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Basic hypergeometric series

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TITLE
BASIC HYPERGEOMETRIC SERIES
BY
John JU-Daum
A PPR O V E D
DATE
(UJ* 2
0 / >
A
7~ ""
A
A F ULS— C _ ^
A
n
Jf. Jf<
A
SUPERVISORY C O M M IT T E E
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R e p ro d u c e d w ith p e rm issio n o f th e co p y rig h t o w n er. F u rth e r re p ro d u c tio n p ro h ib ite d w ith o u t p e rm iss io n .
BASIC HYPBtGEOMETRIC SERIES
by
John A* Daum
A THESIS
Presented to th e Faeulty of
The Graduate College in th e U niversity of Nebraska
In P a r tia l F ulfillm ent of Requirements
For th e Degree o f Dootor of Philosophy
Department o f Mathematios
Lincoln, Nebraska
July 5, 1941
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UMI Number: DP13726
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CHAPTER I
Introduction
In a l e t t e r to P ro f. G. H. Hardy, p ro f. E. T. B ell oommunioobed
th e follow ing Id e n tity
f
a l-r /L i— ♦
A d M ^ l i M
—
♦ ...♦ - J U l-
1-q*
i - q*J
(i)
^ i - q n
w h io h ^ he found while looking fo r quadratie forme which rep resen t
a l l Integers w ith a t most a f i n i t e number o f exceptions. Among sev( 2)
o ral oonsequenoes o f (1) i s th e theorem
th a t th e number of rep re s­
n >0
en tatio n s o f any in te g er
§4(n) - n£o(n)
in th e form wx + xy ♦ ys + tu
where w, x, *, u
0, y
0
ftmi ^ ( i i )
is
i s 1#hd sun
o f th e r - t h powers of a l l th e d iv iso rs of n* P ro f. Hardy re fe rre d
th e l e t t e r to W. N. B a i l e y ^ who oonstruoted th ree proofs of ( l ) ~
one, a d ire c t algebraic proof and th e o th er two, by means of known
( 3)
r e la tio n s in basio hyper geometrio s e r ie s . L ater, Newman A. H allv
'
proved th a t Bailey* s f i r s t proof oould be made t o depend upon th e
(4)
/ g\
b asio analogue
of one o f Thomae* s two-term re la tio n s
involv­
in g th e functions
3fz .
S t i l l l a t e r , B a i l e y ^ again used basio
s e rie s to prove s t i l l another id e n tity
-\.
4 * (1 - q*")1 1 1 - q‘
>*»>/
yn,o<id
V /(»*•- l ) q ~
1 -
1 - q—
j
«(b - l) q * (l ♦ q**1) )
*
( i - «"»*■
J
'toi oid 1 ( 7>
iAJ:
whioh B ell
had deduoed from th e theoren^for any p o s itiv e odd in ­
te g e r
385808
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N[m « 2wx * xy * y* ♦ su ♦ uxj w, x, y , i > 0, u £ 0]
- ] /8 {>£,£*0 - ( 4 a + 1 ) 5 ,0 0 ♦ 4 5 ,0 0 }
These r e s u lts connecting th e theory o f basio s e rie s w ith a rith ­
m etical theorems suggest th e d e s ir a b ility of making a system atic
study to determine th e extent of th e a p p lic a b ility of basio se rie s
to suoh problems.
Iftth th is o b jectiv e in mind, a ra th e r extensive
survey of th e lit e r a t u r e was made.
I t was found th a t th e number of
av aila b le re la tio n s was lim ited and th a t th e subject of basio se rie s
had reoeived scant a tte n tio n .
Thus, i t seemed d esirab le th a t th e
e n tire theory o f basio se rie s be studied in order to determine wheth­
er o r not fu rth e r useful r e la tio n s i n basio s e rie s oan be obtained.
(A\
Since th e basio s e r ie s ' 7 i s a g e n eralisatio n of th e c la s s ic a l
gaussian hyper geometric s e rie s , i t i s t o be expected th a t c e rta in
analogous re la tio n s w ill appear in both eases.
The fa o t th a t even
th e proofs of th e se analogous r e s u lts are q u ite sim ilar suggests
th e use o f analogous n o ta tio n wherever p o ssib le.
The e h ie f d i f f i ­
c u lty heretofore has been th e lack of a su ita b le analogue fo r th e
Graama-funotio n . However, th e m odified Heine Omega-function in tr o (3)
dueed by H all
seems to answer th e purpose admirably and i s em­
ployed throughout th e follow ing discussion.
Heine o rig in a lly studied th e series^
( l- q jl- q * ) . . . ( l - q n) ( l- q y) ( l - q 177) . . •(l-q^*'*1)
as a g e n eralizatio n of th e hypergeometrio s e rie s .
(g\
Sinoe 0. H. Hatsonv 7
used a m odification of t h i s fu n ctio n to derive th e famous Rogers-Ram( a.\
anujan id e n titie s , 7 Heine* s fun ctio n has been replaoed by th e basio
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s e rie s as defined by Watson.
Using H all*s m odification of th e Heine Omega-funotion* Hatson’ s
d e fin itio n of th e basio s e rie s may be w ritte n :
CO
T Ta r» aa»»»» ar5 z~j m \
( a<) g * X *• ( a r)^,vi>
^h-
— 0(b/ )0(bA) . . . 0 ( b j ) \ 7 0(a, q"’)0(a^q*'). . .0(ayq"')
^
0 (a ,)0 (ft4) . . . 0 ( a y.) Z J 0 (q ™ )0 (b ,q ~ )0 (b ;q " )...0 (b J q*’) ‘
mss
where
|q / < 1,
[zj < 1,
(*)*,»«.* ( l - x ) ( l - x q ) ...( l - x q >’' ' )
oo
T"T 1
«*> . 7T i s
-xq
I * 1-3
The function
r ■ se l
0(x)
,
(*)*•* 1
i s th e m odified Heine Omega-funotion.
If* in (2)
and a ,q ■ a^b, ■ a3bi» . . . » a^b** J$s i* c a lle d *well-pOiaed”
I f a l l but one o f th e p a irs o f param eters hare th e seme preduct* th e
s e rie s i s sa id to be "nearly-poised".
In t h i s d issertatio n * an attempt i s made to oonplete th e se t of
b a sic analogues of th e known re la tio n s involving c la s s ic a l gaussian
hyper geometric s e rie s .
In p a rticu la r* an exhaustive in re s tig a tio n of
th e p o s s ib ility of obtaining fu rth e r re la tio n s sim ilar to t h a t used
by n a il i n proving B e ll’ s id e n tity (1) was made.
The r e la tio n employed by H all was found to be th e basio analogue
o f but one o f many involving th e se rie s
jEJ^ whioh were proved o ri­
g in ally by Thom ae^ and l a t e r o la s s ifie d by W h ip p le ^ ^ .
The com­
p le te se t of th ese analogues i s given here and i t is shown th a t t h is
s e t o f id e n titie s oan be divided in to two c la sses » a l l th e id e n ti­
t i e s of one o lass being equivalent to an Id e n tity whioh oan be obtain­
ed by repeated use of any of -the members o f th e f i r s t c la s s .
Thus,
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H a ll's re la tio n i s proved unique in th e sense th a t any r e s u lt obtained
from any id e n tity of th e se t oan also be obtained frost H all's*
.Analo­
gous remarks apply to th e se t o f re la tio n s involving th e e la e sio a l
se rie s
3Fa * F urther d e ta ils on th is treatm ent of th e basio analogues
o f Zhosiae*s re la tio n s w ill be found in a paper to be published sh o rtly
in th e B u lle tin of th e American Hathematioal Soeiety*
The survey of th e l it e r a t u r e on basio s e rie s revealed th a t no
basio analogues o f Hummer's t h e o r e m , D i x o n ' s t h e o r e m , t h e
th ree-term re la tio n s o f Thornae^6) and of several id e n titie s involving
higher order se rie s have been published*
These problems are tre a te d
se p ara tely , th e r e s u lts being in d icated below*
The r e la tio n
X [a, b; -q / b l = 0(aq /b)0(qyV )0(-q/a)0(-q /b )
L » q /b
1
0(aq)0(-q)0(-q/a/b)0(<$/V b)
was dedueed from th e id e n tity
(15)
a , qj/a, -q|/a, d, e, f j a q /d e f
\/a, V a , a q / d , a q / e , aq / t
m
/ d)°( aq / a)0( aq / f)0( aq / def)
0( aq)0( aq /e f ) 0 ( a q / df)0( aq / de)
I t i s th e basio analogue e f Hummer's theorems'(12)'
[a, b ; -l7 B n i + * - b ) r ( l W z )
* 4 L l+a-b J
P(l-»-a)f,( 1+^2-b)
Since Dixon's theorem, whioh sums a general w ell-poised 3P&,
(14)
oan be derived
by means o f Hummer's theorem, i t was hoped th a t
th e basio analogue of Kramer's theorem would, in th e same way, give
r i s e to a basio analogue o f Dixon's theorem*
U nfortunately, however.
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t h i s is not th e ease.
No analogue of Dixon's theorem i s found* but
th e ourious r e la tio n
Z
7l =tf
/aqj* ^ j )
O(aq")0(bq” )0(oq*)
0( <f+0O(»qw/b )0 (aq 7’*/©) Ibo/ ^
0(a)0(b)0(o)
r P5*0* -* * * * /bo7
o(aq)0(aq /b o )
J
was obtained by employing a m odification of th e method used by Watson in
p ro rin g Dixon's theorem.
