Summation and transformation formulas for elliptic hypergeometric series

21 Pages·0.27 MB·English

by  S. O. Warnaar

SUMMATION AND TRANSFORMATION FORMULAS FOR
ELLIPTIC HYPERGEOMETRIC SERIES
0
0
0 S.O. WARNAAR
2

n Abstract. Usingmatrixinversionanddeterminantevaluationtechniqueswe
a prove several summation and transformation formulas for terminating, bal-
J anced, very-well-poised,elliptichypergeometricseries.

3

]
A
Q 1. Introduction
h. In the preface to their book “Basic Hypergeometric Series” [17], Gasper and
t Rahman refer to the enchanting nature of the theory of q-series or basic hypergeo-
a
metric series as the highly infectious “q-disease”. Indeed, from the time of Heine,
m
about a century and a half ago, till the present day, many researchers have been
[
pursuing the task of finding q-analogues of classicalresults in the theory of special

1 functions, orthogonal polynomials and hypergeometric series. It thus seems some-
v whatsurprisingthatwhatappearstobethenextnaturallineofresearch,replacing
6
“q-analogue”by “elliptic analogue”, has so-far found very few practitioners.
0
The elliptic (or “modular”) analogues of hypergeometric series were introduced
0
1 by FrenkelandTuraev[15] intheir study ofelliptic 6j-symbols. These 6j-symbols,
0 whichcorrespondto certainelliptic solutionofthe Yang–Baxterequationfoundby
0 Baxter[3] and Date et al. [11], can be expressedin terms of elliptic generalizations
0
of terminating, balanced, very-well-poised φ series. Moreover, the tetrahedral
/ 10 9
h symmetryoftheelliptic6jsymbolsimpliesanellipticanalogueofthefamousBailey
t transformationfor φ series. Sofar,theonlyfollowupontheworkofFrenkeland
a 10 9
m Turaev appears to be the paper [38] by Spiridonov and Zhedanov, who presented
severalcontiguous relationsfor the elliptic analogueof φ seriesand who studied
: 10 9
v an elliptic version of Wilson’s [40] family of biorthogonal rational functions. As
i
X an independent development towards elliptic analogues, we should also mention
the work by Ruijsenaars [35] and Felder and Varchenko [14] we studied an elliptic
r
a variant of the q-gamma function.
The aim of this paper is to prove several further results for elliptic hypergeo-
metric series. After an introduction to basic and elliptic hypergeometric series in
section2,we use section3to deriveanelliptic matrix inverse. This matrix inverse,
which generalizes a well-known result from the theory of basic series, is used re-
peatedly in section 4 to derive an extensive list of summation and transformation
formulasforterminating,balanced,very-well-poised,elliptichypergeometricseries.
The “q limits” of most of these identities correspond to known results by Gasper
and Rahman, Gessel and Stanton, and Chu. In section 5 we establish an elliptic,
multivariable extension of Jackson’s φ sum associated with the C root system,
8 7 n
generalizingthebasiccaseduetoSchlosser. Ourproofinvolvesanellipticextension
1991 Mathematics Subject Classification. Primary33D15,33D67,33E05; Secondary05A30.
1
2 S.O. WARNAAR
ofa generaldeterminantevaluationby Krattenthaler. We conclude the paper with
a conjectured C Bailey transformation for elliptic hypergeometric series.
n
2. Basic hypergeometric series and their elliptic analogues
Assume |q|<1 and define the q-shifted factorial for all integers n by
∞
(a;q)
(a;q) = (1−aqk) and (a;q) = ∞ .
∞ n (aqn;q)
∞
k=0
Y
Specifically,
n−1(1−aqk) n>0
k=0
(a;q) = 1 n=0
n Q
1/ −k=n0−1(1−aqn+k)=1/(aqn;q)−n n<0.
