hypergeometric functions of two variables

Download Hypergeometric Functions of Two Variables

If you can't read please download the document

Upload: fadil-habibi-danufane

Post on 04-Jan-2016

37 views

Category:

Documents


7 download

DESCRIPTION

Hypergeometric Functions of Two Variables

TRANSCRIPT

  • - 60 -

    for the conti.g1.1Ql1S functions (9.3) . The author does not '

    whether the recurrence formulas for those contiguous functions were : i:. , : , .

    already disc.overed. . ..

    .. . . .. :PROBLEM 1 : Find the formulas for the contiguous

    functions ' : ' ' : l . . ' ' .

    ;- '" 'This 'problem mlght not be very difficult. In fact, the author . /

    the F2 : . . , ; . l . : .. .. . / .. ,. ; I '

    .(at-t)(o(- r')(C(;..1- i'+l)F2

    (oc.-l, 'l/, t'',x,y)

    .. (cl -'t- {'f+ .. (Cit -> p'y}F2 : . - . . ... :. t ; ! -'.. . .: l .. ' . . ' ; . - .. . . .. ....

    { c ' - 2'') cat- (f- (1-x) + c2o{'. - - a'') p iy}x aF2/ d x :. : . ..! \i :-. Y ?;,, r ::.::, i , -;1_ , ; . " j

    ._ "." {

  • - 61 ..

    ~ib. .-.-

    Euler integral representa'tions. The hypergeometric functions ~ I

    :g.f two variables, except for .F 4 , have Euler integral represen-

    . /~ations which are double .~ntegrals but very similar to that of

    ":;~_t.1'' ~ ~ , i , x > . .. THEOREM 10.1: We have .

    ' . . ' ' .

    .. re~> fJ --1 ~-1 i-~-p'~1 ~ . ~ r(~)rc~) ro-~- ~) . u v (1-u-v) (1-xu-yv) dudv

    1~ u, v, 1-u-:v~O I ' i

    : '.1,f. Re ( ~ ) '> 0, . Re ( ~ ') "> 0, Re ( 'I - ~ - (3 ') > 0; ~.'' . .

    ",.j, I '~

    ' ;'j. .. :, :Jno. 2) F 2 ( a( t p , ~ ' , r ' 't I , x ' y)

    ::. :1 ... .. ' ~ ' .. J . i; .! :. : .: ; . I "; .. : ...

    :,:'if Re ( fl ) > 0' Re ( ~ I ) ') 0 , Re ( l - p ) '> 0, Re ( 1 ' - 13' ) >. o; ' .

    :'.{10. 3) . . F 3 (or< ~~ rc~-~-n . H u!-lvP'-1(1-u~v) -~~,~ -l Ct-xu) -~ c'1-~) ~"'dudv ... . . .. ' u,v,1:-u-v_O . ; : . ... , . . ' : .

    J' Re ( p ) ') 0 ~ ~e ( ~ 1 ) :> O , Re ( ~ - ~ .. ~ ' ) > O ~ .. . :. ; . ... . . . r .

  • . ~ .

    - 62 -

    . -o

  • - 63 -

    = f (oc. m+n) rcp+m>rr F1

  • - 64 -

    ~ ( "' ) r ' I' ' II > ' This completes the prcof of Theorem 10.1.

    The function F4 does not have any simple integral represen-

    tation such as (10.1), (10.2) and (10.3). Thi~, however, does not . . '.

    . . mean that F4 does not h~ve any double integral representation.

    For example, the following formula is kn.own:

    , r re 11 > F4 (c

  • - 65 -

    From (10.4) we can conclude that F1 is analytically con-

    tinuable and holomorphic for [C - [l, C)O)] x [C - [l, oo)]. Similarly

    (10.3) implies that F3 is holomorphic for [- [l;oo)] x [C- (l,oo)].

    On the other hand> we can prove the following theorem by using (10.4).

    THEOREM 10. 3: We have

    (10.5) Fl (OC, ~, ~', 'lf, x, y)

    (1-x)-p(l-y)-/J' F.1

    ( -o< > p, p ', '(, x/(x-1), y/(y-1)) -ct

    .. (1-x) F1 (ct, J- ~- ~, p', 'l, x/(x-1), (~-y)/(x-1)) ~ .

    .. (1-y) r1 (Q(,~,Y-(3-8',r, (y-x)/(y-1),y/(y-l})

    ..., (1-x) ~-g( -p (1-y) - ~IF 1 ( t - 0( ' r- p - p I ' fl' , "( , x, (y-x) I (y-1) ) -ta l--13' (1-x) (1-y) F1 (r-oc,~, ~-~-{J',, {x-y)/(x-1),y).

    Proof: This theorem can be proved by changing variable u

    respectively by

    u = 1-v, u "" v/[ (1-x)+vxJ, u ... v/[ (1-y)+vy] . ,

    u c (l-v)/(1-vx) and u = (l-v)/(1-vy) .

    Exercise 1. Prove Theorems 10.2 and 10.3.

    It is easily verified that the following six changes of

    variables:

    u D 1-u' , v ... v' , u u' , v . 1-v' , u 1-u' '

    v *" 1-v' , u zo 1-v' , v ""'u' ,

    u ... v' ' v -1-u' I u c lv' , v -lu'

  • - 66 -

    map the square . 0 " u, v ~ l . onto itself and do not change the

    form of the integral (10 .2). By using these transformations we ca ri"' :;~~

    . _pr.ove the :following theorem.

    : THEOREM 10.4: We have . ' , ,. -- .

    (10. 6 > r 2 < , ~, 13 ' , r , r' , x, y > (1 ... x)-"'F2 ("', r-p, f, "I, 'I', x/(x-lL y/(1-x)) (l-y)~F2 (C(~ p, 1'-p',r, 't',x/(l~y),y/(y-1)) : (1-x..:y):.-F

    2(0c, y ;..13, y ! - ~, '/, 't!, x/(x+y-1.), y/(x+y-1) )

    - (1-y) -~If 2 (: > .. r .-p' ,-p, i' >:'( ''y /(yl)-; x/ (1-y)) . (l-x} ~4 ~'-2 ( .2 P', 'j- P; '/ 1, '(. > y'/(1.:.x) > X/ (K-1))

    . . -Cl( I I t ' .. . ., :. . ... .. . - .(1-x-.y) . F2 (ot, -(3 . > r-_ ~> .'t ,1,y/(x+y ... l),x/(x+y-l)).

    It can be,_, shq~n that there .is no change of variable's of the

    form:

    A '.u' + B 'v ,. + C u = Au'+ Bv' + C

    A"u I + B"v I + c" . . v ' . = Au' + Bv ' + C

    which maps the triangie u; v~ .. 1-u-v ~ 0 into itself and which

    does not change the f~rm' of the .. int:egral (10.3). The~efore, it is

    difficult to find any rtransformati on .. formulas" for F 3 '. which are similar to (10.5) and (10 . 6).

    .. ; .

    Let us nex't find some relations between contiguous functions.

    To do this, consider the . representation (10.4) of F 1

    Raisi ng cc , by one we obtain

    rc'+n re y -~-1> r F1 (((+1, p, P', "(, x, Y> 1 . .

    J ot f -2 - A - ~ t a u (1-u) (1-xu) t' (l ~yv) du 0 . .

  • - 67 -

    On the other hand, lowering ~ by one we get

    ::'., r c ' > r < t - o( - i > , . rci-1> Fi

    f 1 .

    ~-1 t--2 - - I '"" u (1-u) (1-xu) f. (1-yu) /3 du

    o If we put

    we have

    r 1 . > (IX ) r (1 - ~) F ( t:< (.f ' /.t I ) a f u (u) du ' . r( ) 1 ' r' IJ ' ~ ' X' y

    0

    1 f(ij.+l) r((--l) F (-'+1 f'. a.' " . ) "" [ u(l-u)-lU(u) du r < ~ > i .... . , ,... , ,~ , 11 , x , y o

    ':}. ;

    .: . . : ' 1 ' f 1 ' -Io U(u)du+

    0 (1-u)-

    1U(u)du

    f(ot) CCX-ot.-1) F (,..., A a' " 1 ) ( 1(1-u):-lU(u)J.u r < J -1 > 1 V' ' , ... ' j.J ' u - , x ' Y = J, 0

    and hence 'i:

    1 olF1 (ot+l, ~ ~ ~, 'J, x, y) ~ (~ --l)F1 (O{, p, ~', i, ~, y ) - ( ~ -1) Fl ( o< , p , ~ ' , ;r -1, _x , y) = 0

    In this computation, we used the. identity r(x+l) = x r l. Then

    lim U(u) -= 0 , . lim U(u) 0 u,+0 U' 1-0

  • - 68 -

    Therefore, integrating both hand members of (10.7) from 0 to 1,

    we get

    (ci -l)J 1

    u-1U(u)du+ (ot.+1-~)J. 1 (l-u)-1U(u)du 0 0 . .

    r1 . 1 p 'y Jo (1-yu) - U(u)du '"" (1 . . l

    + px Ji (1-xu)- U(u)du+ 0 .

    Thia. means that

    gral representations . In the proof of Theorem 10.1, we expanded

    -oc O( -ot' the kernels of integrals (1-xu-yv) and (1-xu) (1-yv) into

    the corresponding double power series in u and v. If we expand '

    these kernels into series of different types, . we shall obtain some

    other expressions of Fp F2, F3 in fonns of certain series. We

    shall present such an expression for Fl. In (10.1), change

    variables by

    u ... s(l-t), v "" st . This maps the square 0 ~ t, s ' 1 into the triangle u, v, 1-u-v ~ o. If s .. O, then u ~ 0 arid v ; 0 for ' , .' . ' . 0 ~ t ' 1. Hence the side: s = O, 0 ~- t ' 1 of t:he square is mapped onto a point': u ... 0, v = 0 . Otherwise. the mapping is one-to:--one. (Se~. Fig~

    I , ,I ; , '

    10.1.)

  • .. 69 -

    t

    u

    Fig. 10.L ;i, ..

    . t

    rep' >r

  • - 70 ..

    f(p)r(p')r([-(3-p') F = i (o

  • ;?{.. .: Barnes integral ~.:::;:;. ~ .;>tlLl..aLlOClS . In Section 4 (Chapter I) we

    iti~e,ved that the function F( o< , p, ~, x) admits an integral repre -;:)~~q,~ation of the form

    ?\Hr:; . ,'.;if; 't .; 0, -1, -2, , where B is the path given by . ');:~!\

  • - 72 -

    - 1 r . r.rrc~'+t> re >< >t 21ti BF\+t,p,~.x) r(t'+t) -t -y dt' ) rC')f(6'> ( , , ) (11. 4 r

  • - 73 .--

    . r(oc+t)C(~) F(o

  • - 74

    (11. 5')

    ( 1 )

    2 s s r(ot+s+t) r o(' , p, t1 t )x O( y -o< 'F 2 (o

  • : '.:. ~

    - 75 -

    (11. 7) F(o

  • .. 76 ..

    In this manner, we obtain

    . Res +cs> = r 1 a nd larg(-x)I < 7t . ' . . ( ' To prove this, it is sufficient I ."._!

    to estimate .. .. ' ' ' ' : :' ' . . -

    :_ rclJ(~(~r!)+s) rs~.

    on OR. We can not use ' stirli~g Is formula dir~,ctiy~ because

    Stirling's formula is not ,valid in a sector 'larg{~)-~l < f. . To , 1 ' ; ~ ' ; , l_ .' , , , ,',; :' I

    1

    ~ , .!

    ~void this difficulty, ' iet us .use the formula .. :1 ! .. _ .. , . . - .. ' . ' ~~- : . ._.~ .. . .

    ... "IC . r

  • - 77 -

    of 'f'(s) on DR.