This r e s u lt i s in te re s tin g in as muoh as I t ex­
p resses vh&t may be oalled a p seudo-basio s e rie s in te rn s of a function
J&i * This seems to be th e f i r s t r e s u lt of i t s type to appear i n th e
theory of basio se rie s.
in a tte s p t i s made to fin d a basio analogue of Thomas' s th ree-term
relatio n ^
p .b - e + l.b - f + l; l l
1+b-o, 1+b-a J
Ta.b.o; 11 f t l - a ^ e J p C f ^ o - b )
*** I e , f J "
e-b )T (f-b )r( l+b-aJH o)
♦ a sim ila r r e la tio n with b
and £
interchanged.
D etails of th e o rig in a l proof are supplied and m analogous formal
procedure i s devised fo r the attack on th e problem a t hand.
problem reduees to th a t of determ ining a fun ctio n f(x )
analogous to th e function
fu n ctio n al re la tio n s .
The
whioh i s
elirz and whioh also s a tis f ie s c e rta in
The method f a i l s but th e s trik in g s im ila ri­
t i e s in th e r e s u lts o f in d iv id u al step s strongly suggests th e e x is t­
ence of some s o rt o f r e la tio n analogous to Thomas's o rig in a l tra n s ­
form ation.
The basio analogues o f th e two formulae
( 16)
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p f x ,y ,s ,- n j 1~l_ /‘’(▼♦VnVC l^x -u y ’Cl+y-tp/X les-u)
i-O I U,T,W
J Z7( 1+yes-uy*( l+a*x-uV’( l-*-**-y-u)r( 1-u)
(S)
ra,
l« V s ,
w*n,
X,
y,
sj
17
7 ** L * / 2,v,w ,l*y*z-u, 1* z+x-u, l+x*-y-u
where
a ■ x*y*z-u » v»w*n-l,
and
(t-bX, (*»s)n p f u - x , u-y, s, -a ; 1 1
(^ (w )^
**"3 l_l-TfB-n, l-W*B-n, u j
p rx ,y ,e ,-n ; 17
i 3 L U,T,W
J
where
J
( t ) ^ - z( s+l)( b+2). . .f, zen-1),
^
are proved and oan be w ritte n
in the follow ing forms*
X p c*y.B,q“>' ; ql
0(vwq7,)0(xq /u ;0 (y q /u )0 ( zq /n ) ^
u, t , w
J ’ 0(yn/u)0(xzq /u;«Asyq /u )0 (q /u )
Ta, q /a, -q /a , wq71, vq71, x ,y ,e j l/u q 71' '
1
*
L
where
V^a,
-\/a, v , w,yzq / « , xtq / u, xyq / u J
a - vwq>,"/ ■ x y i/u , uvwq71"7 ■ atyz,
(u|>l</*A[
and
x ["x.y,B,q‘n ; ql
«
where
l
u .T„
J
M (v /t)* ,* (V *)«.w
X IV * , Vy# B, q_>*} q 7 / --x
*
4
L
».
^
, -
j (4 )
uvw * ay*/q"'*
These re la tio n s , to g eth er with th e equivalents o f (4*) obtained by p e r­
muting x ,y , s o r u,v,w or by reversing th e order of s a n a tio n complete
th i s sectio n o f the theory of basio s e rie s .
The ourious formula
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m4 (/s)
4(fi*r*&)
(5)
* $ { r ) 4 { S ) 4 ( « - /3) # ( * y * r* S )
deduced by B ailey
gated.
(155
from c e rta in r e s u lts in basio s e rie s i s in v e s ti­
An independent proof, enploying Liouville* s theorem^
e l l i p t i c funotion, i s given.
fo r
This proof e ffe c tiv e ly removes r e s t r i c ­
tiv e conditions is^osed by Bailey* s method.
I t i s proved th a t (5) i s
equivalent t o th e v e il known formulae re la tin g produots o f four Theta(17)
functions as given by Jaeobi
• Finally* i t i s shown th a t (5) may
be w ritte n in the form
y 3 )^33(d * S ^* y)
-a
where
These funotions ip are an inqportant group of functions belonging to
th e class of doubly p erio d ic functions o f th e seoond kind.
They were
o rig in a lly encountered by Jaoobi i n h is study of a o e rta in problem in
Dynamics and are very useful in e sta b lish in g Liouville* s general id ­
e n ti t ie s fo r o e rta in a rith m etical funotions.
U th th e exoeption of Dixon* s theorem* together w ith i t s conse­
quences* and Thomae* s th ree-term transform ations, a ll th e basio ana­
logues o f known re la tio n s in hypergeometric se rie s with argument (+1)
a re now a v a ila b le .
The question of t h e i r usefulness in arith m etic
depends tqpon fu rth e r ted io u s sp e c ia lisa tio n s and a o e rta in amount o f
ingenious an aly sis.
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CHIPTER I I
The Basie Analogues o f Thamae* s Two-Term R elations
The id e n tity
V — C
f i—
Z
u
(
W
>
a
l> q
•Hal
J L
l-q *
- L
. L
f j g a 1
l- q **J A 1-q"
was dedueed from a rith m e tic al considerations by E. T. B e ll.
(1)
™
About
5 years ago, V. N. Bailey^ ^ proved th e r e la tio n
oo
y ( i.q ) ( i^ ) .„ ( iv )
^
( l - .> ( l - q * ) ...( l - q " z )
oo
l-q * "
y
fcjtM iV
from whioh he obtained (1) by d iff e r e n t!a tin g w ith resp ect to s and
(3)
th en p a ttin g z - q. A short tim e l a t e r H all
gore an a lte rn a te
p roof o f (2) by simply sp e c ia liz in g th e param eters in a re la tio n be­
tween basio s e rie s .
Bell* b id e n tity (1) iiqplies in te re s tin g arith m etical theorems*
one of whioh i s th e curious r e s u lt
l£ n ■ wxtaqrtyz+su; w,x, z ,u > 0 ? y > 0] * ^ A(n ) - n £ jn )
where MfJ denotes th e number of se ts (w*x*y*z*u) of integers* subject
to th e conditions indioated* s a tis fy in g th e s ta te d equation in whioh
n
i s ana rb itra ry constant in teg er > 0
r - t h powers of a l l th e d iv iso rs o f
without proof by Liouville^
n.
and ^ ( n )
i s th e sum o f th e
This theorem* whioh was sta te d
has re c e n tly been proved by P ro f. B. T.
(9 )
B e ll' ' who employed an id e n tity between doubly periodio fu notions o f
th e f i r s t and second kinds.
Obviously* i t i s d esirab le th a t fu rth e r
r e la tio n s sim ila r to (1) or (2) be obtained.
The id e n tity whioh H all
employed to o b tain Bailey* s re la tio n i s th e b a sic analogue o f bu t one
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o f many id e n titie s involving th e s e rie s
3PZ whioh were proved o r l-
and l a t e r c la s s ifie d by Whipple
The object o f t h i s chapter i s to in v e stig a te th e p o s s ib ility o f
system atically obtaining new r e s u lts sim ilar to Bailey*s by employ­
in g th e complete se t of basio analogues o f Thomae* s two-term tra n s ­
form ations.
Typical of Thomae* s transform ations i s th e follow ing
b .o ; i ]
®#f
where
_
s r w * ) n » * o ) 3
s » evf-a-b-o
We f i r s t e s ta b lis h th e complete s e t o f basio analogues o f Thomae* 8
rela tio n s* using a n o ta tio n analogous t o th a t used by Whipple.
Let
I - ( a , b, e, d, e, f )
be a m atrix whose elements form an
a rb itra ry perm utation of th e in te g e rs
0* 1* 2, 3, 4* 5,
and l e t
ra , rh , rc , r4 , re , rf be six numbers swoh th a t t h e i r produet i s u n ity .
Define param eters
A and B by
Bw ■ r^q / r „
and funotions
^p
and
by
k 4 bc* A e b c •
$ ? ( a ; b, o) - O C ^ O C B ^ X K ^ )
aj, bo*. oo); ■ 0.........
(A g ^
- A -., i i
0(B^ )0 (B rfc )
*
^
yX Pk ade * Aadf*
k fb c * k i * f
Bce
k a eef»
f i k abc
oo
o{S)o(e)
o ( * ) o (p)o(y)
0(xq1')0 (/aq” )0 (/q ^)
^
----------- 1----------- ——s ,
0<q, *')0($q»)0<*q’>)
*I < 1
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Considering a l l p o ssib le m atrices I* i t i s p o ssib le to form 60 ^ p f s
and 60 ^ n #s*
There are 90 fundamental re la tio n s between o e rta in p a irs o f th e
funotions i^p.
The proof o f th ese relatio n s* whioh oan a l l be w rit­
te n in th e form
a; b, o) -
!?p(a; e, f ) ,
(3)
i s almost id e n tic a l with th a t of th e o rig in a l ease as given by B a ile y ^ ^ ,
subject to o e rta in analogous conditions.
The only e s s e n tia l differenee
i s i n th e replacement of th e Gamma-funotion throughout by th e modified
Heine Omega-funotion*
The proof i s as follow s:
°(4it>c)°(*efa<)°(Kief)
T [ K b * K ix f K k J K e f l
0(Bb i)Q(BGi)
B ^ t Bca
- V * W >W W >) 0 ( W ,>
Z_> 0 (q ^ )0 (B b jq*')0(Btd q^)
J
„
> |5 d
00
y <KW ) o l A f b s D l K j 1
Z -^ ° ( qV,+')0(B^Bcd
* 0(Bba<r)0{Bc»<C)
yt=a
oa
■
K
b
c
*
?