With the usual condensedQnotation
(a ,...,a ;q) =(a ;q) ···(a ;q)
1 m n 1 n m n
we can define an φ basic hypergeometric series as [17]
r+1 r
∞
a ,a ,...,a (a ,a ,...,a ;q)
φ 1 2 r+1;q,z = 1 2 r+1 kzk.
r+1 r
b ,...,b (q,b ,...,b ;q)
1 r 1 r k
h i Xk=0
Hereitisassumedthattheb aresuchthatnoneofthetermsinthedenominatorof
i
theright-handsidevanishes. Whenoneofthea isoftheformq−n(nanonnegative
i
integer) the infinite sum over k can be replaced by a sum from 0 to n. In this case
the series is said to be terminating. A φ series is called balanced if b ...b =
r+1 r 1 r
qa ...a and z = q. A φ series is said to be very-well-poised if a q =
1 r+1 r+1 r 1
a b =···=a b and a =−a =qa1/2. In this paper we exclusively deal with
2 1 r+1 r 2 3 1
balanced, very-wellpoised series (or rather, their elliptic analogues)and departing
fromthe standardnotationof GasperandRahman’s book we usethe abbreviation
a ,qa1/2,−qa1/2,a ,...,a
1 1 1 4 r+1
W (a ;a ,...,a ;q)= φ ;q,q
r+1 r 1 4 r+1 r+1 r a1/2,−a1/2,qa /a ,...,qa /a
1 1 1 4 1 r+1
h i
∞ 1−a q2k (a ,a ,...,a ;q) qk
1 1 4 r+1 k
= ,
1−a (q,a q/a ,...,a q/a ;q)
1 1 4 1 r+1 k
k=0
X
where we always assume the parameters in the argument of W to obey the
r+1 r
relation (a ...a )2 =ar−3qr−5.
4 r+1 1
OneofthedeepestresultsinthetheoryofbasichypergeometricseriesisBailey’s
transformation [2], [17, Eq. (III.28)]
(2.1) W (a;b,c,d,e,f,g,q−n;q)
10 9
(aq,aq/ef,λq/e,λq/f;q)
= n W (λ;λb/a,λc/a,λd/a,e,f,g,q−n;q),
10 9
(aq/e,aq/f,λq/ef,λq;q)
n
where
bcdefg =a3qn+2 and λ=a2q/bcd.
This identity contains many well-known transformation and summation theorems
forbasicseriesasspecialcases. Forexample,settingcd=aq(sothatλb/a=1)and
ELLIPTIC HYPERGEOMETRIC SERIES 3
thenreplacinge,f,g byc,d,egivesJackson’sq-analogueofDougall’s F sum[22],
7 6
[17, Eq. (II.22)]
(aq,aq/bc,aq/bd,aq/cd;q)
(2.2) W (a;b,c,d,e,q−n;q)= n ,
8 7
(aq/b,aq/c,aq/d,aq/bcd;q)
n
where
bcde=a2qn+1.
To introduce the elliptic analogues of basic hypergeometric series we need the
elliptic function
(2.3) E(x)=E(x;p)=(x;p) (p/x;p) ,
∞ ∞
for |p|<1. Some elementary properties of E are
(2.4) E(x)=−xE(1/x)=E(p/x)
and the (quasi)periodicity
(2.5) E(x)=(−x)kp(k2)E(xpk),
which follows by iterating (2.4).
Usingdefinition(2.3)onecandefineanellipticanalogueoftheq-shiftedfactorial
by
n−1E(aqk) n>0
k=0
(2.6) (a;q,p) = 1 n=0
n Q
1/ −k=n0−1E(aqn+k)=1/(aqn;q,p)−n n<0.
Note that E(x;0) = 1−x aQnd hence (a;q,0)n = (a;q)n. Again we use condensed
notation, setting
(a ,...,a ;q,p) =(a ;q,p) ···(a ;q,p) .