    Exercise 1. Complete the proof of Theorem 11.4. . ;

    At :.the:. ~nd of Se~tion 5, we mentioned that t:Wo functions

    (-x)F(' ,9'+1-()' ,o t c -p -n, t - -o('-n, t - -p' -n. For example,

  • - 78 - ;

    R f(ot+s)C(oc:'+t)r(G+s)C([i'+t) f(- )r(-t)(- )s(- )t es rci+s+t) s x y s = o( -m {( ~ . . ta-~1 -n ~

  • ., .- 79/ -

    J~;~ ;. Systems of partial differential equations.. The function . .. .. . . . . . ' - ... .. .. . .

    :'.rt~: , ~, r. x) satisfies the differential ~quation

    x (1 - x)y" +: [ t .- ~o( + ~ + 1):2C.l.Y ' ; .:. .ot..p y .. ~ 0 .; ' ' ' '.... " " ,. I ; - ., ' . ,. ' ' ~ : _,

    11; ~.I!: . ... ' , , ' ': : : I , , , '

    ft.)}~ hypergeometric functions Fj (j ' = 1, 2, 3, 4) also satisfy

    j1y13_tems of pa,~tial . diffe.rentia.l eq.up:;:ipns . ,,,._., ... .- : .: It ' . . . .

    . i :

    : THEOREM 12.1: F1

    , F2 , F3 and F4 satisfy respectiyely : . .'. . j . : ~. I .' , ,. . ~ . ., : . .

    :,'following systems of partial differential equatfons ': . ~ . :: .: . . :

    '.~c:i2 ~ 2 > ,.:::\:,;,_.'-' .

    { : ~~ ~: ~= ~~ :;; :: ; : ~::~:::: ~ ~ ~ ~~~ ~:~i~~ ~\.'o . { x (1-x)r~~ys+( 1,-(ct+~~l)x] ~ .. -~yq : .. ~~z";.: '({ < .~."_

    y(l-y)txys+[ t-(6'+p'+l)y']q-~xp-'~' z = o ~,

    {

    x(l-x)r+ys~i'-

  • - 80 -

    ) . (oCm+n+l>(~.m+l) r2 o, .

    . .. . , .~2 .. - 1ce+r+ o() F 2 "'.' .o ,

    {

    8(B+t+1-l}F3 - x(0+oe)(8+p)F3 = 0 ,

    . ,,. ,(8+t+r.-1)F3 - :y(g>+oe')('+ 1J'F3 = O ..

    ece+r-1>F4 -xCB+,+Q(}(8+,+ tJ>F4 ... o ,

    'fCCf+ '-l)F4 -y(8+'+~)(8+t+P>F4 ""o . From these the systems (12.2), (12.3) and (12~4f can be easily

    derived. ~ .. . : I. "

  • 81 . -.

    ~;: : . Systems of total differential equations. Before '~tudylng the

    ;~i: :. hsystems of . par~ial ~~ff~;enti~l. equafions which . Fj :: satisfy, : ~e

    [:~h~ll explain some aspe_cts of total differential equations. ;~{ ~ .. [.~i\,; ~i' Let us co_nsider. a .system of tota~ ,differential eqli.ations : of

    ;t~~~ form:

    ~zj = .~j '.~'.~ ,z1 ' . ~- ,zn>,~ + gj (x,y ,~1 , . ,~zn~~-! . . . . . ... . . . . - . . .. . .,.. (j i,l,,n),

    ,.

    ,~~'.~~ere f 1 ; .. , f 0 , g1 , , g0 are functions of x, y, z1 , ;~~0::~ ~hich ai;e .. define,d .and co~t;inuously ._differentiable in a domain ::i~: ' n+2 ~~;.~- contained in ,R _If w~ .. C?OQ.sider ; x,:_: Y - ~s independent ~#"~ \t:.~riabl~-~: and ,~ 1 , : , zn as : dependent variables (i. e ; unknown

    ~'.:tti~ceicms of. ;_ ~, y)., ' then the system. (13 .1) is equivalent to the . . f,~~~: :t . :system . -. :, .:: r= ..

    .:- . ~ . ~ , - _i :. ~ ....

    . . , lzj/ax == . fj (x, y, z1 ,_ . ~, z0 ),

    ~zj/'t>y ,;;. ; gj(x,y~ z1; ~ zn)

    . ;, ; . . : ... .. _i, ,: .. :

    ... ~. ; : ! zj .. 'j (x,. y) .... (j .a:; l ~ . i . :. , ' n)

    I 0)

    . ( . ~ : ' ,. . l 1" ,. l '

    :~.~hich is defined and twice co~tintlousiy differentiable in a domain' ;;,,; . '' 2 . :,D c R are independent ilii~j; :of the order of i .differ~nti,.ation with : respect to x . arid "; O

    y: i.e. ;

    . i).2 'f ~ ~ : ' ~2 'f j .~ x ~ y .. a y a x (j - 1, n)

  • - 82 -

    From (13.2) we derive

    (j = 1, , n).

    Hence

    and hence n ._, n

    (13.3) ~fj/"dy + Ei_gk 'd fj/dzk "" ;>gj/ax + ~ fk dgj/dzk

    (j = 1, , n).

    This condition holds for (x, y, ".

    there exists a unique solution of (13.1) which is defined and

  • - 83 -

    .' .twice contiriuously differentiable. in a neighborhood of (x0

    , y0

    )

    ~nd satisfies the initial condition

    .(13 .4) (j 1 > , n)

    DEFINITION 13.1: A system of total differential equations of

    the form (13.1) is called completely integrable if (13 . 3) is satis-

    . tied. .. :~

    !..

    We shall now consider the case when f. and g. are all J J

    holomorphic in x, y, z1 , , z . n

    THEOREM 13.2: Suppose that fj and gj are holomorphic at

    ,

  • .. 84 '

    Moreover if we put

    w ... (.() 11' , ln [~.11 z = . Q = ....... . z n

    we can write the system (13.5) as

    dZ = J2. Z

    The condition of integrability becomes

    .: zk[-af.k/jy+ ~ f 'kghk] = t zklag .k/ax+ t_g.hfhkl. ks:1 J h=l J k=l l: J h=l J J

    j = 1, n

    Since these are identities, we obtain

    (13.6) n

    ~fjk/;,y+ ). f.h~k = h=i J

    n ()g .. ktax+ _Lg.hfhk

    J h=l J

    j, k = 1, n .

    As it is well known, solutions of linear ordinary differential

    equations exist in an interval or a domain where coeff.icie nts of

    the equations are continuous or holomorphic. The same fact holds

    for a system of linear total differential equations. We shall

    state such a result for the holomo;rphic case.

    THEOREM 13.3: Suppose that fjk and gjk are all holomorphic

    in D and satisfy the condition (13.6) there. Let l

    zj .. 'f j (x, y) (j "" 1, ... n) , be a solution of (13.5) which is holomorphic at (xo, Yo). Then

  • >.1'f111 ..... ,'i'n starting at

    - 85 -

    are analyticatty continuable along any Eath in D

    It.is clear that the set of all solutions of a system of

    linear total differential equations is a vector space. Since

    solutions are uniquely determined by their initial values, the

    dimension of this vector space is n

    iHEOREM 13 . 4: The solution space of (13.5) is an n-dimensional

    vector space.

    Consider n ~olutions of (13.5): ~.'I .; ' , -1 J. : ' : .. ' - ~ . '

    (j , k "" 1, p n)

    and the determinant obtained from these solutions:

    ,, 11 'f ln

    ..

    In the sawe way as for linear ordinary differential equation~) we

    derive

    n . n dA ss ( L_fjjdx + ~g . dy)d

    jml j""l JJ

    or n n

    d(log A) ... ,L.fjj dx + L,gj. dy jml j..:l J

    THEOREM 13. 5: We have

  • - 86

    . .

    . (13. 7) J(x,y) n n

    A(x ,y) ... '1(x0,y 0) exp[ L fj. (s, t)ds+ 2:.g .. (s, t}d:t]; 0

    ,y0

    ) J=l J J=l JJ

    We have completely similar theorems for the general system of

    total differential equations

    m ~zj ~ : 2',fjk(x1 , ~ ,x ,z1 ,. ,z )dxk ' j l, ,n . . . . kl .. m n .

    Exercise 13 . 1: Show th~t, if we write' ' system (13 . 5) in the

    . form

    dZ "" Jl Z ,

    then the integrability condition (13 .. 6) can be written as

  • - 87 -

    Transformation of the differential equations for F. J

    into -systems of total differential equations. We saw that the systems

    !: .

    :,~f partial differential equations satisfied by F j are ~ritten in

    ;the form

    {

    Al r + A2s + A3t + A4p + Asq + A6z

    Blr + B2S + B3t + B4P + Bsq -i- B6z

    = 0'

    .where Aj and Bj are holomorphic in . x and y. A3 --= B1 = 0 for

    F1 , F2 and F3 . These equations are solv~ble with respect to r

    ' and t so that we obtain

    where a. and b . are rationa~ functions in x . and y. J 3 -

    Let us differentiate-- the first equation with respect to -y : ' -

    ;J r I";) y . = a 1 J s I ~ y + _s ~a 1/ ~ y + a 2 d p I 'd y + p d a 2 / 'd y I I

    Thus we derive . ' ; ' .. :(14.2) ,-s/';)x ;1- a 1ds/'dy ... . (da1/;;)y+.a2)s + a 3 t + . C>a2t~y)p

    + (aa3/dy+ a 4)q + (7'a-,/'dy)z -'

    Similarly, by differentiating the second equation of (14.1) with

    respect to x, we obtain

    (14. 3) b1 ~s/"ax+ ~s/~y = (()b1/dx+b3)s+b2r+(db2/~x +b4 )p + (3b

    3/ax)q+ (;)b4/ax)z.

  • . the left-ltand members .. o{ (14 .2) .and (14. 3) are linear forms in

    .} _s/":i!' . and . "'ds/''dy ~ The determinant obtained from the coef-

    .ficients of these linear forms is . . .

    We distinguish two cases:

    Case I : 1 - a1b1 0 ,

    Case II: l a1b1 ~ 0

    Case t: tri th~s case, we can eliminate dS/';>x and ' 'ds/"'Jy from

    (14.2) and (14.3) at the same time to obtain a linear relation

    between r, s, t, p, q, z :

    (14.4) c1r + _c2s + c3t + c4p + c5q + c6z O. Suppose that from (14.1) and (14.4) we derive

    whet~ p and olj' . . J definition w~ have

    r O(lp + cC.2q + 3Z '

    8 ~1P ~ ~2q + .P3z ,

    t f 1p + 12q + Y3z ,

    Yj are rational functions in x and y. By

    ~~/ax P. , ~ .z/3y ~ ,

    and herice

    Ctp/t)x .

  • - 89 -

    Thus we obtain a system of total differential equations:

    (14. s) { :: : dq =

    p dx + q dy ,

    (ot.lp+ ol.2q+oc.3z)dx + (~lp+ ~2q+ e3z)dy,

    ( ~ 1 p + $ 2 q + /3 3 z ) dx + ( ~ 1 P + ;r 2 q + ;y-3 z ) d y

    In vectorial notations, we can write this system as

    z 0 J.x dy z

    d p ~ .x3dx+~3dy ~1dx:+P1 dy ~dx+P2dy p q p3dx+t3dY ~1dx+a1dy ~2c1x+12dy q

    In general, this system may not be completely integrable. If this

    system is completely integrable, then the solution space is a

    three-dimensional vector space.

    Case II: In this case, we can express -as/ ~x and ()s/6y as

    linear forms in r, s, t, p, q, z:

    c1r+c

    2s+c

    3t+c

    4p+c

    5q+c6z,

    d1

    r + d2s + d3 t + d4p + d5q + d6z

    where coefficients are rational functions of x and y. Inserting

    : (14 . 1) into the right-hand members of these equations, we 6t;-L

    f . ~StdX = o

  • - 90 -

    dq/C>x = s ,

    as/'Jx = 0(.1 s +otzP +

  • - 91 -

    System for F2: The function F2 satisfies (12.2). Hence

    a = 1

    y/(1-x) , b1 = x/(1-y) ' . 1 - a 1 b1 -/J 0 . .