Z ^ O ( ^ 0 ( b ZB c7 ^ 7 k ! J
X' . ^ba^Kbc*
KbA*1
B ttB catfU dbc
J
y -7
>»=<?
f
f
o ( W
"Z X -i
) Q ( A
^ ) ( W
O tq^')
■»*»/-
mi/ .
u
,>
CKBtJ q Y W 0(B^ r / W ( W ' V ,>"
O t B ^ J O t B c y j J o i B ^ B c r ’/ij^ O C q ’"*')
»/ .
*"*** .< .
i- '* ' ■
-
0(Bta q 7 W ° ( Bc* q V ^ J ( A ^ )" 0 ( A ^ ) 0 ( A ^ ) ( ^
* o < qw/)0(Bto/ ^ 0 ( B ^ / A j ^
?AM=<ZJ ttZO
O U ^ 'M B f a ^ a q ^ f c )
Z_ aO( q ^ /)0(Bi>a/^ ^ 0 (B c a /W 0 (B A a B c a ^ n/Adb}
m-C
«■’’’*
q
Kb<* Kb& Ke&
Bt&Bca <?/±Jbe. ,
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.
y
° ( ^ n> o ( W ) ° ( W ) < K B « X > ( B f t,)
<X % t) ° C W ° < W .
7n" ^
■ ^
Aebc)
T
0 (B c d )0 (Bf d )
~^Sef* *cef» A te f 5 Ajb& J
^
J
g ea#
whioh gives r e la tio n (3)*
The argument req u ires th a t th e double se rie s
be absolutely ooaver gent.
I t i s s u ffic ie n t th a t
1 > BfcaBM / A j k » q r *‘> 0
whore r
i s a p o s itiv e in te g e r.
Then, th e double s e rie s may be comp­
ared with th e product o f two absolutely convergent s e r ie s .
I t i s obvious from th e above proof th a t th e 90 i d e n titie s (3) are
a l l equivalent in th e sense th a t any one o f th ese id e n titie s may be
obtained d ire e tly from any o th er by proper determ ination of th e
The corresponding re la tio n s involving th e funotions $ n
r* s.
are a l l equiv­
a le n t to those involving th e S^p’ s — one s e t being obtainable from
th e e th e r by rep lacin g the
r* s
by t h e i r reo ip ro o als.
As an exmqple o f th e use o f re la tio n s (5 ), consider th e id e n tity
l b o btain th e id e n tity
x
_ 0 ( 6 ) 0 (e )0 ( 8e/*0 y)
^
385808
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*
whioh was used by Hall to o btain Bailey* s id e n tity (2).
For a fix ed value of
a, (3) rep resen ts 15 id e n titie s between
o e rta in of th e funotions & < • ) .
4*5)
3 .5 )
2 .5 )
# p (0 ;
# p (0 j
S?p(0;
^ p (0 ;
# p (o ;
s F p (0 5
^ p (o ;
<&p ( o ;
5 ? p (0 ;
2 ,3 )
1 ,3 )
1 ,2 )
1 ,2 )
1 ,4 )
2 ,4 )
1 ,3 )
1 ,4 )
3 ,4 )
For
a - 0, th ese id e n titie s are
<&>(o; 1 ,5 ) m
m
£ p < ° * 1 ,4 ) m
^ p ( 0 j 3 ,4 ) -
i^p(® )
s £ p (o ;
^ p (0 ;
# p (© ;
^ p (0 »
# p (0 j
2 ,5 )
2 ,4 )
3 ,4 )
2 ,3 )
1 .3 )
1 ,2 )
( 4)
From th is* i t i s o le a r t h a t we h am a second awt e f id e n titie s o f th e
fern
^ P ( * » *» « ) •
+* * )
(6 )
sd b ject to o e rta in oowdi t i ene obtained tsnm th e s e applied i n proving
(3)*
R elation (5) rep resen ts 189 t f d r o l w l id e n titie s .
Typical o f
r e s u lts obtained are
x
3
L
&0£
_ 0 (^ 0 (6 )0 (& £ /? y )
r
J
<K^A/3)<K^)0C5y^) 3 i _
7 i V /7
J
and
r r^A^y,
, £/«;<*7 o (& /^ o (£ A g )o(*) t
3^iL &M-y»2e/*p J oiSe/fTy)0{5e/*j5)o(3)
SfeMpJ/f3
te/ftrje/oc/i
J(r)
I t i s p o ssib le t o o btain Bailey* s id e n tity from e ith e r (6) o r (7 ) .
This elaaaifiaa& ion
o f th e basio analogues of Thcnae* s two-
term re la tio n s indioates* then, th a t out o f th e e n tire se t of id e n tl-
The m ateria l in t h i s ohapter forms, th e b a s is of a paper, by th e
author* ahioh has bead aooepted fo r p u b lio atio n by th e B u lle tin of
th e daeriean Mathematical Society*
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t i e s , only
two
are e s s e n tia lly d is tin c t.
Any f a rth e r r e s u lts oan he
obtained from t h i s source only by sp e c ia liz a tio n of th e p a r o u t e r 8 in
H a ll's id e n tity or i n r e la tio n (S) o r ( ? ) .
Analogous statem ents oan be made concerning th e corresponding
re la tio n s between th e gw en ali sort hypergeoswtrie funotions
^
given by Khipple.
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as
CHAPTER I I I
A BASIC ABALOGGE OP KUHMER'S THEOREM
About one hm dred years ago* B. B. Banner
a,
bj
1+a-b
(i)
J ~ r ( y * * x \i+ * /z
whioh i s known as Busmer* a theorem.
i l l th e formulae involving th e Gaussian functions
rFs
whioh
hare knows analogues in th e theory of basio s e rie s are those in whioh
th e argument i s equal to u n ity .
Apparently, th ere has been no suooess-
fu l attempt to e sta b lis h th e b a sie m alogue of m y id e n tity involving
a fu nction
P ( - l) .
However, a b a sie analogue o f Etauer*s theorem
does e x is t, being, i n fo o t, a very elx p le eonsequenoe o f a known form­
u la.
Before e sta b lish in g t h i s analogue, we note t h a t Kunaer* s o rig in a l
theorem cannot be a p a rtic u la r ease of Gauss's theorem, although Gauss* s
theorem w ill
sub
th e se rie s
S im ila rity , th e b asie analogues of Gauss* s theorem w ill y ie ld th e r e s u lt
b;
q/b*l _ 0(oq /b) 0( q /b* )
aq/b
J
0( q /b)0(a$/b)
T his, however, i s not an analogue e f (1) i n th e s t r i o t sense; f i r s t .
because th e argument
q /b2,
i s formed from th e param eters o r elements
i n exactly th e same way as th e argument i s formed fo r a basio se rie s
corresponding t o th e type
P(+l)
and secondly, because th e r ig h t mm
b e r i s not the analogue of th e r ig h t member o f (1 ).
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■wdi
-it
The basio analogue o f Kuaner*8 th eo rea ire now/yestablished.
k„
We
. (16>
^
Ta,
<j/a,
*
-<j/a,
d,
e,
f J
aq / def 1
Va# a q / d , a q / e , a q /f
Va,
J
- °(*^ / d)°( a<i A )°( *q A M »g / )
0( aq)0(aq /e f ) 0 ( a q /d f) 0 ( aq /d e )
In th is put
• • Va,
i f * . *5
* ' _
f » -Va,
d • b
and obtain
-q A7 _ o(*q A)<Kq/a)oC«^/»)°(-q A )
aq /b
J
whioh i s th e analogne r r f e r w i t o .
from th e analogue
( 8)
0( aq)0<-2)0(-<pV^)0(
Vote th a t i t cannot bo obtained
0 —as»a thw orw .
I f we differentiate (5) with reapeat to b , we ob tain a ourious
id e n tity ;
tO
5
v l- '
" c
w
-
(
t A f
*9/*)?,>»
(
?
[
_
£
. .
* ? < * >
C ^ L l-b q ^
b
J
bj
CO
_ a-1 T T (l-a q " )(l+ q r) ( W ,V b 2-) V
(l+ a q ^ /b ) q”
' * U (l-eq * H)l-a q " /b )(le q V b ) Z _ , 1 -a q ^ /b * ) ( l-aq Y b ) ( leqY b)
fo r
Jb/
oO
>
jq ^
I f , i n (5 a ), we p u t
a » 0, we o b tain
n-_l
0>
b < . q / b r ( y r -
i l ]
+ -
j
l TT (W*)
^
ClaqYb) *
f 7 __ £
ltq V b
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(4)
I f , in t h i s , we l e t
b—►!,
and get
whioh give8
as & p a r tia l oheek on (3a)*
Obviously, other re la tio n s oan be obtain­
ed from e ith e r (3a) or (4 ) .
Only tedious sp e c ia lisa tio n s w ill in d io ate
whether o r not re la tio n (3a) is p lie s in te re s tin g arithm etioal theorems*
The basio analogue of Summer* s theorem immediately suggests th e
(1 3 )
p o s s ib ility of th e existence of a basio analogue of Dixon* s theoremv
In th e theory o f th e Gaussian h y p e r g e o m e tric s e rie s , i t was shown by
Watson
th a t Dixon* 8 theorem, whioh sums a general w ell-poised
with u n it argument, oan be derived from th e theorems o f Summer and Gauss
by means o f an interchange in th e order of summation*
This method, when applied to th e basio analogues, does not lead to
an analogue o f Dixon* s theorem.
The d if f ic u lty i s due to an e s s e n tia l
d ifference in the p ro p e rtie s of th e Omega and Gamma fu n o tio n s.