1 m n 1 n m n
Manyofthe relationssatisfiedbytheq-shiftedfactorials(see(I.7)–(I.30)of[17])
trivially generalize to the elliptic case. Here we only list those identities needed
later. The proofs merely require manipulation of the definition of (a;q,p) ;
n
(2.7a) (aq−n;q,p) =(q/a;q,p) (−a/q)nq−(n2)
n n
(2.7b) (aq−n;q,p) =(q/a;q,p) (a;q,p) q−nk/(q1−k/a;q,p)
k n k n
(2.7c) (aqn;q,p) =(aqk;q,p) (a;q,p) /(a;q,p) =(aq;q,p) /(a;q,p)
k n k n n+k n
(2.7d) (a;q,p)n−k =(a;q,p)n(−q1−n/a)kq(k2)/(q1−n/a;q,p)k
(2.7e) (a;q,p) =(a,aq,...,aqk−1;qk,p) .
kn n
Finally we will need the identity
(2.8) (a;q,p) =(−a)nkpn(k2)qk(n2)(apk;q,p) ,
n n
which follows from (2.5) and (2.6).
AfterthesepreliminarieswecometoFrenkelandTuraev’sdefinitionofbalanced,
very-well-poised,elliptic (or modular) hypergeometric series [15],
∞ E(a q2k) (a ,a ,...,a ;q;p) qk
1 1 4 r+1 k
(2.9) ω (a ;a ,...,a ;q,p)= ,
r+1 r 1 4 r+1
E(a ) (q,a q/a ,...,a q/a ;q,p)
1 1 4 1 r+1 k
k=0
X
4 S.O. WARNAAR
where (a ...a )2 = ar−3qr−5. Following [15] we will stay clear of any conver-
4 r+1 1
genceproblemsbydemandingterminatingseries,i.e.,oneofthea (i=4,...,r+1)
i
isoftheformq−n withnanonnegativeinteger. RemarkthatbyE(x;p)E(−x;p)=
E(x2;p2) the above ratio of two elliptic E-functions can be written as
(qa1/2,−qa1/2;q,p1/2)
1 1 k.
(a1/2,−a1/2;q,p1/2)
1 1 k
Hence in the p → 0 limit we recover the usual definition of a balanced, very-well-
poised, basic hypergeometric series.
An important result of Frenkel and Turaev is the elliptic analogue of Bailey’s
transformation (2.1).
Theorem 2.1. Let bcdefg =a3qn+2 and λ=a2q/bcd. Then
(2.10) ω (a;b,c,d,e,f,g,q−n;q,p)
10 9
(aq,aq/ef,λq/e,λq/f;q,p)
= n ω (λ;λb/a,λc/a,λd/a,e,f,g,q−n;q,p).
10 9
(aq/e,aq/f,λq/ef,λq;q,p)
n
Of course we can again specialize cd=aq to arrive at an elliptic Jacksonsum.
Corollary 2.2. For a2qn+1 =bcde there holds
(aq,aq/bc,aq/bd,aq/cd;q,p)
(2.11) ω (a;b,c,d,e,q−n;q,p)= n .
8 7
(aq/b,aq/c,aq/d,aq/bcd;q,p)
n
3. A matrix inverse
Beforederivingnewsummationandtransformationformulasforelliptichyperge-
ometricseriesweneedto prepareseveral,mostlyelementary,resultsforthe elliptic
functionE ofequation(2.3). Thiswillresultinamatrixinversethatwillbe atthe
heart of all results of the subsequent section.
The obvious starting point is the well-known addition formula [39]
(3.1) E(ux)E(u/x)E(vy)E(v/y)−E(uy)E(u/y)E(vx)E(v/x)
v
= E(xy)E(x/y)E(uv)E(u/v).
x
By iteratingthis equationone readilyderivesthe followinglemma, which forp=0
reducesto a resultofMacdonald,firstpublished byBhatnagarandMilne [4, Thm.
2.27].