    System for F3: In this case, from (12 . 3) we derive

    al ... -y/x(l-x) ' b = l. -x/y(l-y) ' l - a 1b1 t 0 . System for F4: From (12.4) we derive

    a1 = 2y/(l-x-y) , b1 "" 2x/(l-x-y) '...,. 1 - ~1b1 1= 0

    This shows that (12.2), (12.3) and (12.4) belong to Case II.

    Sy using the procedure of deriving systems of total differential

    equations, we obtain such systems for F1 , F2 , F3 . and F4 . It can

    be verified also that these systems for F. are all comp.letely . . J

    integrable. The proof of this fact will be left to the readers .

    . :?hus we come. to. th.~ fo11owin~ conc1usio~.

    THEOREM 14.1: The dimension of the sol~tion space of the .: ::_ .

    l_,::: . . system for F1 is three, while the dimensions of the s~lution

    . ,: .. . _.,.:, .. . . ' spaces of the syste~s for F2 ,.F3 . . =- .. . .

    and '

    are all four~ .

    Let us examine the system for , . F. in more detai,1. - Differen-. . . . . . .:. l. '.>

    tiating the first equation of (12.1) with respect to y we obtain

    . ,or r I ~ :

    .x(l-K)cls/C>x ,+ y(l~~). c}s/Jy! LY+l~(o(+p~2)x}s .-~yt - (oC+.l)~q = 0 . ,' I '. . '. , . - - ' ' ' '

    Similarly, differenti~ting the second equation of (12.1) we obtain

    .y(l-y)~s/~y+x(l.-y)as/ax+ [+1-(0(.+~'+2)y]s - ~'xr - (+l)p'p = o.

    To eliminate as/~x and ds/~y from these two equations, we

  • - 92

    multiply .the fir~t equation by 1-y and th~ second equation by

    1-x and we subtract one from the ocher. Then we get

    [ (+ 1) (1-y )- ( +e+2 )x (1-y) -

  • - 93 . .,; .

    (yx)s by ~q-~'p, we obtain the third equation of (i4.7).

    Exercise 1. Prove that

    (x-y)s ~'p +pq = 0

    has a solution of the form

    where f is an arbitrary function. In particular, if we take '1-( ) = (cx,m) l m

  • - 94 -

    This implies that any solution of the system for F1 is regular

    analytic in

    c ~ c-{x. = o~ u {x = 1} u/y = o~ v{y = qu{x = Y) However, solutions are in general multiple-valued functions .

    y

    x /

    In the same way, if we write the systems for F2 , F3 and F4 in

    the form

    r = a 1s + a2p + a3q + a4z ,

    t = b1s + b2p + b3q + h4z

    ~s/ ~x = '1 s + o

  • - 95 ..

    y . y . y

    I\_

    Remark 1. A natural compactification of the complex plane C . .

    is the Riemann sphere which is the unit sphere of real dimension

    two, i.e. s2 This can be regarded as the complex projective line

    P1 . On the other hand, C x ( has two natural compactifications.

    The projective plane IP2 is one of them. The other is e1 x 1P1

    Both of them are compact c.omplex manifolds, but they are not bi-

    holomorphically equivalent. This means .that there is no biholomor-

    phic mapping from one into the other. They have, however, a common

    modification and hence they are equivalent in a certain sense.

    Since coefficients of the systems for F. are rational in x and J

    y, these systems are well d~fined in the compactifications of

    ( ~ . In various cases, a choice between two compactifications of

    C l(. C is not a serious problem.

    Remark 2. The Gauss differential equation

    x(l-y')y" + [-(+~+l)x}y' - 'Jlpy = 0

    is ~educed to a system

  • - 96

    ( 14. 8) . f yz'' z/x , l :c _ xi3 Y + (-1_-_r + r-1)(-8-1] z

    x-1 x x-1 '

    if we put

    z = xy' .

    Note that

    z ' xy" + y '

    and hence

    x(l-x)z' = x[x(l-x)y'I + (1-x)xy'

    a:o(~xy - LY-(o

  • - 97 -

    +cyaz/dy] = yt + q . . *Y

    Therefore, in a way similar to that in the case of the Gauss equa-

    tions, we derive from (14.7) the following system:

    z z

    d ~)z/)x = {A~x + Bgy_ +C dx +ok-+Ed(x-y)) x';>z/Clx , A y . x-1 y-1 x-y

    where

    y ~z/ ~y y ;Jz/ 'dy

    A

    0

    0

    0

    O

    0

    0

    E m 0

    0

    1

    1-1+13'

    -(3'

    0

    0

    0

    -~ . (!>'

    0

    0

    0

    0

    0

    0

    , B

    D =

    0

    0

    0

    0

    0

    0

    0

    0

    0

    0

    l

    -~

    1-)'+ ~

    0

    0

    -p i-oc-jl' -

    ,

    ,

    Note that the coefficients of this system have simple poles at

    x = 0 , x = 1 , y a 0 , y ... 1 and y .. x

  • 98

    CHAPTER III

    Monodromy Group of the Gauss Differential Equation

    15. Euler transform. In Section 3 (Chapter I) we derived the

    Euler integral representation of the function F(oc , p, y, x):

    (3 .1) 1 _ rcr> f "-1 _ r-~-1 _ --cc

    F(o(,e, '(,x) - r(~)r(l-~) Ou (1 u) (1 xu) du.

    The kernel of the integral is -ar

    (1-xu) Dropping the constant

    factor r

  • - 99 -

    i . 4'-1 L(y(x)) = L((l - xu) ) f(u) du, c where

    L((l~~m)~-l) = x(L-x)[(,\-l)(A-2)u2 (1-xu).\-J] . ,\ 2 + rr-(O(+~+l)xJ[-()\-l)u(l-xu) - ]

    ' .' , ~ ' A.-1

    .. al~(l-xu) .

    Let us write L((l-xu)A-l) in the form

    .\-1 A-3 L((l-xu) ) "' (1-xu) H(x, u, ...\),

    where

    H(x,u, ~) 2 2 = (~-1) (~-2)u x(l-x} - (A-l)u[ t-(oe.+p+l)x] (1-xu) - 0

  • - 100 -

    L(y(x)) = o(l (l-xu)-2 [-(cc.+l)xu(l-u)+ (1-xu)(~u-~)} j(u) du c

    i --2 4-l o( [-(oc+l)x(l-xu) u(l-u)f(u) + (1-xu) (tu-f1)

  • - 101 -

    (i) the path joining 0 to 1 if Re~> 0, Re( - {J) :> O;

    (ii) the path joining -oo to 0 if Re~> 0, Re(o O;

    (iii) the path joining 1 to +oo if Re( -(J)> 0, Re(o0 .

    On the. other hand, Jacobi showed that the following three

    curves satisfy our requirements under certain conditions on the

    parameters:

    (iv) the path joining 0 to 1/x :

    (v) the path joining 1 to l/x ;

    (vi) the path joining oo to l/x .

    In order to find the conditions on the parameters that these three

    curves satisfy our requirements, we must examine not only the

    cor.dition (15.4'), but also the assumption that

    y'(x) =Jc

    y"(x) = L ~ ~-1

    ax (1-xu) '(u) du '

    . In deriving the conditions on ~ , l and C, we actually assumed .that the order of integration and differentiation can be inter-

    changed. This requirement must be satisfied by the three curves

    (iv), (v) and (vi). Note that we have

    d ff (x) J f '(x) ...L d; ja . F(x,u)du = . a i)x F(x ,u)du+ f' (x)F(x,f(x))

    Therefore, if F(x,f(x)) ~ 0, we get

  • - 102 -

    . .

    d Jf(x) Jf (x) ~ d; . F(x,u)du = ~x F(x,u)du

    a . a-

    If we have F(x,f(x)) = 0 with f(x) ~ l/x and F(x,u) = . ~-1

    (1-xu) (u) , then the formula for y' (x) is verified .. On the

    other hand, if we have F(x, f(x)) = O for f(x) = 1/x and 3 . ~-1 .

    F(x,u) = ~ (1-xu)

  • - 103 ...

    proved the following result:

    THEOREM 15 .1: If we put

    (15.10) U(x,u) u 6-1 Cl-u) 1-~-l(l-xu)-

    then the following integrals are solutions of trw hypf'rgeometr i. c

    differential equation (15.2):

    (15.11)

    FOl(x) t

    = i U(x,u)du 0

    . so F~0 (x) = 00 U(x,u)du

    F100(x) = J1tA U(x,u)du fl/x

    F 1 (x) = J. U(x,u)du .Ox O

    ~1/x

    F i(x) = U(x,u)du tx-

    if Rep >0, Re(!-/3) > 0 ,

    if Re,,>O,Re(oO '

    if Re('(-{0>0, Re(oO,

    if Re~ > 0, Re Q( < 1 ,

    F1 (x) = J: 00 U(x,u)du if Re(c.X+l-,r) > O, Re O. Then x is in

    the upper half-plane and l{x is in the lower half-plane. More

    precisely, we make the convention:

  • 0

  • - 105 -

    -- branch of U(x,u), it is sufficient to det~rmine the argument of

    ~ each factor u, 1-u and 1-xu of the function U on th~ paths

    of integration. We shall. fix those arguments in the following

    manner:

    {i) For F 01' arg u = o, arg(l-u) = O, - 7t ~ a rg ( 1-xu) ~ 0 ; "

    (ii) for FooO' arg u ,., 'lt', arg(l-u) = 0, 0 ~ arg(l-xu) ~ 7t ;

    (iii) for F loo' arg. u = 0, arg (1-u) :.: -'It", -7t ~arg(l-xu) ~ 0;

    (iv) for F ol' x

    -7t ~ arg u ~ 0, o-a arg(l-u) "'1t , arg(l-xu) = O; (v) for F 1, -1t ~ arg u "' 0, 0 ~ arg(l-u) ' 1l7, -lt ~ arg(l-xu) ~ O; . 1-

    x

    (vi) for Fl , -1t ~ arg u ~ 0, O" arg(l-u) ~ 1'C, arg(l-xu) = - 7t . -oo x

    As we have shown before, the integral F01

    (x) is the Euler

    integral representation of the function F(, p, r, x) multiplied

    by a constant:

    rep> r< r-A > re r>

    In other words, we have

    F () = rcl}>rF 01 x r < "> - , ,.. , 1 , x If we make the change of variable

    u .. v(v-1) -l (1 ~ v ~ O) ,

    the second integral F~o becomes

    So . -1 ~-1 - , -1 r-~-1 -1 :..' -~v

    F040 (x) = [v(v-1) ] [l-v(v-1) J [l-xv(v-1) ] 2 1 (v~l)

  • - 106 -

    Let us suppose that arg v = 0 and arg(l-v) = 0 as v goes

    from 0 to 1. Then, since arg u =7t, we must have

    -1 . 1ti. -1 v(v -1) = e v(l-v)

    and hence

    [v(y-l)-ll ~-1 ... e1d..((J-1) "~-1 (i-v)-S+l .

    On the other hand, arg(l-u) = O, -1 arg(l-v) = 0 and 1 - v(v-1) a (1-v)-l imply that

    .Cl - v(v-1)-11 cr-s-1 "" (1-v) r+p+1.