A s lig h t
m odification of t h i s method does, however, lead to a ra th e r strange
r e s u lt whioh we now develop*
«o
0(aq")0(bq*)0(eq»)
0( <f^0( aq*"-*)0( aq /bo)
aqf_*£i> 0(aq**w)0 (a q /b c )
b o /q
* 0(aq™yb)0(aq»*J/o)
0( q**1) 0( aqlh+*) 0( aq
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s p oj aqT,)0(bq1*)0(oq>>)
/aq?»
“Z a O C ^ 'JO C a q ^ tX a q /b o )
_
0 (« ^ H*l)0<l>cfwl")0( Oq1^
(bo/ 9
A 0 (1 » ^ )0 < o q ^ M a if^ O C q ™ ) f c J
>n^
\ 1’ \ f i|K H ")0( ' > < n o ( « 0
m & ^ 0 ( a q ” )0(bqP)0(oqO_____________ / aq \P
" Z a / jO C q ^ 'X K q ^ O C a q A o J O C a q ^ ) U ® )
VtsO
/ * *)**
/a q y
0(a q /b o )0 (aq fH‘)0(%,f<0 V*®/
q
*
\ ^ 0 ( a<j >^ ) 0 ( a q " ) Q ( q - ^
" ^ 0 ( a q /bo)0(aqM )0(qp+') (bo/
0(a)O(bq**)0(oqy)
>tfziO
/M p *
" 271-0
- .Z
i o ( i'« ) o ( f " ) o ( H /i» > ) < K ^ “ *'") U i
fn- O
-
/®q V**
»
0(q"+l)0(a)0(q-P)0( a q ^ O
-g[*»
*
- ‘I1’*'']
z^ ' L
•3 1’**
-*
- V70**)0^ 1^0* ^
/aqy*<K<>/a)0(-<s/a)Q(aq^ 1)0(-qf*)
^ 0 ( a q / b o ) 0 ( a q r ') 0(qhO ( b j ^ - q ^ a q ^ o W V ^ O C V i q p+)
0(«)0(b)0(o)
" Z -* CKaq)0(aq/bo)
<* qP* > < l t * | * e m - q i * ^ ‘)0( V iq ^* )
^ 0(a)0(b)0(o)
^ [b»
0(aq)0(aq Ao) 3*1 ^
•*
qA ,
-<1 »
- q /a
W
®qAol
J
I n th e analysis* we hare used th e b asio analogues of th e theorems
o f Gauss and Kroner*
le a s e , we hscre
( - , A . . ^ , * ¥*>
4 - O (q-0O (aq-A )O (aq> -/o)
( 6)
„ 0(a)0(b)0(e)
t ft,
0( aq)G( aq /b e ) 3& t j.
o,
-q J
<j/a,
aqA cl
-qA
J
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subjeot to o e rta in conditions whioh are neoessary in order to ju s tif y
th e interchange in order of summation.
The arguments re q u ire s t h a t th e double se rie s be absolutely con­
vergent.
Suppose th a t
r
i s a p o s itiv e in te g e r and th a t
2f t l
b < q
,
whioh im plies
e < q
,
a>q
yr*4-
^be > 1
Consider now th e two s e rie s
( 6)
and
(7)
if
These s e rie s are both
i/b c
< l/q .
Form-
ing t h e i r product, we
ee
pO
r w)
Z
L 0( q ^ ')o ( <f ♦' )<K q
*2.0
*15*
Sinoe
b < q
,
^ ^ flC q ^ ^ O
/
" (0 ) v
we h ere, from th e p ro p e rtie s o f th e Omega-
funotion
otb ,” ™) < t r ' " )
<
S im ilarly ,
0 (c q ^
Sow, sinoe
( 8)
<
0( 0)
a > q*r t \
0 ( a q ^ w" ) >
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Hence, i f th e param eters and q
are assumed to be p o s itiv e and
s a tis f y th e shove conditions, th e double s e rie s in question s i l l be
ab so lutely convergent
— being g re a ter in absolute value th an th e
product of two ab so lu tely convergent series*
The conditions sta te d
above are s u ffic ie n t.
I t should be noted th a t th e l e f t member o f ( l ) i s a modified
basio s e rie s somewhat sim ila r to some o f th e q-hyper geometric s e rie s
studied by Jackson^ ^ .
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CHAPTER IV
Atteapted proof o f Thomae* s T hree-tern R elation
The r e la tio n
- [a , b, 0 } 1]
** L
e. f J
rb ,b -e * l,b -f+ l; 1]
rte -T > )T (f-b )r(l* b -a r(° ) 3 til l*b-o, 1+b-a J
* a sim ila r expression w ith b
and o
interchanged
was o rig in a lly proved by Thomas^ ^ .
An attem pt i s made here to prove th e b a sic analogue o f th e above
id e n tity by using th e method analogous to th a t used in proving th e
o rig in a l id e n tity .
As a guide, we supply th e complete proof of th e
o rig in a l id e n tity .
Consider th e in te g ra l
a+g^besV X o+s)
r(e * « )rtf* s )
a*
■LoO
where th e p ath o f in te g ra tio n i s curved so as to separate th e in o reasing and decreasing sequenoe of p o les.
rad iu s p
Let
C be th e sem i-oirole o f
on th e rig h t of th e imaginary axis w ith center a t th e
o rig in , and suppose th a t
o f th e distance o f
P ~*m in such a way th a t th e lower bound
0 from poles of /"’( - s )
i s d e fin ite ly p o s itiv e .
Then using S t i r l i n g 's formula, we have
r ( - « ) r ( 8)r(b+«)/-*( o+s)
P(9*B)fXt*8)
JT*ir* ^(a»s)/T(b+a)r(o»8)_______
P ( e+ 8 )r(f+ s)rt l+ s) sin( - 7Ts)
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- 21-
tiTTS
as
{si—
f+f-3-b-Ci-<a*b*c-e-f-i -7rjj{s)l
on th e imaginary a x is .
verges, and th e in te g ra l around
E(aabeo-e-f) < 0.
Thus th e o rig in a l in te g ra l con­
C tends to zero as p —*m when
The in te g ra l is th e re fo re equal to minus
27Ti
tim es th e sum of th e residues of th e integrand a t th e po les on th e
rig h t of th e contour.
(The minus si go. i s due to th e fa o t th a t in
th i s oase, th e d ire c tio n of progress along th e contour i s negative*)
S im ilarly, i f th e contour i s taken on th e l e f t of th e imaginary
a x is , th e in te g ra l i s equal to
27Ti
tim es th e sum of th e residues
o f th e integrand a t th e poles on th e l e f t of th e oontour.
For th e poles on th e rig h t of th e imaginary axis, are have
Utrs
®
(j/2 rri)
- s ) r ( a+8)f1(t>+B)[,( oas)
ds / 7(ees)/l[ff’s)
~ ~ l^ R ( n « 0 ,l,2 ,...)
-ioO
i t th e pole
s • n
Hits fT(n*'s)/1('bos)/7(oe8)
r ’(e*-s)/l[f+s)
where I j / ( - s )
i s an in te g ra l function.
Hy. -
- * lJr*
ht
Henee
r(a * n )r(t* n )r(o » n )
r(9+ n)r(t+ n)
« - F( a+n)r,(b+nyT( o+n)
r(V n )P (f* n )
1
nt
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p(f*n)r(*+ n)r{°*n)
f
Thus
1
/>(a)p(b)p(o) ^ f a , b. 0 7
r(e)r(f) ^AL e , f J
.
ioO
f l ins ri-*VK»**)r{*+a)rK<»*)
y
-Lea
r(«OrO>)r(°) ,-ra» b» «1
A e * s)/X f* s)
8“
/*( « ) / * ( * ) ? AL e , f
J
For th e poles on th e l e f t of th e imaginary axis*
<CO
-Led
How,
E (-a-n) - l i »
/ 1( e # - s ) r ( f + s )
L
- f
(liC f]
^ r j( ,a + a ) J
ihrfooOn ( am)/"^ b -a-n jf’f o-a-n) ( -1)”
---------------■ e
n!
r ( e -a -n j/^ f-a -n )
-s) • 7r/sin7Ts.
Hence
/^ (b -a -n ) •
c -irv
_ ( - l) V ( l- h e a M b - a )
Z7( l-b aafn ) sin (b -a)
/"'(l-b+a+n)
/Tr( o-a-n) *
_ ( -l)1’/ ’’ ( 1-oea) f \ o-a)
(-1 )V
/r7( 1-of a*n)
l-oeasn) sin( o-a)
r 7( e-a-n) ■
- ( - l ) > ( l- e + a y t(e-a)
(-tfv
/^ (l-e fa fn )
/r7( 1-e+a+n) sin( e-a)
/^ (f-a -n ) »
(-1 ?TT . . i - i T r
r*( 1 -fta tn ) sin( f - a) /"*(1-f* atn)
Therefore,
00
V7iota P{b~B)P( e-a)/^ a*n)/*(l-b+ajf^ l-o+a)/^( l-e+ atn )/^ 1-fea+n)
1
/ T(e-aV T( f - a y 7(l-e + a )T (l-f+a)/, (l-b+a^n)/7(l-o e a tn )
I t -a-n) ^ e
»r,
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1
n!