Lemma 3.1. For n a nonnegative integer and a ,b ,c ,d (j = 0,...,n) indeter-
j j j j
minates there holds
n
(3.2) b /c E(a b )E(a /b )E(c d )E(c /d )
k k k k k k k k k k
k=0
X
k−1 n
× E(a c )E(a /c )E(b d )E(b /d ) E(a d )E(a /d )E(b c )E(b /c )
j j j j j j j j j j j j j j j j
j=0 j=k+1
Y Y
n n
= E(a c )E(a /c )E(b d )E(b /d )− E(a d )E(a /d )E(b c )E(b /c ).
j j j j j j j j j j j j j j j j
j=0 j=0
Y Y
ELLIPTIC HYPERGEOMETRIC SERIES 5
Proof. Wecarryoutinductiononn. Forn=0thelemmaisnothingbut(3.1)with
u = a , v = b , x = c and y = d . Now write (3.2) as L = R and assume this
0 0 0 0 n n
to hold for n≤m−1. With the abbreviations
f =E(a b )E(a /b )E(c d )E(c /d )
j j j j j j j j j
g =E(a c )E(a /c )E(b d )E(b /d )
j j j j j j j j j
h =E(a d )E(a /d )E(b c )E(b /c )
j j j j j j j j j
we then have
m−1
b
m
L =h L + f g
m m m−1 m j
c
m
j=0
Y
m−1 m m−1
b
m
=h g − h + f g
m j j m j
c
m
j=0 j=0 j=0
Y Y Y
m m
= g − h =R ,
j j m
j=0 j=0
Y Y
where in the second line we have used the induction hypothesis and in the third
line the addition formula (3.1) in the form h +b f /c =g .
m m m m m
Making the substitutions
a →(abd2)1/2, b →(ab/c2)1/2, c →(ab)1/2rj, d →(a/b)1/2qj
j j j j
and using the definition of the elliptic analogue of the q-shifted factorial (2.6) and
the relations (2.7) it follows that
n E(aqkrk)E(bq−krk)(a/c,c/b;q,p) (abd,1/d;r,p) qk
k k
E(a)E(b) (cr,abr/c;r,p) (q/bd,adq;q,p)
k k
k=0
X
E(c)E(ab/c)E(ad)E(bd) (a/c,bq−n/c;q,p) (abd,dr−n;r,p)
n+1 n+1
= 1− .
E(a)E(b)E(cd)E(abd/c) (bdq−n,ad;q,p) (r−n/c,ab/c;r,p)
(cid:18) n+1 n+1(cid:19)
For p = 0 this corresponds to [18, Eq. (2.7)] of Gasper and Rahman. Important
will be the specialization obtained by choosing d=rn,
n E(aqkrk)E(bq−krk) (a/c,c/b,q,p) (abrn,r−n;r,p) qk
k k
(3.3)
E(a)E(b) (cr,abr/c;r,p) (qr−n/b,aqrn;q,p)
k k
k=0
X
E(c)E(ab/c)E(arn)E(brn)
= .
E(a)E(b)E(crn)E(abrn/c)
WewillusethisidentityinthenextsectiontoproveTheorem4.1. Nowitisneeded
to obtain the following pair of infinite-dimensional, lower-triangularmatrices, that
are inverses of each other
(aqkrk,qkr−k/b;q,p)
n−k
f =
n,k (r,abr2k+1;r,p)
n−k
f−1 =(−1)n−kq(n−2k)E(aqkrk)E(qkr−k/b) (aqk+1rn,qk+1r−n/b;q,p)n−k.
n,k E(aqnrn)E(qnr−n/b) (r,abrn+k;r,p)
n−k
For p = 0 this is [8, Eqs. (4.4) and (4.5)], [16, Eqs. (3.2) and (3.3)] and [24, Eq.
(4.3)]. To derive it from (3.3)we follow [17, Sec. 3.6]and set c=1 followedby the
6 S.O. WARNAAR
replacements n → n−l, k → k−l, a → aqlrl and b → bq−lrl. By (2.7) one then
finds the desired orthogonality relation
n
(3.4) f−1f =δ
n,k k,l n,l
k=l
X
withf andf−1 as givenabove. Finally replacingr →qr and using (2.7)yields the
new inverse pair
E(abq2rk) (aqn;q,p) (ab,q−rn;qr,p)
(3.5a) f = rk k qrk
n,k E(ab) (bq1−n;q,p) (qr,abqrn+r;qr,p)
rk k
(b;q,p) E(aq(r+1)k)E(bq(r−1)k)
(3.5b) f−1 = rn
n,k (aq;q,p) E(a)E(b)
rn
(a,1/b;q,p) (abqrn,q−rn;qr,p)
× k k qk.