    If we suppose that

    (15.13) 0 ~ atg[l - (lx)v] ~ 7C,

    then we get

    -1 -0(. -oe. 0( (1 - xv(v-1) ] = [1 - (1-x}v] (1-v) ,

    since

    -1 . . ._l l - xv(v-1) = (1 - (1-x)v] (1-v)

    and

    0 ~ arg(l-xu) ~ 1c

    The assumption (15.13) is justified by the convention: -1t. <

    arg(l-x) < O. Note that we assumed at the beginning that 0 <

    arg x < TI:. Thus we have

    J 1 1Ci

  • .. 107 -

    1d(~-1> rca > r F000 (x) = e r(Q(+p+l-) F(ot, (3,o rn-"'~ -1& =e . rp-+l) (-x) F(~-)'+1 , ~,/J-.+1,1/x)

    . . 0

    ) -1 ~-1 -1 Y-~-1 -ct. 1-x F 1 (x) = [x (1-(1-x)v)] [1-x (1~(1-x)v)] [l - (1 - (1-x)v)] (-7)dv 1- 1 . x .

    Jl -1 . /3-1 ri -1 'r-P- 1 -CK _:1

    -= 0

    [x (1-(1-x)v)] [e x (1-x)(l-v)] [(1-x)v] x (1-x)dv .

    e id. f 1 xl-Y (1-x) r-o

  • - 108 -

    e7ti.(-p-l) f(l-~)r(r-O)(l-x)r-~-~F(Y-~ -~ 1-~-~1 1-x) rcr--~+_1> . , ' ,

    5 -1 -1 (3-1 -1 -1 (-~-1 -1 -o( -2 -1 . F 1 (x) = (v x ) (1-v x ) (1-v ) (-v x )dv -~ . 1 . x f,

    1 -1 -1 (3-1 Jti -1 -1 r-p-i -ti . -1 -at -2 -1 a: (v x ) [e (1-vx)v x ] [e (1-v)v ] v x dv 0 . .

    f,1 t-~ ~~ 1 - t-~-1 u(t+oe-p-1)

    x v ("v) (1-xv) e dv 0

    ~ eiti.(r+Dl.-~-l) C(Q{+~(I~r51 -0(> x1-~F(p-r+l,+1.;.,2-t,x).

    We shall summarize these results in Table 15.1.

    The integral of the form (15.1) is transformed by the change

    of variable

    u = l/f

    into

    (15 .14)

    where

    'l'< 5) .= -(~, ~ )1-A t

  • 109 -

    is called the Euler transform.

    The integral b . . .

    J 8-1 ~-(1-1 '-ot a u (1-u) . . (1-xu) du , \ where a, b = 0, 1, 1/x, &0 , is transformed by u = 1/5 into

    d . .

    J ~-r r-~-1 -~

    const. c s (1- s) (x- s) ds where c, d = 0, 1, x, oa . From this we derive the following

    theorem.

    THEOREM 15.2: If we put

    then

    (15.15-1) f: ECx, S )d5 = const. (-x)-"F(o

  • .. 110 -

    are all solutions of the hypergeometric differential equation

    (15.2).

    Supposing that Im x > O, taking the paths shown in the

    figure given below and fixing branches of E(x, 5), we can verify

    the formulas (15 . 15).

  • .. 111 -

    16. Connection formulas and monodromy group of the Gauss dif-

    ferential equati~ Suppose that

    (16.1) 0

  • R

    We sha~l fix a

    this branch is

    - 112 -

    / /

    C1

    branch

    -r 0

    ul (x.,.., u)

    single-valued .on

    r

    of

    CRr

    ' ' '

    1-r

    the function U(x,

    and its interior.

    u) so that

    This

    condition is equivalent to the condition that arg u, arg(l-u)

    and arg(l-xu) change cbntinuously on the curve ~r Hence u1 (x, u) is uniquely determined by the conditions:

    (16.4) arg u =it' , arg(l-u) = 0 , 0 ~ arg(l-xu) ~7t

    on the segment c1 Under the assumptions (16.4), we observe that

    (i) on ~l' arg u decreases from 7C to 0,

    (ii) on c2'

    arg (1-u) starts from 0 and comes back to 0

    after taking negative values,

    arg(l-xu} changes continuously from a positive

    value to a negative value;

    arg u = 0, arg(l-u) = 0, and arg(l-xu) varies

    between -1t and 0 and takes a negative value

    at u 1-r ;

    (iii) on 12, arg u = 0 at u = 1-r and l+r, arg(l-u) changes from 0 to -:rt

    -1t: :.;; arg (1-xu) ' 0 ;

  • . 113 -

    (iv) on C3, arg u ::: 0, arg(l-u) = -'7t, - 7t 1i: arg (1-xu) ' 0 ; (v) on r-3, arg u changes from 0 to 7t and comes back to

    the initial value (i.e. 7t. ),

    arg(l-u) changes from -11: to 0,

    arg(l-xu) increases and arrives at the initial

    positive value.

    By virtue of Cauchy's theorem, we have

    i ul (x, u) du = 0 . CRr

    Let r tend to zero and let R tend to infinity, then

    and

    and

    lim J . u1(x,u)du = 0

    1j (j = 1, 2 , 3)

    lim f u1 (x, u) du = F000 (x) , cl

    lim I ul (x, u) du = F 01 (x) , . c2

    . lim ( u1

    (x , u) du = F ~ (x) le . J.O> . 3 .

    Thus .we obtain the following relation:

    F000 Cx) + F01 (x) +. F100 (x) = o

    Consider next three integrals F 000 , F 1 ox case, we shall use the following curves :

    In this

  • cl :

    -r-1 : c2 :

    ~2

    C3

    lJ :

    - 114 -

    a line -R ' u ~ -r, where R > r > 0 , i9

    a circular arc u = re , where - 1t~ 6 ~ arg(l/x) < 0 ,

    a line u :;;; f exp(i arg(l/x)) , r ~ f ~ lxt-1-r , .J -1 i6 -1 -1 .

    a semi-circle u = x +re , -n:+ arg(x ) ~ e ~ arg(x ) >: . -1

    a line u = f exp(i .arg(l/x)) , tx l + r ~ f" R , -i0 -1

    a circular arc u = Re , -arg (x ) ~ 6 ~ 7t:.

    Determine u1 (x, u) uniquely by taking

    -1C < arg u,.. arg(x-1) < O ,

    0 < arg(l-u) < n:..

    arg(l-xu) . = 0

    on the line segment c2

    . Letting r -> 0 and R --> +oo , we

    obtain

    and

    -2rcift F + F + -2.1tiotF 0 e ooO .l. e 1 = . ox xi)()

    -R

    >,, 1/x . "" ...

    1"3 "-

    Similarly we derive'---.._. ____ _____ ,.....

    F 01 + F 1 - F 1 = 0 lx- 0-

    1r

    Rlxl/x

    :).

  • Tab

    le 1

    5.1

    inte

    gra

    l arg

    u arg

    (l-u) arg

    (l-xu

    ) tran

    sform

    ation

    id

    en

    tificatio

    n

    ---------1

    . F 01 (x

    )'"' s 0

    0 [ .;c, OJ

    r(l3)r(

    -ft) F(ae ~

    y x

    ) r

    ' , ,

    0 0

    F.,0 (x

    ) ~ 7

    t 0

    [ 0' 1t]

    um

    v/(v

    l) e'1

    ti(i'l) f

    } (-x)-oCF(ot,ot-1+1,_~+l,1/x) -

    7t

    1 -

    ---

    1/x

    ..

    .. F l (x)=

    = ~

    [-7c:,OJ

    [ 0 ;n:]

    0 u

    =v

    /x

    -7ti~ r ro-ocL rrn+1f-r+1) , , -= e rroc+~+fl'-r+1) Fl(.,~' f3 ,ot+~+ ~ -r+1, 1-x, 1-y).

    As the first integral (18.8-1) has six expressions, (Theorem 10.3,

    p.65), each integral has also six expressions. Therefore, alto-

    gether, we have sixty expressions of solutions of (12.1).

    Remark 1: The integral (18.3) is transformed by the change

    of variable

    u = 1/v

  • ?able 18.1

    integral arg u arg(l-u) arg(l-xu) arg(l-yu) transformation

    1

    ~ 0

    0 0 [ -7r",0] [-7t,O] .

    0

    ~ 1t 0 [0,7r] [ 0 .7t] uv/(v-1) IO

    00

    ) 0 -~ [-7t, OJ [-7(,0) u l/v 1

    1/x

    t [ -'Jt, O] [ 0 ,7r] 0 [0,1t] uv/x l/y

    ~ 0

    [-7,0) [ 0 ,1t] . [-7t,O] 0 uv/y

    l/x

    ~ [-7t,O] [ 0, 7t] [-1t,0] [0,1t] u = l/[x+(l-x)v] 1

    l/y

    5. [-Jt,0] [ 0 ;rr.] [ -1t,-O] [ -lr, OJ u = l/[y+(l-y)v] 1 00

    } [-1t,O] [0,7tl -1t [O,ir] u l/(xv) l/x co

    ~ [-'JC, O] [ 0 ;re] [ -1[,0] -1t u l/(yv) 1/y l/y .

    it [ -7[, 7r] [ 0 ,iC] [ -l:', 0] [0,1r] u = 1/[y+(x-y)v]

  • Table 18.2

    integral arg v arg(v-1) arg(v-x) arg(v-y) restrictions

    1

    ~ 0 -TC [-7t,O] I . [ -1r, O] Re (p+f3' -r) > -1 ! Re(t - cx) > 0 0 0 I

    l -1t -7C [-'It:~ O] ,.

    [ -7C,O] Re(~+~' -1) > -1 I ! Re> 0

    60 ! 00 I

    ~ . 0 0 [-1t,O} [ -7t,OJ Re(1-) > 0 Re I( > 0

    I I . ~ x I I Re(/3+~' -)') > -1 ) [ 0, n:J I [ 0, 1C} (-7,0] [O ,211:] i I ! Re /3 < 1

    0 !

    y j

    ~ [0,7&] I [ 0 ;n:] [ -lt, 1C] [-7t,0] Re(~+~ -r) > -1 Re /!,' < 1

    0 i

    x . I ~ [ 0, 11:] [ 0, 7t} [-'it,O] [ 0,27ti

    Re (If-)> 0 I

    I Re fJ < 1 1 I

    I y ! Re(Y-ot) >0 ~ [0,Jr] [0,7t'] [ -1t,lt] (-7C,O] Re~'< 1 ' 1 I i 60 I ! I Re~< 1 i [0,'!r] ' [ 0, 1t] [0,1t] [ 0 '211:} Re Pi. > 0 x

    . co Re /3' < 1 I [ 0, 1t] i (0,7['] [ -'7t, 1t] [ -7r ,O] Re ot. > 0 y

    y Re ~

  • - 133 -

    into r V(x, y, v}dv , where

    ~ B' -i Y-cc-1 -/3 - a V(x, y, v) vt'' (v-1) (v-x) . (v-y) P ,

    and

    c l/b , d l/a

    Thus we derive ten integrals of V(x, y,v) from (18.8-1}""(18.8-

    10). These ten integrals are solutions of (F1).

    THEOREM 18.2: The integrals

    . fd ,....'2'-J J~Ol~l -A --' (18.15) c v P (v-1) , (v-x) l"'(v-y) dv are solutions of the system (12.2) under the respective restric-

    tions given in Table 18.2, where c and d are any two points of

    O, 1, x, y and oo The paths of integration are taken as

    indicated by Fig. 18.4 and branches of integrand are determined

    as indicated by Table 18.2.

    Fig. 18.~

    In Fig. 18.4 and Table 18.2, we assumed that

    { 0 < arg y < arg x < 1C , 0 < arg(y-1) < arg(x-1) < 7t:

  • - 134 -

    Remark 2. In order to guarantee the convergence of the

    integrals (18.3) and (18.15) we must assume certain conditions

    on the parameters orcoe>r F (a(~ A' > . . r 1. ,,..,,.. ,y,x,y

    if none of ae. , X-o

  • - 135 -

    I divergent integrals is essentially based on. the concept of

    analytic continuation. For example 1 f 0 U(x, y, u)du

    is originally defined for 0

  • - 136 - .

    19. Connection formula and monodromy representations for the

    system (12.1) satisfied by F1 . In this section, we shall briefly

    explain connection formulas amcng solutions of the system (12.1)

    and monodromy representations for the system (12.1).