- 60 -
- *
^ ^ ) / n(b“ a)/7(o-a)
fa. 1-e+a, 1-f+ al
r(e-a )T (f-a )
3<l [ l-b*a, 1-o+a J
R(-b-n) - lim
(b*») * )
1
f^ rt(« + b + n ) J
( a + b ^ f l - ^ y -***1^?**)
r(*+«V'(3P*»)
±i7^b-n)p(b^n)/T( a-b-n)p( c-b-n)
( -l)*
/"*( e-b-n)/^ f-b -n )
e
fifth p ( bfn)p( »*b-n)p{ o-b-n)
~~Z'
•b-n)P( f-b-n)
nf
1
nf
But
r ( a-b-n) -
_ (-lfP ( a-b)P( 1-a+b)
< - l) V
P( 1- atb+n) sin7T( a-b) /~,(l-a*b+n)
P (o-b-n)
_ ( - l f / 7( 0.b M l-ofb )
(-D 7T
/_I(l-o*b+n) sinB( o-b)
/",(l-o*b*n)
P ( e-b-n)
_ (-l)V (e-b)r(l-e+b)
(-D >
P( 1-e+b+n) sin^t e-b) /''(l-e+b+n)
/"*( f-b -n )
Z7 ( 1-f+b+n) slnP{ f-b)
P( 1-f+b+n)
eo
^R (-b-n)
■>
1=C
ry,hP( a-b)r(o-b)p(b) /»(l-a+b)r( 1-o-b)
r(b»n)T( l-a*b+n)f*( l-f»b+n)
“ « r(e-b)f(f-b)
r (b )n l-afb)r( l-f+b)>(l-«»b^n)A 1-o+b+n)
X fi(-b -n ) -
»=o "* ’
/7(b)rl( *»b)/7( °’b )
^ ( e -b ^ f-b )
£ f b » l-»»b,
***- i
1-f+ b l
1-atb, l-c*b
J
S im ilarly
f " R(-o-n) - ^
T‘=°
cr ( 8W * - 0)P(b-o) c p*. 1-e+o, l- f * o l
/*(e -o )rtf-o )
3 l
l-b*-o J
Henoe, ne have
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-i 00
. .
rim T(aV7(b-a)/1(o-a)
fa,
n
P(e-a)ft f-a )
3hi [
♦
nvc
!^c
♦ e
1-e+a,
1 -f+ a l
I-**®* 1-o+aJ
r(^)T(a-b)r(o.b) P fb, l-e*b, 1-f+bl
T C e-bhf.b)
1-a+b, 1-o+bJ
a-oV ^b-e)
fo,
T ( e -c )r (f- o )
3 /i L
1-e+o,
l-a+o,
Equating re s u lts , we hare (m ultiplying f i r s t by
e
</7"S
1-f+ol
1-b+oJ
+LVS
and using e
in integrand)
7 O T T
j- [a,
T ( a rr*(b-e)n[o-a)
(b
? aL
/*<•- a ) f ( f - a )
1-e+a,
1-b+a,
-trt+^Wb)/K a-b)f( o-b) - lb,
* e
1-f+a]
1-o+aJ
1-e+b, 1-f+bl
V (® -b)/i(f‘-'b)
JL L
1-a+b, 1-o+bJ „
/^(e-oyXf-o)
- fc,
^1 [
1-e+c, 1-f+ol
1-a+o, 1—
b+oj #
'Tiour'
-M*r( a)r(b)H °) r- [a, b , o7
*----7 % e)r(t)------3 1 L e, f J
~ina
(M ultiplying by e
and
using e'
in integrand)
mr(a)TOi-*)r%°-*) p f®» 1-e*a» l-f*a7
" /"(e-a)T(f-a)
*** L 1-b+a, 1-o+a J
*6
(b)p( a-b)r(o-b)
r(e-b)Tfr" ‘ %
r7f-b)
p ( 0y*( a-oV ^b-o)
r(e-c)T(f-c)
p>* 1-e+b. 1-f+bl
L 1-a+b, 1-o+bJ
j- fo, 1-e+o, 1-f+ol
‘x L
1-a+o, 1-b+oJ .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Henee,
" m r lt)
p. [""&» b , o1
-■ S * L o , f J
- 2 i s i n ^ ( b . a » ^ l ± L ^ J b * 1' e+b* a- f+b]
r(e -b )r(f-b )
1 -a fb , 1-o+bJ
. 2 i . m r n > . tj P ( » ) n t ^ W t - < » )
Y 7(e -o )/X f-o )
r
l-* + o ,
i *fW
1-b+oJ
Sinoe
^ ( a J f X l- a ) ■ .I.. ?T.., o r
7
s in zTa
sinTT(b-a) ■
sin7Ta ■ ..... ■
vt^ y ■.
r{a)F{l-*)
___ -------------
F( a-b)r( 1 -a+b)
sin 7T( 0 -a) » ----
.
■v ,
P { a^o)r(1-a+o)
we hare, dividing by
n o > Y \ o)
2 ^i
r-fa , b, o l
-a) 3 *r L e, f J
.T(b)r(o-b)
"/‘I(e-b)P(f-bV'( 1 -a+b)
p-fb, 1 -e+b, 1 -f+bl
|_ 1“*#b» 1 -o+bJ
r to ^ C b -o )
+/^( e-o)r( f-oV tl+ o-a)
_ fo, 1-e+c, 1 -f+ o l
L 1-a+o, 1-b+oJ
Henoe
p-
fa, b, cl rd-^rCayXfVXo^b)
L e,f J rKe-bJ/Xf-bjrCl+b-ayCo)
/(l-a M erC fH b -o )
np-oy’Cf-oy’Cl+o-ajrtb)
n», b-e+l, b-f+ll
|_
1 +b-a]
jo, o-e+1 , o-f+ll
1 +o-b, l+o-aj
We tu r n now to th e corresponding problem i n baeio s e r ie s to
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In d icate th e o rig in o f th e d if f ic u ltie s involved.
The analysis here
i s p u rely formal — questions o f convergence are not discussed.
Consider th e in te g ra l
f +tj\ OCl/s)0( as) 0(bs)0( os)
L
where
f" J
o(.-.roTf«T—
*•
C i s the u n it o iro le with oenter a t th e o rig in and indented so
as to separate the two se ts o f poles — one se t approaching th e o rig in
and th e oth er in creasin g in absolute value.
f(s)
i s a reg u la r func­
tio n in th e f i n i t e plane whioh i s to be properly determined i n order
t o ju s tif y c e rta in steps in th e an aly sis.
The integrand has poles*
s • q (m ** 0*1* 2, « « ...)
» l/a q n(m -
0, 1, 2, . . . . . )
- l/b q V -
0, 1,2, ......... )
■ l/o q (m *• Of 1» 2$ ••«•»)
Denoting th e residue of th e integrand a t
i 9 by R ( 0 *
we hare
0 (l/s)0 (b s)0 (o s)
- . J*. « '
13m
o i e s"m r s j
£
^
1-asq
\ 0(l/s)0(bs)0(os)
. %7T ( l - $ +l)
f(s)
(.j/aq -) 3jjgr; "
W a,V "
f( V ,q ’
.
>
f t ( 1 -a sq ') IT(l-aaq"')
">*-o
Hsm«i
#0
0 (e s)0 (fs)
o W « r )o (t/* t)
f ( l / . q- ' ) ° ( ^
) 0< V ' t f , ) 0 ^
{- V ' i >
: ) ( . 3/ a f ) ...... 1
0( e/a< T )0 (f/aq ’)
Hip
,V(
(-ifq
,
o (^ )o (f/^ )
^
ot4w')
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or
•» W - f J
ik v m
/ . ^ . r w
>
A
f (v « 4
m
v V
)
(-i)
—
Vow wo her*
0(aq A M * / * )
(Kaq^j/b)
( 1-aq / b ) ( l- a q V b ) .. . . ( l- b /a ) ( l- b q / a ) . . .
(l-q)(l»<J*)«»»*
(1-aq / b ) ( l- a q Y b ) ...( l- * q V b ) ( l- V a ) ( l - b q / a ) * . .
( .ir (v .T 4 - ^ ( i- 4 ) ( i- o ( l-b /a q ^ )( 1-b/aq"'1) . . . ( 1-b/aq) ( 1-b/a) ( 1-bq / a ) . . .
•TO
,
-vrt
- (-1 ) ( l A ) 4
” 0 (V » O
or
Sim ilarly*
o(«AT) - (- i f ( v ' o f .r^ o W A W o A )
0 (a q "7 o )
0 (e /^ r) 0 ( a |’ / « )
0(aq / f )
H o bo ®
»
o ( ^ ’ )o(<tg A )tx t/« )o (« 4 /< < )°('v / » ) ° ( « « r 7 « ) o ( > 'r 7 f ) /» f r
0(<f")0(aq v/«)0(Vl«)0(»q A ) 0 ( f A ) 0 ( lu f "/>>)0(«u1’" '/ o ) l*bo/
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,
-
.7 * *
-rvt y n O » \n )
(-1 ) a q
—
*•
, , ^
'0 / ~ n *
0( aq /b )0 ( aq /o)Q ( aq^)0( a ^ 7 e ) 0 ( aj™ 7f)
/ ef Y
0(q~")0( a)0( aq /e )0 ( aq / f ) 0 ( aq™ 7b)0( aq~7o) I abc/
. 0( a)0(b/a)0( q/a)
0 (q /a )0 (f/a )
S im ilarly ,
H(V*b<n -
*(VW*> X
x 00>)<Kqfr)0(</b)
0 (V b )0 (f/b )
0 (b q /a)0 (b q /o )0 (b ^ » )0 (b q "w7 a ) 0 ( b f ^ /f)
0(<T/)0(b)0(bq /e )0 (b q /f)0 (b < f,7a)0(b< f‘7 ro)
/e f f
Ubo /
/
-*1
H (l/oq~)- ■■
' - - j -------- f O /o f O *
cq
0 (o )0 (« /e)0 (b /c)
0(oc/a)0(oql</b )0(oq>^0(oq>,v*/e)0(o< f*/f)
* 0 (a /o )0 (f/e )
/ e f V”
0( <F*)0(o)0( O q/e)0(oq/f)0(o<f*/a)0(oq"7b)l **>«/
Also,
s (r)-iiJ-+J7"
0( as)0(bs)0(es)
— '*
f ( .) o (7 ° g y a ,) ( . . f ) 7 r ^ v
o (e s)0 (fs)
/ I 1—q / s
7»S^
oO
0 (e s )0 (f8}
f o T l- q V s )
^ (< 2 o (b £ 2 o (o £ ) r j i ( i - < r <') _
o (.f')o (fq ')
T
rd -q * ”)usm#/
fta-<r")
nsd
w
(-1) q
^ 0( a<T)0(b<T) 0( °<T)
q *(q )o ^ ')o (a q ^ )o (f^ )
rm -tnCn**])
(-D q - <r
^
E r y s r — '(q > *
vSSo)
w 0 (a) 0 (b)0 (o)
* 0(e)0(f)
0(o)0(f)0(aq^)0(b^)0(oq")
(** 'T
0(q^)0(a)0(b)0(e)o(aq^)0(f<r)l«bo/
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Now, i f
f(s)
oould be determined so that
( - i f . '" '
( - if< j" " ' < r*?:i,f(]A q~5 • • LWi
TTC.