(qr,abqr;qr,p) (q1−rn/b,aqrn+1;q,p)
k k
This last pair of inverse matrices will be used repeatedly in the next section. We
note that it also follows by the (simultaneous) substitutions a →ab, b →aqi and
i
c →qri in the following elliptic analogue of a result due to Krattenthaler [24, Eq
i
(1.5)].
Lemma 3.2. Let a and b ,c (i∈Z) be indeterminates (such that c 6=c for i6=j
i i i j
and ac c 6=1 for i,j ∈Z). Then (3.4) holds with
i j
n−1E(c b )E(ac /b )
j=k k j k j
f =
n,k n c E(ac c )E(c /c )
Qj=k+1 j k j k j
and Q
f−1 = E(ckbk)E(ack/bk) nj=k+1E(cnbj)E(acn/bj).
n,k E(cnbn)E(acn/bn)Qjn=−k1cjE(acncj)E(cn/cj)
Proof. Sinceforn=l(3.4)clearlyholdswQemayassumen>linthefollowing. Now
letn>0in(3.2)andmakethereplacementsn→n−l,k →k−l anda →a1/2c ,
j l
b → a1/2c , c → b /a1/2, d → a1/2c . Noting that, in particular, a = d
j n+l j j+l j j+l 0 0
and b = d so that the right-hand side of (3.2) vanishes, and after performing a
n n
fewtrivialities,onefinds(3.4)(withn>l)withf andf−1givenbyLemma3.2.
4. Summation and transformation formulas
Ourapproachtoelliptichypergeometricsummationandtransformationformulas
is a standard one in the context of basic hypergeometric series, see e.g., [9, 10, 19,
20, 24, 34]. Given a pair of infinite-dimensional, lower-triangular matrices f and
f−1, i.e., a pair of matrices such that (3.4) holds, the following two statements are
equivalent
n
(4.1) f a =b
n,k k n
k=0
X
and
n
(4.2) f−1b =a .
n,k k n
k=0
X
ELLIPTIC HYPERGEOMETRIC SERIES 7
Our first example of how this may be usefully applied arises by noting that,
thanks to (3.3), equation (4.2) holds for the pair of matrices given in (3.5) with
(bq;q,p) (c,ab/c;qr,p)
rn n
a =
n (a;q,p) (abqr/c,cqr;qr,p)
rn n
(a/c,c/b,q,p) (qr,abqr;qr,p)
n n
b = .
n (cqr,abqr/c;qr,p) (a,1/b;q,p)
n n
Hence also (4.1) holds leading to
n E(abq2rk) (bq,aqn;q,p) (ab,c,ab/c,q−rn;qr,p)
rk k qrk
E(ab) (a,bq1−n;q,p) (qr,abqr/c,cqr,abqrn+r;qr,p)
rk k
k=0
X
(a/c,c/b,q,p) (qr,abqr;qr,p)
n n
= .
(cqr,abqr/c;qr,p) (a,1/b;q,p)
n n
By (2.7) we may alternatively write this in hypergeometric notation as follows.
Theorem 4.1. For r a positive integer there holds
ω (ab;c,ab/c,bq,bq2,...,bqr,aqn,aqn+1,...,aqn+r−1,q−rn;qr,p)
2r+6 2r+5
(a/c,c/b,q,p) (qr,abqr;qr,p)
n n
= .
(cqr,abqr/c;qr,p) (a,1/b;q,p)
n n
When r =1 this corresponds to the specialization bc=a of the elliptic Jackson
sum (2.11).
With Theorem 4.1 at hand we are prepared for the proof of the following qua-
dratic transformation.