    To begin with, let us consider the ten integrals (18.15). We

    suppose that these ten integrals are well defined at the same time.

    This is possible under a suitable assumption on the parameters ,

    p , ~ ' and 't The paths of integration for these integrals are shown by Fig. 18.4. (Cf. Theorem 18.2, p.133.) We shall use

    the same idea as in Section 16 (Chapter III) to find connection

    formulas among solutions of (12.1). The basic idea is

    (i} to take a closed curve consisting of parts of these paths

    of integration and small circular arcs and large circular arcs sc

    that Cauchy's integral theorem can be applied, and then

    (ii) to let the radius of small circular arcs tend to zero and to

    let the radius of large circular arcs tend to infinity.

    By using various closed curves of this kind, we can find more

    than thirty relations among the ten integrals (18.15). For example:

    if we use the closed curve given by Fig. 19 .1, we get

    JO V dv + 1 ~ V dv + f, 00 V dv = 0 to 0 1

    It is known that the system (12.1) has only three linearly inde-

    pendent solutions. (Cf. Theorem 14.1> p.91.) This means that

    there are only seven independent relations. As these relations

  • - 137 -

    show, any three of the ten integrals (18.15) are not necessarily

    linearly independent. It can be shown that three integrals

    (vdv, f: Vdv and are linearly independent.

    0 1

    ~ .

    \

    Fig. 19 .1

    We shall now proceed to the monodromy for the system (12. :_).

    In Section 14, it was explained that the system (12.1) has the

    singular set which is the union of the five lines-:

    (19 .1)

    and that any solution of the system (12.1) is regular analytic

    in the domain

    (19 .2)

    (Cf. Section 14, p.94.) In the same. way as we defined the funda-

    mental group 1tl (D, x0) of D C - { 0> 1 J with the base point x0 in Section 6 (Chapter I), we can define the fundamental group

    TC l (J) , {x0 , y 0)) of ~- with a base point (x0 , y 0). Then

    taking a fundamental system of solutions of (12.1), we can define

    the monodromy representation with respect to this fundamental

    system in the same way as we defined monodromy representations

  • - 138 -

    of the Gauss differential equation (6.1). (Cf. Section 6,

    Chapter I.) The monodromy representation thus defined is a homo-

    morphism of 7t 1

    (JJ , (x0 , y 0)) into GL (3, C).

    It is known that 7t1(D, x

    0) is a free group generated by

    two elements. (See Section 17, Chapter III.) However,

    1t"1 (~, (x0 , y0)) is more complicated. It is true that 7r1 (oi@, (x0

    , y0)) is generated by five elements. To find these

    ~ive elements, consider a complex line in C x IC which passes

    through the base point (x0 , y0) and intersects with the five

    singular lines (19.1). This means that this line is not parallel

    to any of these five singular lines (19.1). Furthermore assume

    that this line does not go through four points (0, 0), (0, 1),

    (1, O) and (1, 1). These four points are. intersection-points

    between the singular lines (19.1). This complex line can be

    identified with the complex plane C. Then the intersection-

    points with five singular lines (19.1) are represented by five ..

    points A1 , A2 ; A3 , . A4 and AS on th.is complex plane. The base

    point (x0 , y0) is also represented by a point B on this .com-

    plex plane . The intersection of ~ and this complex line is

    identified with C. - {A1 , ~ ", A5}. The fundamental group

    1t.1 (C -,{A1 , ,As}, B) of _c - {Al' , Asl is a free group generated by five elements. Those generators are represented by

    five loops surrounding . A1

    , A ' 5

    respectively. Let us denote

    these loops by .l 1 , , ./. 5 . (See Fig. 19.2~) . It can be proved

  • - 139 -

    that '7r 1 (~, (x0 , y0)) is generated. by five elements correspond-ing to these five loops.

    Fig. 19.2

    Although ' n:1 (C - {A1 , , A5}, B) is a free group,

    1ti ( .!; , Cxo, y 0)) . is not a free group. - In other words, there

    are relation$ among the five generators. To see this, let us

    consider i. 2 and l. 3 . Note that the complex line intersects

    with the singular lines x = l and y - 0 at A2 and A3 re-

    spectively. Suppose that the base point (xo, Yo) is in a

    neighborhood of (1, O). The point (1, 0) is the intersection-

    point of the two singular lines x = 1 and y "" 0. More precise-

    ly speaking, suppose that (x0 ~ y0) is in the domain

    (19.3) 0

  • - 140

    loop can be considered as a circle s2 defined by

    i6 Sz X = 1 + (x0 -1) e > y "'" y O .

    The loop .I, 3

    can be deformed into a loop lying in the line

    x = x 0 , and this deformed loop can be considered as a circle

    83 defined by

    83 = x - XO , y Yoe

    18

    The product of s2 and s3 is a torus contained in ~ ' and

    82 and S3 are two circles on this torus. It is well known that

    52 and S3 are commutative on the torus. In other words, s2s3

    is homo topic to S3S2 on th~ to~s. Therefore, S2S3 is homo-

    topic to S3S2 in ~ This means that l.2 L3 is homo topic

    to l3 l.2 in . ~ Thus we conclude that c i 21 and [ .l 31 are commutative in the group 7r

    1 (,J, (x

    0, y

    0)). This shows that

    this group is not a free group. Similarly, . [ 1.1

    ] and [ .i 4]

    are commutative in 7t 1 (~, (x

    0, y 0)) . . There are other relations.

    For example, there are. _ relations among

    These relations arise from the fact that three singular lines

    x ... 0, y = 0 and x = y intersect at one point (0, 0).

    In order to compute the monodromy representation with respect

    to the fundamental system

    Jx V(x, y, v)dv , 0 . J y V (x , y, v) dv , 0 . .. .

    1 f V(x, y, v)dv O

    of the system (12 .1), let us first consider three loops l l' i 2 and l 5. The line y = Yo intersects with three singular lines

  • - 141 -

    x ... 0, x = 1 and y = x at . (0, y 0r, (l, Yo) and

  • - 142 -

    This is due to the fact that arg v and arg(v-x) change to

    arg v+ 27ri and arg(v-x) + 27t'i, but other arguments do not

    change after the analytic continuation along .li. Let us next continue the integral

    (19.4) f: V(x, y, v)dV along the loop li We must deform the path of integration so that x does not go across the path. Hence at the end of analy-

    tic continuation the path of integration becomes a path as shown

    by Fig. 19.4.

    0

    Fig. 19.4

    Note that the path shown by 19.4 can be further deformed to a

    path as shown by Fig. 19.5~

    y

    0 Fig. 19 . 5

    Taking the change of arg(v-x) into consideration, we conclude

    that the integral (19.4) becomes

  • - 143 -

    21ti~ 5x f y {e- -1) Vdv + Vdv . 0 0

    Similarly the integral

    50

    1 V(x, y, v)dv

    becomes

    x 1 . (e2~ -1) J Vdv + J Vdv

    0 0

    after the analytic continuation along ./. l. Therefore the circuit matrix along li is

    e 27t1cs-n 0 0 -2~ e . -1 l 0

    e-~-1 0 1

    In the same way we get other four matrices corresponding to the

    other four genera tors of 7rl ~ , (x0 , y 0)). By these ma trices

    the monodromy representation with respect to the fund~mental

    system

    f x Vdv, . 0 ...

    is completely determined.

    ~ y

    J Vdv, . 0

  • - 14!~

    20 . General solutions of the systems (12.2)~ (12.3) and (12.4).

    In this section, we shall derive general solutions of the systems

    of partial differential equations which are satisfied by F2, F3

    and F4

    respectively. We shall first consider the system (12.2)

    which is satisfied by F 2 :

    (12.2) {

    x(l-x)r - xys + [t c (ot+~+l)x] p - ~yq -Ill.~ z - 0 ~

    y(l-y)t -xys+ ri'-(oc.+p'+l)y]q-{3'xp -p'z 0 . Let us make the change of variable

    (20 .1)

    Then the system (12.2) is transformed into

    {20.2)

    x(l-x)r' -xys '+[ 2A+1-(2A+fl+O{+p+l)x] p' -().+Myq'

    -[(.A+p+oc.)(A.+~)-A(A+Y-l)x-1]z' = 0,

    y (1-y) t I -xys '+ [ 2fl+ )'' - (7p.+A+~+ ~'+ 1) y] q I - ~+~' )xp I

    -[ (.\.+f+o

  • - 145 -

    z z'

    The second case of (20.4) yields the transformation

    (20.5) t-r z x z'

    and the transformed system (20.2) is obtained from (12.2) by

    replacing Q(. ' ~ ' -~'. '( , . l ' by oC+l-t " f3 +1- r , ~., 2- '( , Y' respectively. Therefore, this system admits a solution

    (20.6) z' F2 (0(+1-l'_,. ~+1-~, Ji', 2-r, y', x, y).

    From this we obta~n a_ solution of (12 .2): 1-J . .

    (2 0. 7) z = x F 2 ( oc. + 1- , ~ + 1-t , f , 2 - l , l ' , x, y) Similarly, the third case of (20.4) yields a solution of (12.2):

    (20.8)

    and the four-th case of {20.4) yields a solution of (12.2): 1-1 1-r' ,

    c20.9) z = x y F2(+2-r-1, ~+1-r, 1J'+1-y', 2-r, 2-r, x, y) . . Thus we obtain the following theorem.

    THEOREM 20.1: A general solution of -the system (12.2) is

    given by

    (20.10) z = AF2 (Q(, p. f, r, J',x,y) + Bx1 -)'F2 (o+1-l,~+l-t,~' ,2-1, ' ,x,y) + cy1-r'F2

  • - 146 -

    (12.2)" by the change of variables:

    (20.11) x - 1/ ~ ' y = 1/71' ' Z = s~ 'ld. Z I

    In fact, by this transformation, the syscem

    (12.3) .. { x_

  • - 147 -

    representation in Section 11 (Chapter II). (See Theorem 11.3,

    p.74.)

    Let us now consider the system (12.4) which is satisfied by

    F4 :

    {

    x(l-x)r - y2t - 2xys + [r-(+~+l)x]p - (D(+(i+l)yq -lz = 0 , (12 .4) 2

    y(l-y)t -x .~ - 2xys + [ r' - ~+~+l)y]q - (a(+~+l)xp - o. and }J-- by

    (20~16) A. (X + r -1) = O, )A(p.+ t'-1) = 0

    The equations (20.16) yield

    . { A.= 0 ,

    f- = 0,

    {A=l-t, {.A=O, \ ~- o , fo = 1- r', {

    ~-1-1,

    f'-= i- r'.

    Hence in the same way as we - derive~ Theorem 20.1, we obtain the

    following theorem:

    THEOREM 20.3: A general solution of the system (12.4) is

    given by

  • - 148 -

    (20.17) z AF4 (ot, p, Y, 't'~x,y) 1-1. R. I +Bx F4 ((l(-i+1,,..-r+l,2-~1 ,x,y)

    + cy1- r' F 4

    (.- J'+l, 13- r'+1,1,2-r' ,x,y)

    + Dxl-yl-t'F4 (o

  • - 149 -

    21. Euler transform in double integral. In the previous sec-

    tion, we found four linearly independent solutions for each of

    the systems (12.2), (12.3), and (12.4). However, we would be

    unable to calculate the monodromy representations of those systems

    with respect to these fundamental systems of solutions. In

    Section 19, we explained how to use the simple integral represen-

    tation of Euler for computing circuit matrices of the system

    (12 .1) which is satisfied by F 1

    A similar method based on the

    double integral representations of Euler may yield monodromy

    representations for the systems (12.2) and (12.3) which are

    satisfied respectively by F2 and F3, although, to the author's

    knowledge, nobody has ever tried to compute the monodromy repre-

    sentations for the system (12.2) and (12.3).