( - if .— q ^ f d /o T )
-
(- D V ^
& r
we would hare
V 'n / , /
^
“ *
° ( a)° (V a )0 (« /a) T fa, a q / e , a q / f ;
0 (« /a )0 (f/a )
A L a q /b . a q /o
i.^O O O tX v'bM c/b)
e f/a b o ]
-*
fb , b q / e , b q / f j ef/abo 1
bq / a, bq / o
y 1R(l/o<T^ » “iirc
J
^ 8)° (b/ ° ) i f®» 0<l/« » 0<1 A » ef/abo 1
0 (e /a )0 (f/o )
•*•«■L oq / a , oq / b
J
_ 0(a)0(b)0( o) £ fa, b , oj ef/abo ]
£ * ( i n • o('.')o'(f) M L
t
J
V «/
which are exactly analogous to th e oorresponding r e s u lts in th e o rig in a l
case*
Then, i f th e in te g ra l oonrerges, th e desired r e s u lt would follow*
i l l attem pts to construct a function
conditions hare failed*
f(z ) sa tis fy in g th e
above
This, o f course, does not n e c e ssa rily in p ly
th a t th e b a sic analogue does no t exist*
There are two p o ssib le con­
clu sions, e ith e r th e method i s not applicable, or th e basic s e rie s in ­
volved must be modified in much th e same way as th e s e rie s involved
in th e basio analogue o f Dixon* s theorem.
I t seems probable, however, th a t th e batsio analogue o f th e th re e term re la tio n does exist*
I f so, th a t p o rtio n of the theory connected
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w ith th e numerous re s u lts obtained by Thomae would be oonplete and a
more general statem ent concerning th e p o s s ib ility o f obtaining fu rth e r
r e s u lts useful i n arithm etic eould be made*
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CHAPTER V
The Basic Analogues o f Some Transformations of a W ell-Poised ^
The fo Honing m ateria l i s presented mainly to f i l l an obvious gap
in th e general theory of basio hypergeometric s e rie s .
O riginally* i t
was hoped th a t th e re s u lts might be susceptible to sim p lific a tio n s
which mould produce in te re s tin g arithm etic ad relations* b u t such does
not seem to be th e ease.
Us e sta b lis h here th e basio analogues of th e follow ing th ree
i d e n titie s due to B ailey
(22) (17)
* where
uevevr - x+y+s-n+1
y, s, -n ; 11
J
u, t , w
r(
y*( 1+y-u)p{ l-n -u )r(l-n -u )
( 1)
/"*(l* y -n -u )f( le*-n-u)P( leys t-u ) r( 1-u)
7
where
j_
a • yes-n-u • vrsv-x-1,
y, *, -n ; l l
u, ▼, vr
f 1(Temen)f,(le x -u )p (le y -u y , (les-tt)
J ^(leyez/ 1(ley es-u )r(l+ afx -u )/’7(le*sy-u)/7(l-u )
a», le a * /2, wen, ven,
a * /2,
where
t
x
,
y
»
z
}
( 2)
ll
w* 1+yfz-u, Aet+x-u, 1+xey-u
a* • x+y+s-u - -vewen-1.
(3)
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■where
(z)^
*(z+l)(z+2)...(sen-1).
•en published'
pi
The basio analogue of (1) has been
' but, for the sake
of completeness ire prove i t here, lb hare.(* >
T K
* 7L
<j/a» - q /a»
d»
«
*
f
*
e
\/a, -Va, aq / d, a q / e , a q / f , a q / g ,
» b
J
a2 qV def ghj
aq /h
-*
^ 0(aq /f)0 (a q /g )0 (a q /h )0 ( aq/fgh) _ faq/de,
f,
g,
hj
0( aq)0( aq / f g)0( aq / fh)0( aq / gfr)
a q /d , a q /e , fg l/a
ql (4)
J
^ 0(aq/d)0(aq/e)<Kaq/f)0(aq/g)0(aq/h)0(a^q*/defgh)0(fgj/aq)
0(aq)0(aq /de)0(f)0(g)0(h)0(a*-q /d fg h M a ^ q /e fg h )
*
T fa q /g h , a q /fh , a q /fg , a^qVdefgh; q~l
*^3 L
aq/fgh, a*"q*/dfgh, a^qYefgh J
/
I f in (4), aq/de,
f , g, or h is of the fora -1^
q#iere
1 is a posi­
tiv e integer, the seoond series on the right vanishes and we obtain
Ta, q/a, -q/a,
* * 7 |_
\/a, -Va,
d
,
e
,
f
,
g
, h; aW d efghl
a q /d , a q /e , a q /f , a q /g , aq /h
-*
Q(aq/f)0(aq /g)0(aq /h )0( aq /fgh) ^ Faq/de#
S# bj
g l (5)
0( aq)0( aq / f g)0( aq / fh)0( aq / gh)
[
aq /d , a q /e , fg l/a J
provided only that the series on the right terminates end the series on
the le ft converges.
Ta, l* t/z9
7^" [
o
(5) is the basio analogue of
,
d
,
e
,
f
,
g
J
*/2, l+a-o, 1+a-d, l+a-e. ++a-f, 1+a-g
l
l
J
Pi 1*a-e)f( !•»a-f )r( 1+*-g)r( 1+a-e-f-g)
r ( loa)P( l+a-f-g)r( 1-fa-g-eV^ 1+a-e-f)
f l+a-o-d,
e ,
f
,
g
^ [
1+a-o, 1+a-d, eefeg-a
}
ll
J
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I t transform s a w ell-poised
in to a Saalsohutsian
provided th a t
th e l a t t e r s e rie s term inates*
The v$3 in (5) term inates i f one of
form q ^ .
If
h u t, i f one o f
aq / de, f , g, or h
i s o f th e
i s of t h i s form* idle 8$7 does not term inate;
a q /d e
f , g, h
n ates as w ell as th e A
then th e g&t term i­
i s of t h i s form ( q ^ ) ,
.
Hence, th e re are two d is tin o t oases to be
considered*
Suppose th a t
__ «-i
uvwq
- ays
and put f i r s t
ti ■ q
u » fjft/a
a q /d ee -■ x
v -» a q / d
f - y
w » a q /e
a * n / x q » y j/u q
s - g
Then (5) beoomes
-*1
*» y* *# q.
r-
4
J
U» ▼» w
x)°(™
-
x
A)0( V W 1'' )
*
0(y/uq*'* )0( a/uq”"()0(y*q /u )0 ( q /u )
[a, <|/a, -q /a , t/ x ,
*
/a , V a,
v,
v /x ,
w,
y
,
s/uq11''
which i s th e basio analogue of (l)*
s
,
(6)
q"M
;
, y/uq*"' , y s q / u
x q /u l
J,
In t h i s case, both s e rie s terminate*
To obtain th e second case, we pu t
xtrwq~‘ * ays
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-o *-
and
sq /do ■ q->*
▼ ■ aq /d
h■x
w * aq / e
g« y
u • f g l/ a
f * t
Then (S) becomes
rx. y. X.
M
U.
«
V,
J
■
^
8
; ,1
A )o(yq A )o(«g /» )
0(y*q /u)0(x*q /u)0(xyq /«)0(q /u)
fa*, <j/a», -qVa*, wq71,
71
Va#, -Va*. v,
vrfaere a* • nrq17*1» xya/u
Vq*,
x
,
w, yzq /u ,
y
( 7)
,
z
J
l/uq™'1!
xzq / u, xyq/tt
J
and urwq’", « xye.
t $7 in (8) converge, I t i s neoessary th a t
In order t h a t th e s e rie s
jq,~>lj •
ju ( >
This i s th e
b asio analogue o f (2 ).
To prove th e basio analogue of (3 ), *e h a r e ^
J, £
W u]
” * ]-
(s,
Also, by (1)
Tt/ z, V x j ^ l
V'q” w;(TWq1'/ s ) i ? 'l
J" '4 t ™* / , , c l £ L
But, i f
v r .
uvw ■ xyn/q11' 1,
-^ E * y A js ]
and henoe (8) beoomes
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J
4TW
"1
3/qM'V; TwtfH
x» yJS l X
l/q ^ *
(9)
_
r
|V*# Vy; x y ^A l ^ [▼/*» V * ;^ !
"^ L
Jl9,L
U
li A
/ q r- ' z
J
We now proceed, to equate o o e ffio ien ts of ‘S’” on th e two sid es o f (9).