Theorem 4.2. Let bcd = aq and ef = a2q2n+1. When g = a/b or g = a/e, there
holds
n E(aq3k) (b,c,d;q,p) (e,f,q−2n;q2,p)
k k qk
E(a) (aq2/b,aq2/c,aq2/d;q2,p) (aq/e,aq/f,aq2n+1;q,p)
k k
k=0
X
(aq2,a2q2/bce,a2q2/bdeg,agq2/cd;q2,p)
n
=
(a2q2/beg,agq2/c,aq2/d,a2q2/bcde;q2,p)
n
× ω (ag/c;a/c,gq2/c,beg/a,d,f,g,q−2n;q2,p).
10 9
For p = 0 and g = a/b this identity reduces to a transformation of Gasper and
Rahman [18, Eq. (5.14)]. We should also remark that the left-hand side does not
dependong sothatthetwodifferentcasesactuallycorrespondtoatransformation
ofthe right-handside. Indeed, the equalityoftheg =a/bandg =a/einstances of
the right-handside is animmediate consequenceofthe ω transformation(2.10).
10 9
Proof. Given the previous remark we only need to prove the g = a/b case of the
theorem. Starting point is again the pair of inverse matrices (3.5) in which we set
r=2. The crux of the proof is the observation that equation (4.1) holds, with
(b/d,bdq;q2,p)
(4.3) a = n ω (ad;ad/c,c,dq,dq2,aq/b,abq2n,q−2n;q2,p)
n (adq2,aq/d;q2,p) 10 9
n
and
(q2,abq2,aq/b;q2,p) (a/c,c/d,dq;q,p)
n n
(4.4) b = .
n (a,1/b,bq;q,p) (cq2,adq2/c,aq/d;q2,p)
n n
8 S.O. WARNAAR
Assumingthisistrue,weimmediatelyrecognize(4.2)asTheorem4.2(withg =a/b)
under the simultaneous replacements b → a/c,c → c/d,d → dq,e → aq/b and
f →abq2n.
Of course, it remains to show (4.1) with the above pair of a and b . Writing
n n
a = a inaccordancewiththedefinitionof ω ,westartwiththetrivialities
n j n,j 10 9
P n n k n n n n−j
(4.5) b = f a = f a = f a = f a .
n n,k k n,k k,j n,k k,j n,k+j k+j,j
k=0 k=0j=0 j=0k=j j=0k=0
X XX XX XX
Using the explicit expressions for f and a as well as the relations in (2.7)this
n,k n,k
becomes
n (dq,aqn;q,p) (abq2;q2,p) (ad,c,ad/c,aq/b,q−2n;q2,p) (bq2/d)j
2j 2j j
b =
n (a,bq1−n;q,p) (ad;q2,p) (q2,adq2/c,cq2,aq/d,abq2n+2;q2,p)
2j 2j j
j=0
X
× ω (abq4j;b/d,bdq2j+1,aqn+2j,aqn+2j+1,q−2n+2j;q2,p).
8 7
By the elliptic Jacksonsum (2.11) we can sum the ω , and after the usual simpli-
8 7
fications we find
(abq2,aq/b;q2,p) (1/d,dq;q,p)
n n
b =
n (adq2,aq/d;q2,p) (1/b,bq;q,p)
n n
× ω (ad;c,ad/c,dq,dq2,aqn,aqn+1,q−2n;q2,p).
10 9
By the r=2 case of Theorem 4.1 the ω can be summed yielding the expression
10 9
for b given in (4.4).
n
AresultverysimilartothatofTheorem4.2isthefollowingcubictransformation,
which, for f = a/b, provides an elliptic analogue of [18, Eq. (3.6)] by Gasper and
Rahman.
Theorem 4.3. Let bcd = aq and de = a2q3n+1. Then for f = a/b or f = a/e
there holds
n E(aq4k) (b,c;q,p) (d;q,p) (e,q−3n;q3,p)
k 2k k qk
E(a) (aq3/b,aq3/c;q3,p) (aq/d;q,p) (aq/e,aq3n+1;q,p)
k 2k k
k=0
X
(aq3,a2q3/bce,a2q3/bdef,afq3/cd;q3,p)
n
=
(a2q3/bef,aq3f/c,aq3/d,a2q3/bcde;q3,p)
n
× ω (af/c;a/c,fq3/c,bef/a,d,dq,f,q−3n;q3,p).