    First of all, we must generalize Cauchy's integral theorem.

    Such a generalization was given by H. Poincar~ as follows: Let

    f (x~ y) be a holomorphic function of x and y in a domain D.

    Let S (CD) be .a closed smooth surface of real dimension two.

    If there exists a set V of real dimension three such that

    (i) V C D ,

    (ii) the boundary of V is S ,

    then

    ff f (x, y )dx dy 0 s

    In Section 10 (Chapter II), we derived the double integral

  • - 150 -

    representations of Euler for F1 , F 2 and F3 . (S" e Th ''' rcm 10.1

    on p.61.) We showed that the three integrals

    (21.1) J J u '3-lv 13' -l (1-u-v) r-,-~, -l (1-xu-yv) at du dv , u,v,1-u-v~O

    and

    (2.13) r r ~1 ~-1 ~-~-$'-1 - -ot' } } u v (1-u-v) (1-xu} (1-yv) du dv u , v , 1-u -v~ 0

    are solutions of (12.1), (12.2) and (12.3) respectively, if these

    'integrals are convergent. The method of the Euler transform which

    was explained in Chapter III suggests, as in Section 18, that we

    may replace the domains of integration

    u, v, 1-u-v ? 0 and 0 .~ u ~ 1, 0 ~ v . ~ 1

    by other suitable domains of integrations so that the integrals

    thus obtained also satisfy the systems (12.1) , (12.2) or (12.3).

    Actually we cari find .more than four solutions of (12 . 2) and (12 .. 3)

    in this manner. However, any new solution of (12 .1). can not . be

    obtained by this method.

  • - 151 -

    22. Characterization of the systems of partial differential

    equations satisfied by F1

    , F2 , r 3 and F4 . Consider first a

    diferential equation of the second order

    (22 . 1) y"+p(x)y'+q(x)y = 0 .

    Suppose that x "" a (:f co ) ls a singular point of (22 .1). This

    means that x = a is a singular point of p(x) or q(x) or

    both. The point x = a is called a regular singular point of

    (22.1) if x =a is at most a simple pole of p(x) and at most

    a double pole of q(x). Suppose that x ""a is a regular sin-

    gular point of (22.1). Then the equation (22.1) can be written

    in the form

    (22.1 '} 2 (x-a) y" + (x-a)P(x)y' + Q (x)y = 0 ,

    where P(x) and Q(x) are halomorphic at x = a. As the general

    theory of linear differential equations guarantees, the differen-

    tial equation (22.1') admits a solution of the form

    y = (x-a/

  • - 152 .

    Let f 1 and f 2 be the exponents of (22. l ') at x. = a. If f

    1 - f 2 I integer, then (22. l ') admits two solution.:>

    (22 .2) - f 1 f 2

    (x-a) 'fi(x), - - ex-a) cr 2 Cx) ,

    where 0, the constant E may be zero. A regular singular

    point x a is called logarithmic if b = 1. A system of

    solutions (22.2) or (22.3) forms a fundamental system of solutions

    of (22.1'). This fundamental system is called a canonical system

    of solutions of (22~1') at x a.

    -Suppose next that p(x) and q(x) are holomorphic in a

    domain

    If we make the change of the independent variable:

    x ... its

    then the differential equation (22.1) becomes

  • - 153 -

    (22 . l") 2 2 2 1 1

    d y/d5 + (~--2 p(l/~))dy/d~ +-:;z;-q(l/~)y = 0. ~ l .

    We shall call the point at x ~ ~ a regular singular point of

    (22.1) if 5 - 0 is a regular singular point of (22 . 1"). The indicial equation of (22. l") at ~ = 0 is called the indicial

    equation of (22.1) at x oo The exponents of (22.1") at ~ ,. O

    are called the exponents of (22 .1) at x oo If x oo is a

    regular singular point of (22.1), then x = ~ is a zero of p(x) of multiplicity at least one and .a zero of q(x) of multiplicity

    at least two. Therefore, the differential equation (22.1) can be

    written as

    (22. l" f) 2 x y"+xP(x)y' +Q(x)y - 0 ,

    where P(x) and Q(x) are holomorphic at x = oo The indicial

    equation of (22.1"') at x oo is given by

    ~(A+l) - )..P( oo) + Q{oo) - 0

    The differential equation (22.1) is called an equation of

    Fuchsian type, if all singularities of (22.1) are regular singular

    points including the point at infinity. The Gauss differential

    equation:

    (22.4) . "+{ ! + +@-1+1} f + ot./J . - 0 Y x x-1 Y x(l-x) y

    is an equation of Fuchsian type which has three regular singular

    points at x - O, 1 and oo The indicial equations and the

    exponents at these three singular points are as follows:

  • - 154 -

    singular indicial equation exponents point

    0 A_(;t-1) + 't A= 0 0, 1-Y

    -

    1 ~(~-1) +(cl.+~ -l+l);\ = 0 o, t--P

    00 ).(~ +l) - (oc.+~+l)A + o(.~ ... 0 o( , , i

    If 1 f: integer, then

    F(oc.,~,y,x), 1-l'

    x F(cC-f+l, ~-+1, 2-r, x)

    form a canonical system of solutions of (22.4) at x "" 0. If

    "t-0{-ft:f integer, then '--, .

    F(,p, oe.+p+1-r, ~-x), (1-x) F('(-oc, 1-13, d'--~+1, 1-x)

    form a canonical system of solutions of (22.4) at x z 1. If

    c< - p :/- integer, then x-oeF(, D(-1+1, Ol.-p+l, l/x), x-~F(~, p-t+l, p-oc+l, l/x)

    form a canonical system of (22.4) at x = oo . . .

    A second order equation of Fuchsian type with three regular

    singular points at x = a, b and oo can be written as 2

    .,.1 .+ Ax+B , + Cx +Dx+E y = O ' (x-a) (x-b)y { )2 ( b)2 ' x-a x-

    (22.5)

    where A, B, C, D . and E are constants. The indicial equations

    of (22.5) at x = a, b and oo are respectively given by

  • lSS

    Let fl' f 2 be the exponents at x a, ~1 , ~ 2 the exponents at x '"' b and 't" 1 , T 2 the exponents at x = oo Then the

    relations between roots and coefficients of a quadratic equation

    yield

    r1 + f2 -1

    1 - (Aa+B) (a-b) , f1 r2 - (Ca2+Da+E) (a-b)-2 ,

    er l + ~2 -1

    = 1 - (Ab+B) (b-a) , dj_ '2 - (Ch2+Db+E) (b-a) - 2 ,

    't'l + '(2 .. - l+A , "tl t'2 - c .

    From these relations, we obtain

    {22.6) r i + r 2 + ~ i + 0 2 + 't' 1 + 't" 2 ... 1 This relation is called the Riemann or Fuchs relation.

    Suppost! that two points a and b (a + b, a :;. oo , b :! oo ) and six complex numbers fl' f2' ~1 , ~2 , 'C1 and :i:z satis-fying (22.6) are given. Then a second order differential equation

    of Fuchsian type with three regular singular points at x = a, b,

    and oo is uniquely determined in such a way that its exponents

    at x = a , b, and 00 a re f l, fl and If" l, If' l and L' l , -c

    2 respectively. In fact1 such a differential equation is given

    by

    (22. 7). 11+ 1 2+ 1 2 t (1- f - f 1- 0. - ~ )

    Y x-a x-b Y

    ( f 1 r2 (a-b) is-1 ~2 {a-b)) y

    + "C'l 't2 + x - a ... x - b (x-a) (x-b) = O

    The differential equation (22 . 7) is called Riemann's equation, and

    the set of all solutions of (22.7} are denoted by

  • ~

    - 156 -

    b

    (22. 8) x

    The notation (22.8) is due to Riemann and it is called Riemann's

    P-function. The set of all solutions of the Gauss differential

    equati~n (22 .4) is thus given by

    1 ca

    (22.9) y ... {o

    p 0 0 ct x

    1-)' r-- ~ p Now we shall explain the characterization of Riemann's dif~

    ferential equation due to Riemann. To do t his, for simplicity, sup -

    pose that none of the singular points a, b and oo is logarithmic .

    Let j be the set of all solutions of Riemann's differential equa-

    tion (22. 7). This means that j denotes Riemann's P-function

    (22.8). Let D be the domain D = S - ta, b, co}, where S is the

    Riemann sphere . Then .; (i) Every function in

    multiple-valued.

    ~ has the following properties :

    ~ is analytic in D, but it may be

    (ii) For every point x0

    in D, there are two branches f 1 and

    2

    of two functions . or one in g. which are defined in a neighbor-I

    hood of x0

    and which satisfy the condition

    ; 0 .

  • .. 157 -

    A function f defined i _n a neighborhood of x0

    is a branch of a

    function in J. if and only if f is a linear combination of f1

    and 2 .

    (iii) In a neighborhood of a, there are two branches of functions

    in ~ which are of the form

    where ~ and 11

    (x .. a)f1_ t1 (x) '

  • - 158 -

    lw = p

    An outline of the proof is as follows: Let y(x) be an

    arbitrary branch of a function in p which is defined in a

    neighborhood of x0

    . Then the condition (ii) implies that y, f 1

    and f 2 are linearly dependent and hence

    y(x) f1

    (x) f2

    (x)

    y' (x) fi(x) fz(x) = 0

    . y" (x) f'{-(x) f'2 (x) in a neighborhood of XO. This relation can be written as

    y"(x)+p(x)y'(x)+q(x)y = 0

    which is a linear differential equation. By using (ii), we can

    prove that p(x) and q(x) are holomorphic at x0 and that they

    do not depend on the choice of branches f 1 and 2 . This means

    that p(x) and q(x) are uniquely determined by $ . On the

    other hand, the condition (iii) assures that x = a, b and 00

    are regular singular points . of this differential equation. This

    proves Theorem 22.1.

    Remark. If some of a, b and oo , say a, is logarithmic,

    then the first part of condition (iii) should be replaced by the

    following condition: .,

    In a neighborhood of x = a there are two branches of func-

  • - 159 -

    tions in Jr which are of the form:

    P1 (x-a) '(

    1 (x) ,

    where Tl and

  • - 160

    f 1 (x,. y), f 2 (x, y) and f 3 (x, y) of functions in j

    1 such that

    fl f2 f 3

    r/: 0 at

    and that a function f (x, y) defined in a neighborhood of (x0

    , y0

    )

    is a branch of a function in 31

    if and only if f is a linear

    combination of 1 , 2 and f 3 ;

    (iii-1) for any point (0, y0), where y

    0 r/: 0, 1, 60 , there are

    three branches of functions in jl which are of the form:

    ~-1+1 . 'f 1 (x , y) ,

  • - 161 -

    three branches of functions in ).1 which are of the form:

    t-ct.- ~ -(y-1) . 't' 3

    (x, y) , -'f 1 (x, y) ' ,... -

    where t 2 and 'f'J are holomorphic at (iii-6) for any point (x0, oo), where x 0 :f 0, 1, oo, there are

    three branches of functions in '}1

    which are of the form:

    -~ ..... . -13' - - -Y. X1 (x, y), . y . X2 Cx, y), y . x.3cx, y), - - -where x1 , x. 2 and x.3 are holomorphic at (x0 , oo) ; {iii-7) for any point (x

    0, y

    0) , where x

    0 y

    0 .; 0, 1, oo, there

    are three branches of functions in ~ 1

    which are of the form:

    ~l (x, y) , ~2 (x, y) , (x-y) '/-,-~, f 3 (x, y) , where ; 1 , ; 2 and t 3 . are holomorphic at . (x0 , y0)

    Picard proved that, . if 1' 1 satisfies (i), (ii) and (iii) and

    some additional hypotheses, then ;. 1 is the . set of all solutions

    of the system (12 .1). We .. do not know clearly what additional

    hypotheses must be required. Picard as well as Appell and Kampe de

    Feriet gave the hypotheses in a _very vague fo~ in their works.