Wa hare
f r ( ^ r ( y k r y]
I
( V
£ ^ 0 / £ ^ ^ ' ' )%, s
(«v*o/<r'«)*,l«
> .
J
t
( V ^ k r ( V 'y k i- /i y \ V ? r - c ’( T A ^ a ( V t ) , . s
_
£=o%zo
^V-5
'
*
V^V^7 (V x )* r(V y k r(V * )» ,* (V * )is fx y S rs*>Y''hS
'
(q )^ (V ^ (v H )J U / ^
P u t N « r+s.
v ? r < r <4 . r t y y ^ W
V
f " » U
A
( V ( V*)g,r ( V y ^ r (
r
/
A lj
(q )f,r(u)f,r(^)yJA'-y(l/q^ ,*)f,/«'-r
-i—
/ TV
,f
7 <y v
t
/
J
■
(2^*7^ ^
'u • J
Equating o o e ffio ien ts of ^ /V ,110 o b tain
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- 36 -
V
*xh A f U , r (
*
/
i / q" ’'*)?,*-*
( 10)
_
( ^ ( ^ y U ^ r iV g ) ? ,r
/a ^
How we have
( V
( i - * ) ( i - » q ) . . . ( W " >w)
( * ) j> v
" ( W
) ( i - , r ’~ v . . ( i . I f ^
( - if( .y »
*
*j ^ t v v v
But
( 0 ?)w * ( l-* )( - z q ) ...( l- z < f * ‘)
/
, v"
-n
T itilr i)
/ , /
-n -i
- (-1 ) z q 2. (V * q
\
) ?l>u
Haaoe
(* W
( - « '( • > , , .
- / q>*^N (V' , (1« ) ?1,
/ , \'1+r -rt-r frVwr+sVrl /
,.
, (■!) g q_______ (V «q
( l / « r ‘) » , r
Using these r e s u lts , (10) becomes
q V W * ) y , ~ f 7 ( x)Slr (y)g,r( g)giv>( q~^)f,y> # qy
(q' " ^ t o
(q)y,Y-(u)^r(T)^r<w)^r»
« **qV<(T/ gV * (V * )9 ^ ^
(*/*)g,r(*/y)j>* ( « k r ( g “* ) S,r
# qr
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idler e
uvw » xy*/ q1*1.
This i s th e {dialogue of (3) and i s a new resu lt*
oheeked num erically w ith
and n » 1.
x - y • z - l/z ,
u ■ 2* ▼ • 2 # w - ]/32
Obviously* id e n tity (11) has a number o f equivalents ob­
ta in e d by permuting x, y, z or
of summation.
(11) has been
u* v , w or by reversing th e order
Several transform ations o f a w ell poised
oan be
obtained by using (6) or (7) with (11)*
0
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CHJPTER 71
A T h eta-Id en tity Due to B ailey*
In & reoent paper, B a ile y ^ 6^ gay© th e i d e n t i c
4 (*) $ (p * y ) %(?*$) 4 («*y ♦*)
- A {/3)4(*+ y)4(«+ Z )4(p+ r*S)
(i)
V ) 4 (« -p )4 (« */s+y*Z) .
♦
This id e n tity was obtained from o e rta in r e la tio n s involving basio
se rie s and i s subjeot to o e rta in oonditions which are made noessary
by Bailey* s an aly sis,
lb giro here an independent proof o f (1) and
then show th a t (1) i s equivalent to Jacobi* s fundamental formula( 18}
To prove (1) by L iouville* s method, l e t
Lf(oi) - 4(p+7)l$?(p+&)
- 4 (p )$ (p +y*$yih u ♦ 7 ) $(*+'b)/i% (<*)'$3(&+Y+H)
-
4 ( 7 ) 4 ( D $ (* -p )
and oonsider <p(c<)
as a function o f X .
I t i s e a s ily shown th a t ^(o<0
i f ( <rt)
is
+p*r+£)/ih (x)i4(*+t+i>)
has periods 7T and V T % Hence,
a doubly p erio d ic fbnotion with periods 7T and Tr'i"' t and
i t s only p o ssib le s in g u la ritie s in th e fundamental period p a ra lle lo ­
gram are at th e p o in ts 7T/2* TTf/z and 7T/2* 777/2- Y • 6 * —^ + m77^/
2
where n , m are properly selected in te g e rs.
This id e n tity , as given in Bailey* s peper, i s inoorreot — d if ­
fe rin g in th e sign of one o f th e terms.
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At <x- 7T/& 7ir/zt we hare
h
)ti(p*r*r> 2%<(nv)&W*l)
A (7*3)
[* v V f . V" ' J]
. , lq-w # iH p ) U e * y + i) M iW ,< .n
# (f *3)
Z^(s* Jr/2*7rr/z) - ( i / q y*e*Z)z4(s)
^ ( z + n-/2*7rr/2) - ( ] / q /v e‘Z) 4 ( * ) .
4(*+ r+ 5)
A V ) $ < .S ) U - M M > + t)
4 (x-*3)
q
'
& /« * .'‘'’ i d / s ’"
* i / qw . « » * '
1
t>rfr*s)
Since th ese would be th e c o e ffic ie n ts o f
Laurent expansion of the function
l/(tX- /7/2 -
TTf/z)
in th e
th e re i s no pole at
d • TT/z* 7Tf/z*
Sinoe ‘f (o(*£2.) *
we hare at c* » 7r/ 2+ TTT/ 2-T-
l/q'A
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.
- i -'/v
Also,
?hM
L
_
1 * * 7T/2 + r r r M - i - &
40)
(V<jfc«*, , ’,,d]tv'g/ *«A)
4 (r*S)
.
j / q '* * '" '" 4'
-*
Therefore, ^?(o()
has no pole a t o( * <7"/2+ ZTr/2 -/-S .
Hence ^ (c<)
is
a doubly p eriodio meroaorphio fm o tio n w ith so poles i n a o e ll and
by L iouvllle* s theorem^®^, i s a const ant.
low
<p((3) - ih{p*Y)ih(p+&)
ih (/?)
(/?+J)
fy({3)2%{/3+7+3)
# (y) # (<3)
Q)
y»7» X)
- 0
Thus*
z% iM)'%{fi+r)'ib{(3*S)&3U*Y+Z)
• Z % ^3 )A (/7 + ^ )^ + y )2 ^ (« :+ 5 )
Before proving th e equivalence of (1) and Jacobi* s formula* we
develop a few a u x ilia ry formulae which are im e d ia te equivalents of
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-41-
(1 ).
I n (1)# P » t
w ■ (S+Y+&
w* * «(
x ■ <*♦ S
x* - /j ♦ y
Y m oi+ 7
yf m id is'
z » (3
I* •
(2)
Then we hare
2w ■ - w, +x, +y, +2f
2w* ■ - w*-xsyas
2x ■
w '-x'+yf+z’
2x* ■
wwxfy+s
2y ■
w*+x, -y*+z,
2y* «
w»-x-y+z
2z -
w^+x’+y^z*
2s* »
w*x*y-z
(5)
and (1) becomes
&(w, ) z & * ) 4 ( y , )4 < « '») - 2 i ( w ) ^ ( x ) ^ ( y ) ^ ( z )
♦ 4"(y-w*)
Increasing
x, y, z, w (and thus also
*-**) # (»• -*)
w+wr*)
(4)
x* , y f , **, w*) by TT/ 2,
(4) becomes
A < « , >2&*, )1& 7, )1& «, > - 4 w 4 ( * ) 4 < y ) 4 ( ‘ )
(5)
Increasing a l l th e v a ria b le s of (4) and (S) by 7TT/2, -we o b tain
£<w, )A(x*)A(y’ )4 (« , > - 4 .0 ) 4 ( *)4 ( y ) 4 ( «)-4 (y -» ')
-*)4 ( ^ T o
i> - ( V ) 4 '( i') ^ y , ) l|( * ') - 4(w)4-(*)£?(y)2f(r)
♦
y -v )jf(* .w < )^i» » -« )lj'(iin r')
From th ese four r e s u lts , we o b tain th e followings
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(7)
- 42 -
[4]»
♦ [3]* * [4] .♦ [3]
C l]'
♦ [23* - Cl] ♦ C2]
[2]*
+ [3]* - [2] ♦ [2]
[4]*
♦ [1]* » [4] ♦ [1]
(8)
and
[83* - [13* - [3] - I I ]
(9)
[43* ♦ [23* - [4] - [2]
whore
[r]=
l£(w ) 2£(x)i^(y)zfy(*)
and
[r]*3
)? £ (* ')•
To show th a t Jacobi* s fundamental form ula oan be derived from
Bailey*$ id e n tity
i t i s s u ffic ie n t to show th a t
--
% (f)T> ,t**l)% k+ S)0;{/H 7i!S) + l*z{p)&<*+*)&&+f)&(p+r+Z)
(A)
and
* Z%>(«)&((3+y)7b({S+2)1%(#+/+?)
oan be obtained from Bailey* s id e n tity by elementary transform ations
(adding h a lf periods to arguments).
In th e n o tatio n o f JI.J.S . Smith,
(23)
( a) and (B) oan be w ritte n
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(A*) [3] - [4] ■
[ l ] * ♦ [2]*
(B») [3] ♦ [4] -
[3]* ♦ [4]*
witore
w* *
“x-/3*?
y m {3 +S
w* - ft+Y+S
x» - <*♦<*
y» - <*■+ $
Z m of +Y+& t* m p
Aiding (A*) and (B*) we o b tain Jaeobi*s formula
2(3] -
[1]* ♦ [2]* ♦ [3]» ♦ (4]*
10 f i r s t prove
ik(<*)ih(p+y)tii{(i+s)Th(?t+r+f)
AMAjfi ♦ /) &(p+2)A(**y*S)
-
(A)
■- A f t ) A t e +7) A u+2) A (p+y+i) * A i^ A f c + y ^ z f a * * ) zfa+ r+ S).
