K,_,,+ (2 (max {t--.t,}) 1/2) ^(x,) "’’+m’-

_dxt...dx,

i=1

! r(a+’’’+’’’+’")r(b+’’’+’’’+’’")

_{l,> lb>-O.}

-2 (+m,)...(an+mn)

Consequently, for theLaurieellahypergeometrie^function

F(c ’0,

byasimilardevelopment^wehave the result

F(c

^")

(a;b;c,...,cn;za,...,z,)

2

fi o ofit-t-,(max{t, ,n})z--/t.,_a+ (2(max{t ^tn}, 1/2)

r,)r) ^v, ,...,

i=1 i=1

fl0F1

^(ci,iti

^{dtl dtn}

i=1

Rea>l, Reb>-l.

ForAppellfunctionstheseread

(.

b,.;

, ) r()

r()r(- + )

"D,D2

l+Rec>Rea>^1"

(10)

(11)

(12)

F2(a;bl,b;cl,c2;z,x2)

12 r(.) (tt2)(’-s)12exp (-

max

{tx, t2}) {1El (bl"

^el;

Zltl)} {1El (b2"

c2;

z2t2)} dtdt2

Rea

>

(13)

F4(a;b;c,c2;cl,z2)

Rea>1, Reb>0.

(14)

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