    The principle of the proof is the same as in the case of Riemann's

    equations. Let z(x, y) be an arbitrary branch of a function in

    3r 1 which is defined in a neighborhood of a point (xo~ _Yo> oB i Then the condition (ii) yields

    z zl z2 ZJ z zl z2 Z3 p P1 P2 P3 p P1 p P3

    - 0 , 2

    - 0, q ql q2

  • - 162 -

    z zl z2 z 3

    p P1 P2 P3 = 0 ,

    q ql q2 q3

    t tl t'l t ~ i where

    p "'Jz/'"Jx, q ")z/'dy, 2 2 r = ~ z/ ax , ..,. t d2Z/dy2

    and

    p. a tlz./-ax, . J J

    These three relations can be written as

    {

    r _ ... at1

    (x, y)p+ oc2 (x, y)q + at3 (x, y)z,

    s a ~1 (x, y)p + ~2(x, y)q + p3 (x, y)z' t =

    1 (x, y)p+ 1

    2(x, y)q+

    3(x, y)z.

    After very long calculations, Picard showed that this system coin-

    cides with (12.1).

    PROBLEM 1. Complete the work of Picard.

    Riemann's point of view ca~ be applied to the systems (12.2),

    (12.3) and (12.4). Goursat applied the same principle ,~o the

    systems -(12.2) and (12.3) . However, the author does not know

    whether or not the same principle has ever been applied to th

    system (12 .4).

    PROBLEM 2. Complete. the work of Goursat . l;

    PROBLEM 3. Deri~e th~ system for F4- (i.e. the system (12 .4)) by using Riemann's point of view.

  • - 163 -

    CHAPTER V

    Automorphic Functions, Reducibility

    and

    Generalizations

    23. Automorphic functions. It is well known that the elliptic

    function was discovered by Gauss, although he did not publish his

    discovery. Then Abel and Jacobi rediscovered the elliptic function

    independently . The discovery of the elliptic function opened an

    epoch in the history of mathematics. Indeed, this discovery had

    influences not only on the analysis, but also upon all fields of

    mathematics . Gauss also found a new function which is related to

    the elliptic function, quadratic forms, and arithmetico-geometric

    mean. This new function is called the elliptic modular function.

    It was Dedekind who rediscovered the elliptic modular function .

    This is the earliest example of automorphic functions.

    In 1972, Schwarz derived, from the Gauss differential function,

    automorphic functions other than the elliptic modular function.

    Ihen Fuchs tried to generalize Schwarz's results to more general

    linear differential equations of the second order. Stimulated by

    the work of Fuchs, Poincare founded the general theory of automor-

    ~hic functions and called a kind of automorphic functions the

    ~uchsian function.

    We shall now explain how we can deri ve automorphic funct i ons

  • - 164

    from a linear ordinary equation of the second order

    (23 .1) y"+p(x)y'+q(x)y = 0

    Suppose that (23 . 1) is a Fuchsian equation with regular singular

    point~ at a1 , a 2 , , an. Set D - S ~ {a1 , , aJ. Let 'fi (x)

    and ' 2 (x) be linearly independent solutions of (23.1) and let

    / ., : 7tl (D, x0

    ) -+ GL(2, C)

    be the monodromy representation of (23.1) with respect to the

    fundamental system of solutions

    (23.2) [ :~ l We denote by T the set of linear fractional transformations

    ( ) az+ b t z .. cz+ d (ad - be + O) , and define a homomorphism r : GL(2, C) ~ T by

    Then the composite map '1: of is a homomorphism of 1Ll (D, x0) into T. Let us denote by G the image of 7r1 {D, x0) under the

    map "t 0 r : G "" 1: o r (rr1 (D, x0))

    ' G is a subgroup of T. Let l be a loop in D '"lt x0

    rf

    ffLl = [: :].

    then fundamental system (23.2) becomes

  • - 165 -

    by analytic continuation along .1..

    Consider the ratio

    f(x) = f 1 (x)/ ~2 (x)

    Then f (x) becomes

    8 f1(x)+b~2(x) _ af(x)+b C'f'l {x) + d

  • 0 166 Q

    Example l ~ Let .:1 = t and G ={ tn; t (z) = z + 2n7'Ci~ n

    n E. i }, where 'J: is the set of all integers. Then

    g(z) = e 2

    is an automorphic function relative to G.

    Example 2: Let .A = fC and i

    G "" ~ t (z) = z + m w + n w " ~ m~n 1 . 2

    m. ~ ~ E. ::L where ".!) and U.J r ' 1 2 are complex numbers such that:

    Then automorph:i.c funct:Lons relc.t:tve tc G are e lliptic functioru3

    u.L and w A typical exampls- i2 t he Weierstr ass .J. 2.

    ~lliptic function

    2: I 1n~nE~ ;..

    (lll~n)~(O,O)

    1 ~~ . 2

    1 ... =mr , =rt'' ) ,... "'"'l ~2

    Schwarz cons:i.dered the Gauss differential equation under the

    assumption that all of the paramet ers and '1 a.re real

    numbers . He determined a.11 casES that the image .4 of D = ~ =

    { oll l \ under the map

    is a univalent domain contained in S and hence the inverse func ~

    ti on

    x = g(z)

    of f is single-valued in A We shall summa.rize 'his results<

    THEOREM 23.1: Assume that are all r eal.

    Then

  • 2> 16 7 -

    (i) g(z) isl single~valued ,t;; l.J. and only if

    .A "" ~ 1 - r I 3IZ 1/1. l :s 1 s 2 ~ ... 00 ~ ~ r= lr~q{-i8l '" l/m, m "' 19 2' 0 0 ' ~ v "" l ~ ~ ~ l "" l/n 9 n """ 1:1 2, 00 ~ ~

    where if one of >.. / iA and v is one ll the others a re equal;

    (ii) if

    then g(z) is a rational function;

    (iii) if

    J\ + )4+ i.> = l/ l + l/m + l/n = 1 9

    then there exists a linear fractional transformation t(z) in T

    such that g(t(z)) is either a simply periodic function or a

    doubly periodic function;

    (iv) if

    A+~+ Y"' 1/.1. + 1/m + l/n < 1 9 then there exists a linear fractional transformation t(z) in T

    such that g(t(z)) is an aut:omorphic function defined in jz!

  • - 168 -

    'f r -oo du ~ ~ f,'OO . . du . 1 Jo. Ju(u-l)(u-x)' .. 2. 1 ~u(u-l)(u-x)

    and A is the upper half-plane Im z > 0 ~ and G is the group generated by

    t 1 (z) = z + 2 , z

    2z+ 1

    We remark that

    t .. f 1 Ii Z dv z . 2. -1 Cl-v ) (1-xv )

    It is known that f(x) =

  • - 169 -

    " . L. z

    y"' 2 2 - - -p.

    Y2

    Thus we have

    (23. 3)

    Differentiating both sides, we get

    Yz .. -\yi{p(x)+z"/z'}-;y2 {p'(x)+ [zn/z'] '}

    .\ y 2 { p (x) + z" /z '} 2 - ; y2 { P' (x). + z'" /z' - [z" /z' J 2} .a.

    and hence

    (23.4) yl 0 Ji;y2 {p(x)2 -2p'(x) -2z"'/z'+3[z"/z'] 2 +2p(x)z"/z' }.

    Since y2 is a solution of (23.1), substituting (23.3) and (23.4)

    into (23.1), we get

    \y2

    {p(x) 2 -2p'(x) -2zm/z'+_3[z"/z'J2 +2p(x)z"/z'} 2 . .

    - \y2 {p(x) +p(x)z"/z'}+q(x)y2 -0.

    Dividing both sides by \ y2 , we obtain

    (23.5) z"'/z' -~ [z"/z'J 2 -z{!p(x)2 +ip'(x) -q(x)}. The left-hand side of (23.5), i.e.

    (23.6) zttt/z' -~ [z"/z 1 J2

    is called the Schwarzian derivative of z and often denoted by

    {z, x } The right-hand side of (23.4) or the quantity

    I(x) = tp(x)2 +\p'(x) -q(x)

    is an invariant of (23.1) under a transformation of the form

    (23. 7) y a(x)u

    In other words, if (23.1) is reduced to

    u"+p1 (x)u' +q1 (x)u ., 0

  • 170 -

    by (23.7), we have 2 2

    \ p (x) + 12 p' (x) - q (x) = ~pl (x) + ~ p1 (x) - ql (x) .

    In particular, if we take a (x) = exp { - ~ Jx p (t)dt}, then (23 .1) becomes

    u" - I (x)u = 0 .

    For Riemann's equation with scheme

    a b oo

    we get

    1z, x} = ~[(l-'A.2) a-b+ (l- . 2) b-a +l->'2] l , l x-a /A x-b (x-a) (x-b)

    where

    In particular, for the Gauss equation, we have

    .{z, x l.,. 1 { l-.X2 + 1-..2 - l-X2-.2+v2 } I 2 --;r (x-l)Z x(x-1)

    where

    ~ 2 - ( 1 - '1) 2 ' f-2 = ( - - ~ ) 2 ' }12 - (0( - p ) 2

    About at the same time when the general theory of automorphic

    functions was founded by Poincar~, Picard discovered an example of

    automorphic function of two vili:iables by utilizing the system (12.1} I

    w~ich is satisfied by F1 . Ao we explained before, the system

    (12 .1) has three linearly ir1deeT1d~ni: solutions. Let us denote

    them by

  • - 171

    z1

    (x,y) , z2(x,y), z

    3(x,y)

    and let f 1 be the monodromy representation with respect to the fundamental system

    (23 .8)

    z1(x,y)

    z2(x, y)

    z3(x, y)

    The representation ft is a homomorphism of '7C l (~ , x0

    ) into

    GL(3, G:}. For a loop .2 at x0

    , the homotopy class [ J. ] is an

    then the fundamental system (23.8) becomes

    al bl cl

    a2 b2 c2

    a 3 bJ CJ

    z1

    (x, y)

    z2(x, y)

    z3

    (x, y)

    by the analytic continuation along l. Set

    '

    z1

    (x, y) s f (x, y) "" , z

    3(x, y)

    z2

    (x, y) t = g(x, y) = z

    3 (x, y)

    rhen the functions f(x, y) and g(x, y) become

    alzl + blz2 + clz3 a 1 + b1g+ c 1 = a)f+b3g+c3 8JZ1 + b)z2 + C3Z3

    tnd

    a2zl + b2z2 + c2z3 a 2f +- b2g+ c2 =

    aJzl + b)z2 + C3Z3 a 3 + b3g + C3

  • - 172 -

    respectively by the analytic' continuation along 1 . Let

    x = j' (s, t), y = f

  • - 173 -

    containing a parameter x or two parameters x, y. An int~gral

    of algebraic function is called, an Abelian integral which is: very

    important in the theory of algebraic functions and in the theory

    of algebraic geometry. The theory of automorphic functions has

    become a branch of mathematics related to number theory and alge-

    .braic geometry.

    Finally we shall state an extension of the Schwarzian

    derivative to the case of two variables. _Consider a completely

    integrable system of partial differential equations

    (23. 9) {

    r a ~p+ c(2q + C(3z

    8 ~lp+ ~2q+ ~3z

    t r1p+ Y2q+. Y3z Let z1 , z2 and z3 be linearly independent solutions of (23.9}

    and set

    Then

    - o( 2 '

  • - 174 -

    where

    The four differential expressions on the left-hand sides are a

    generalization of the Schwarzian derivative to a map of. c2 to 2 C : (x, y) ~ (u> v).