To do this* put
pC■
—* e<
p-*[3+ 7T T/2
y -^ y ♦
ttt/%
&—
>&*TTT/z
in th e o rig in a l id e n tity and obtain
( 10)
- M / ) M S ) l f a - /S)&(t<+/WS)
low put
/2
oi-#> o( +
TT
P ^> p
tt/z + ttt/z
♦
Tf-+ Y ♦ ffj/g
S*~> J ♦ 777J/ 2
i n B a ile y 's id e n tity and o btain
i ^ ( « 0 A(/3*$)A(*+y+}) - # ^ ) 4 - ( * + r ) # ( * + / ) 4 (p+r+S)
(11)
* * k ( y ) A( J )
+/£+?+£)
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From (10) and (11), then we get th e da sire d id e n tity (A)*
follow s in n ed iately from (8 ) .
How (B)
Hence, adding (A) and (B) we hare
Jacobi* s fundamental formula*
Conversely, to show th a t Bailey*s id e n tity oan be obtained from
Jaoobi*8 fundamental formula, pu t
w • p+y+S
X
■ <*♦ f
X*
y - <*. * y
E
w* - *
- (S * i
( 12)
y» -
* f3
E*
a P<
$
and w rite
where th e su b serip t r e f e rs to th e determ inations of w, x, y, s
given
in equations (12)*
put
w m1
X m
$
ym
[3
X*
- of
% - o<ep*-re^'
*
(3
♦
i
y* ■e(+y* <T
** -
- cK
and w rite
2 [l)/v - [ I ] ', * [2 ];^ - [3]«¥ e [4]*y
or
-2 [ i ]/¥ - -[i)* ¥ - [2]*¥ e [3]*¥ - [4];¥
o r, sinoe
CDfv
-- W *
[2]*y
- [2 ]k
[s]fy
- [3]!*.
[4]*,
- [4]*,.
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(14)
-so ­
ws
hove
- 2 (l]/ y «
♦ £3]*^ ♦[4]/V
(15)
Now adding (13) and (1 5 ), -we have
t%
- t l ] , , - t* ]J x
whioh i s B a ile y 's identity*
Hextoe, B a ile y 's id e n tity ie equivalent to
Ja o o b i's fundamental formula*
F in a lly , we note th a t i f we m u ltip ly (1) by
we obtain
fiSl W 'tf u , («♦'?’')
(16)
“ Vi13 ( «, -fi) tjhfr-*4 P *'
where* 19>
T) •
Obviously (16) w ill give r i s e to a number of other sim ilar re la tio n s
obtained by in creasin g th e param eters by half-periods*
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CHAPTER 711
Conclusion
The development o f basio se rie s p a r a lle ls th e theory o f th e c la s­
s ic a l gaussian hypergeometric s e rie s in as much as the b asic s e rie s —
as defined by Watson^®) — should be oonsidered as an analogue, ra th e r
th an a g e n eralisa tio n o f th e c la s s ic a l s e rie s .
Thus, re la tio n s among
o la s sio a l hyper geometric se rie s o ften lead one t o suspect th e e x is t­
ence of analogous re la tio n s cmong basio se rie s and in t h i s way, most
o f th e known re la tio n s in basio s e rie s have been discovered.
Though
th is fa o t furnishes a very u sefu l guide in studying basio se rie s, i t
i s of l i t t l e value whenever any but th e sim plest of th e functions of
param eters i s encountered.
This was th e ease in Chapter 17 when we
( 5)
attem pted to fin d a basio analogue o f Thomas* s three-term r e la tio n .
The problem introduoed th e re i s th a t of fin d in g a function analogous
to th e exponential function.
problem introduoed here was th a t of in te g ra ls o f th e
( 24)
( 5)
Barnes
type fo r expressions involving th e function
0( x). I t
A nother
may w ell be th a t the complete so lu tio n o f th e d i f f ic u lt i e s i n
Chap­
t e r 17 depends upon t h is type o f study.
The d i f f ic u lt i e s involved in fin d in g a basio analogue o f Dixon* s
theorem(13)' seem to be muoh more complex. At p resen t, th e re i s no way
o f being c e rta in of th e existence o f such a re la tio n .
And, i f i t does
e x is t, i t i s q uite probable th a t e n tire ly new methods must be develop­
ed.
In general th en , i t can be sa id , th a t th e theory of b asic hypergeometric s e rie s i s not a n a ly tic a lly sa tis fa c to ry since most of th e
av ailable methods depend upon th e analogy w ith the o la s sio a l gaussian
hyper geometric s e rie s .
I t i s hoped, however, th a t th e use of th e
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/ g \
modified Heine Omega-funotion;
as in th e above, w ill serve to es^jha-
siz e th is analogy and sim plify th e n o ta tio n generally*
The question o f ap p licatio n s to Arithmetic i s a d if f ic u lt one*
The
number o f d is tin o t re la tio n s i s lim ited and th ere have been no fu rth e r
id e n titie s suoh as those given by P ro fe sso r B ell which might serve as
in s tru c tiv e examples.
However, a study of a number of paraphrases
(25)
given by P rofessor B ell in a reoent paper
i s planned* The objeot
o f t h i s study w ill be to o b tain these desired examples and attem pt to
c o rre la te them w ith o e rta in re la tio n s in basio series*
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BIBLIOGRAPHY
(1 ).
B ailey, W. N., An Algebraio Id e n tity , J* Load. Math* Soo*, 11,
156-160 (1956)
(2 ).
B e ll, B. I . , Bull* Asa. Math. Soo*, June 1956, 577
(5 ).
H all, Henman A*, An Algebraio Id e n tity , J* Lond Math. Soo*, 11,
276 (1956)
(4)*
B ailey, W* H., Generalized Hypergeometrlo S eriea, Cambridge
Traot Ho. 52, Chcpter V III.
(5 ).
Thomae, J . , Journal fu r Math., 87, 26-75, (1879).
See also
B ailey, W.H., Generalized Hypergeometrlo S eriea,
Cambridge Traot Ho. 52, Chapter I I I .
(6 ).
B ailey, W*H., Some I d e n titie s Conneoted w ith R epresentations o f
Humbers., J . Lond. Math. Soo*, 11, 286 (1956)
( 7 ).
B e ll, E. T ., The Form 2nx ♦ *y + yz ♦ zu * ux, Am* J . o f Math.,
58, 282 (1956)
( 8 ),
Heine, E ., Theorie derKugelfunotionen, 1, 91-125 (1878)
( 9 ).
Matson, G. H., A Hew Proof of th e Rogers-Ramanujan I d e n titie s ,
J* Lond* Math* Soo*, 4, 4—
9 (1929)
(1 0 ). B ailey, V* N., Generalized Hypergeometrio S e ries, Cambridge
Traot Ho. 52, P . l l
(11). Ih ip p le , F.J.W ., A Group of Generalized Hypergeometrio S eriest
R elations between 120 A llied Series of th e Type
F[a, b, o; d, e ], Proc. Lond. Math. Soo. (2)
25, 625-544 (1925)
(12)* Summer, E. E ., Ueber die hyporgeometrisohe Relhe, Journal fu r Math.
15, 59-85 (1856).
See also Bailey* s Traot P . 9
(15). Disoon, A. 0 ., Summation of C ertain S e rie s, Proo. Lond. Math. Soo.,
(1 ), 55, 285-289 (1905)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
. 49 .
(1 4 ). Whtson, G. H., Dixon* b Theorem on Generalized Hypergeometrio
Functions, Proo. Lond., Math. Soo., (2) 22 (1924)
x m i - XXXIII (Records fo r 17 May, 1925)
(1 5 ).
B a ile y ,
W.N., S eries of gypergeometric Types ehioh are I n f in ite In
Both D ireotlone, Q uart. J . o f Math. (Ox) jt
105 (1936)
(1 6 ). "Whittaker o Watson, Modern Analysis, 4th E dition, p . 431
(1 7 ). B ailey, W. N«, Generalized Hypergeometrio S eries, Cambridge Traot
Ho. 52, 65.56 (1935)
(1 8 ). Jacobi, C .G .J., Ges. Math.Wsrke, X p . 505
(19). B e ll, E. T ., A rithm etical Paraphrases, ( I I ) , T ransactions to . Moth.
Soo., ZZ» Ho. 2, 206 (1921)
(2 0 ). L io u rille , J . , Comptes rendus, P a r is , 62, 714 (1866)
(2 1 ). Jackson, F. H., Summation of q.Hypergeometric S eries, Messenger o f
Math., 60, 101, (1921)
(22>. B ailey, W. H ., An Extension of Whipple* s Theorem on W all-Poised
Hypergeometrio S e rie s, proo. Lond. Math. Soo. (2)
31, 505-611 (1930)
(2 3 ). Smith, H .J .S ., Proo. Lond. Math. Soo., I , May 21, 1866 pp 1-12
(2 4 ). Barnes, E. W., A Hew Development o f the Theory of th e Hypergeometrio
Functions, Prop.Lond.Math.Soo. (2) 6, 141 (1908)
(2 5 ). B e ll, E .T ., Doubly P erio d ic Funotions of th e Seoond Hind and the
irith m e tio a l Form xy ♦ sir, to . J . o f M ath., 57,
245 (1935)
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