  • - .115 -

    24 . Reducibility. We shall .consider again a linear differential

    equation

    (24.1) . y" + p(x)y' + q(x)y -= 0

    We shall write the equation (24.1) in a form

    The . expression

    (24. 2)

    d d . .( 2 .) . dxz+p(x) dx +q(x) Y = o .

    d2 ' d -z' + p(x) di"+ q (x) dx

    is a differential operator. It happens that the operator (24.2) is

    decomposed into a product of two operators:

    (24.3) ' ~~ + p(x) ! + q(x) ( ! + s(x)) ( ;-+ r(x)) The right~hand member of (24 .3) means that

    ( ! +s(x)) ( ! + r(x)) ~ d22 + [r(x) + s (X) I! + [r' (X) + s (x);,(x) 1 . . dx < . . .

    A polynomial is said to be reducible if it can be decomposed

    into a product of two polynomials of lower degrees. In general,

    the _r .educibility of polynomia_ls depends on the field to which the

    coefficients of polynomials belong. Similarly, for differential ' ' - ' ' I ~ .. .

    op.erators, the reducibility depends on the coefficient-field. In

    other words, it is necessary to prescribe a field to which .the

  • - 176 -

    coefficients p, q, r and s .. belong, if we discuss the decom-

    position (24.3) of the operator (24.2). We shall restrict our-

    selves to the set of all rational functions C(x). Namely, we

    shall suppose that coefficients of differential operators belong

    to C(x).

    Remark that C(x) is not only closed under the usual addition,

    subtraction, multiplication and division, but also closed under the

    differentiation with respect to x. Such a set is called a dif-

    ferential field. A precise definition of a differential field is

    as follows:

    A set K is called a differential field if

    (i) K is a field in a usual sense;

    {ii) there exists a map D from K into K such that, for any

    a, b E K, we have

    (a) D(a+b) ,.. D(a) +D{b) ,

    (b) D(ab) .. D(a)b + aD_(b)

    The map is called a differentiation. An element c of K is

    called a constant if O(c) = 0.

    Let us return to the operator (24.2). The equation (24.1) or

    the operator (24.2) is said to be reducible if the operator (24.2) admits a decomposition of the form (24~3) in C(x).

    . -Suppose that

    (24.2) admits a decomposition (24.3). Then any solution of

    {24. 4) y' + r (x)y ... 0 is a solution of (24.1). This implies that there exists a non-

  • .. 177 ..

    trivial solution of (24.1) which satisfies a first order linear

    differential equation (24.4). We shall show that the converse is

    also true. Let . ~(x) be a non-trivial solution of (24.1) which

    satisfies (24.4). This means that

    {24. 5) ' " + p ' ' + q ' - 0 and

    (24 .6) ,. + rcr - 0 Differentiating (24.6), we obtain

    (24. 7) 'f" + r 'f' + r 'er - 0 . Subtracting (24.7) from (24.5), we have

    (24.8)

    niminating

    iet

    :o derive

    (p~r) f' + (q~r')1' .. 0 d . ~. . an . J from (24.6) and (24.8), we get

    I . 1 . r I .. q "."' r' - (p-r)r = 0 . . p-r q-r' s .. p - r

    r+ s = p ,

    r' +rs =- q

    his means that (24.2) admits the decomposition (l4.J) .

    THEOREM 24.1: The operator (24.2) is reducible if and only if

    ~ere exists a non-trivial solution of (24 .1) which satisfies a

    lrst order linear differential equation (24.4).

    Consider the Gauss differential equation

  • - 178

    (24.9) x(l-x)y" + [ l'-(cx+~+l)x]y' - ll(p-y: .= 0_. _

    Suppose that (24.9) is reducible. -Then there exis~s a non-trivial

    solution of.(24.9) which satisfies a linear differential equation

    (24.4). Such a solution is given by

    (24.10) y f(x) = const. exp_[- )x r(t) dtl Since r(x) is a rational function in x, we have

    - ~x r(t)dt = rl (x) +~/Li log(x-ai) ,

    where are non-zero constants. Hence

    rl (x) P.1 T (x) = const. e TT (x-a1) From the fact that (24.9). is an equation of Fuchsian type, it

    follows that rl(x) must be a constant. Therefore, we can write

    T (x) in the form (24.10.') -rr P.1 y .. .f (x) = const. 11 (x-a1) On the other hand, the equation (24.9) has its singular points at

    x = 0, x = 1 and x = oo . The exponents at x = 0 are 0 and

    1- '{ ' while the exponents at x - 1 are 0 and r -fJI.- {J

    Therefore, if ai ii 0 and t: l, then P.1 - 1. This implies that

    'f (x) is expressed as one of the following forms: P(x)

    x 1-r P(x) , (24.11) 'f (x) = 't - -~ (x-1) P(x) ,

    x1-

  • - . 179 - .

    while P(x) is a polynomial in x. Let P be of degree n ~ 0.

    Then r

  • 180

    (24.13) [t(x)J 'fCx)

    is a fundamental system of (24.9). Let f be the monodromy representation of (24.9) with respect to the -fundamental system

    (24.13). If J. 0 and l. ' 1 are loops at "xo surrounding x - 0 and x = 1 respectively, then the function '(x) becomes con-st-. T (x> by the analytic continuation :a.iong .t

    0 and 1

    1.

    Hence

    [** *oJ ~ f = , Consequently, the monodromy group consists of lower t!iangular

    matrices. - Conversely, it is . easily verified that, if- the monodromy

    group consists of lower triangular matrices, then f (x) takes one

    of the forms (24.11). This means that (24.9) is reducible. Thus

    we proved the following theorem.

    THEOREM 24.3: The Gauss differential equation (24.9) is

    ~educible if and only if there exists a fundamental system of (24.9)

    with respect to which the monodromy group consists of lower tri

    angular matrices.

    ' .. We shall now explain briefly how to generalize the concept of - ,

    reducibility to linear differential equations of higher orde~s. , . ~ .

    Consider an n-th order equation

    (24.14)

  • - 181

    . or the corresponding differential operator

    dn dn-1 (24. 15) - + p (x) + +

    dxn 1 dxn-1 _ Pn (x) .

    The equation (24.14) or the operator (24.15) is said to be reducible

    if the operator (24.15) admits a decompos~tion dn - -dn-1

    - (24.16) - + p (x) + +p -(x)-dxn 1 c:b:n-1 n -_

    -( d :1 + .. +Pa (x)Y( a;, + : + ~n~(x)) ____ _ dx l _ - -_ l - ') -dx 2 - 4

    T: ..

    where 0 < n1 , n2 < n. The following theorem is a generalization

    of Theorem 24.1.

    THEOREM 24.4: The operator (24.15) is reducible ' if- and -only

    if ther-e exists a non-trivial solution of -(24.14) which satisfies a

    linear differential equation of a!l order lower than n.

    A ._monodromy representation of (24.14) is a -homomorphism f of

    _7rl-_(D, x0

    ) into GL(n,_C), _where D - i$ the. greatest domain in

    whic~ the c~efficients of _(24.14) are holomorphic. A monodromy

    represe_ntation

    0

    * . * }02 ~

    n2

    I

    iihere n1

    and n2

    _ ~re determined by f and they are independent

  • - 182 '.9

    of each matrix of f { 1t1

    (D, x0)). The following theorem is a

    generalization of Theorem 24.3.

    THEOREM 24.5: Supp0se that (24.14) is an equation of Fuchsian . . . . -

    type. Then the equation (24.14) is reducible if and only if there

    exists a reducible monodromy representation of (24.14).

    We shall now proceed to the systems (12.1), (12.2), (12.3) and

    (12.4). The system (12.1) has a form

    (24.17) { r

    Q{1 P + oc.zq + 0(3 z ,

    s - P1 P + ~zq + ~3 z, t - "1 p + y 2q + '13z ,

    while the systems (12.2), (12.3) and (12.4) .have .the form

    (24.18) .{. r ::1s+ a 2p+ a3q + a4z ,

    t b1s+b

    2p+b

    3q+b

    4z.

    In their book, Appell and Kampe de Feriet gave the following defi-

    nition of reducibility. The system (24.17) (or (24.18)) is reducible

    if there exists a non-trivial solution of (24.17) (or (24.18)) which

    satisfies a system of partial differential equations whose solutions

    form a vector space of dimension

  • - 183 -

    reducibility. We do not know any necessary and suffi..cient condi-tions for reducibility which are given explicitly in terms of

    parameters. Appell and Kampe de Feriet gave only a few examples.

    rhe reducibility of moriodromy representations of the systems (12.1)

    ,,..,.(12.4) can be introduced in a natural way. However, it seems to

    is that there is no clear . statement concerning the relation between . . . .

    ~educibility of systems and that of monodromy representations.

    PROBLEM 1: Find a necessary and sufficient condition that the

    :ystems (12.1) "'- (12.4) be reducible. Find also relations between

    educibility of the systems (12.1) ~ (12.4) and that: of their

    onodromy representations.

  • -- 184 -

    25. Generalizations. We defined the hype:rgeometric func.tions of

    two variables. by.utilizing the Gauss function. In a natural way,

    we can int~oduce hypergeometric functions of n variables.

    Lauricella. introduced the following f~ur functions:

    F .(oc.. R ~ v . y . x . x . ) A ' t' l :, , t' n, ' l, , 0 n, 1, , n

    .:>:".:-> .. ( m_) (~ - m -)(A m) ~(A m) 1' l n' n t"l' 1 t'n' n

    I. (~ ,~+ +m0

    ) (l,~) (1,mn)

    - (at ,ml+ +mn) (~,ml+ +mn) ~ m n . L (11,ml) ... ((Y'n,mn) (l,~) .. (1,mn) xl . x a

    and

    (a( m.. + +m ) ( II. m ) ( n. m ) m m ' l n t'l' 1 '"'n' n 1 n

    ' x x '- ('j ,~+ +m0

    )(1,m1) (l,mn) 1 n

    In case when n 2, we have

    . FA = F 2 , F B = F J , F C = F 4 and FD = Fl .

    There are many functions of intermediate types. I

    I

    On the other hand, many functions of one variable are defined '

    as generalization of the Gauss function. The Gauss function has

    the three expressions:

    (i) .. 2:, (Cl( , m) ( ~ , m) xm

    m=O (1,m)(l,m) '

  • - 185

    (li) . . f,' d..:...., r'-{J-1 -0( const .. 1

    u (u-1) . (u-x) . du ,

    (iii) const. r(at+s) r(6+s) r(-s) (-x)s ds. f +ieo

    -ita . re t+s) The second expression can be. written also in the .forlJl .

    (ii') const. fl -1 .,_ -1 -oc u~, (1-u) ~ (lx\l) du

    0 . .

    The series

    F (ac.l, ... I O(p+l ~11 I ~p

    . - 1' "'pl' m x - x

    )

    ., - ( o( m) (..1 m)

    f:o (~ 1 ,m) ~ (~p,m) (l,m) is derived from the expression (i) in a natural manner. More

    ~enerally, we get the following series:

    (at ,m) m p x (o

  • . f ~ioo -1.00

    - 186 -

    p TI rcoc .+s> i=l 1.

    Tr rep .+s) il 1.

    s r(-s)(-x) ds.

    These ideas of generalization of the Gauss function are also

    applicable to generalization of Appell's and Lauricella's functions.

    Finally, we shall talk about the confluence-principle. To

    start with, let us consider the Gauss function F(ac, ~, '(, x).

    Introducing a new parameter to define a function by

    . F(CI(, 1/E., J, x)

    and then taking a limit as e. tends to zero, we obtain a new

    .. function

    lim F(t

  • - 187 -

    regular singular point at x z 0 and an irregular singular point

    at x .. oo Such equations form an important class of equations

    which contains Bessel equation:

    (25.3)

    The equation (25.3) has the Bessel function of order n as a

    solution.

    Humbert obtained from Appell's functions seven functions by

    the confluence-principle. For example,

    , ~ ( ~ ,m) ( 13', n)xtl)rn i!W F 1 (l/t' p, ~ , ~, .x, t.Y) .. }.;J "t ,m+n) (l,m) (l,n) and the function defined by this series admits an integral represen-

    tat ion

    C