10 9
Againwenotethatthetwodifferentcasescorrespondtothe ω transformation
10 9
(2.10) applied to the right-hand side.
Proof. By the above remark we only need a proof for f = a/b. The claim is now
that if we choose r =3 in the pair of matrices (3.5) then (4.1) holds, with
(4.6)
(b2/a;q3,p)
a = n ω (a2/b;ac/b,a/c,aq/b,aq2/b,aq3/b,abq3n,q−3n;q3,p)
n (a2q3/b;q3,p) 10 9
n
and
(q3,abq3;q3,p) (b/c,c;q,p) (aq/b;q,p)
n n 2n
(4.7) b = .
n (a,1/b;q,p) (acq3/b,aq3/c;q3,p) (bq;q,p)
n n 2n
ELLIPTIC HYPERGEOMETRIC SERIES 9
Clearly,ifthisistruewearedonewiththeproofsincewiththesea andb equation
n n
(4.2) corresponds to the f = b/a case of Theorem 4.3 with the replacements b →
b/c,d→aq/b and e→abq3n.
To establish(4.1) with the abovea and b we follow the proofof Theorem4.2.
n n
That is, we again write a = a and use (4.5). Inserting the expressions for
n j n,j
f and a this yields
n,k n,k
P
n (aq/b,aqn;q,p) (abq3;q3,p) (a2/b,ac/b,a/c,q−3n;q3,p) (b2q3/a)j
3j 2j j
b =
n (a,bq1−n;q,p) (a2/b;q3,p) (q3,aq3/c,acq3/b,abq3n+3;q3,p)
3j 2j j
j=0
X
× ω (abq6j;b2/a,aqn+3j,aqn+3j+1,aqn+3j+2,q−3n+3j;q3,p).
8 7
The ω can be summed by (2.11), and after some manipulations involving (2.7)
8 7
we arrive at
(abq3;q3,p) (b/a;q,p) (aq/b;q,p)
n n 2n
b =
n (a2q3/b;q3,p) (1/b;q,p) (bq;q,p)
n n 2n
× ω (a2/b;ac/b,a/c,aq/b,aq2/b,aq3/b,aqn,aqn+1,aqn+2,q−3n;q3,p).
12 11
According to Theorem 4.1 with r = 3 the ω can be summed to yield (4.7) as
12 11
claimed.
Theorems 4.2 and 4.3 imply several other quadratic and cubic summation and
transformation formulas.
The most obvious ones arise when we demand that g = 1 in Theorem 4.2 or
f =1 in Theorem 4.3.
Corollary 4.4. Let bcd=aq and ef =a2q2n+1. When b=a or e=a there holds
n E(aq3k) (b,c,d;q,p) (e,f,q−2n;q2,p)
k k qk
E(a) (aq2/b,aq2/c,aq2/d;q2,p) (aq/e,aq/f,aq2n+1;q,p)
k k
k=0
X
(aq2,a2q2/bce,a2q2/bde,aq2/cd;q2,p)
n
= .
(a2q2/be,aq2/c,aq2/d,a2q2/bcde;q2,p)
n
For p = 0 and b = a this is a summation of Gessel and Stanton [19, Eq. (1.4)],
and for p = 0, e = a it corresponds to [33, Eq. (1.9), b → q−2n] by Rahman and
[10, Eq. (5.1d)] by Chu.
Corollary 4.5. Let bcd=aq and de=a2q3n+1. When b=a or e=a there holds
n E(aq4k) (b,c;q,p) (d;q,p) (e,q−3n;q3,p)
k 2k k qk
E(a) (aq3/b,aq3/c;q3,p) (aq/d;q,p) (aq/e,aq3n+1;q,p)
k 2k k
k=0
X
(aq3,a2q3/bce,a2q3/bde,aq3/cd;q3,p)
n
= .
(a2q3/be,aq3/c,aq3/d,a2q3/bcde;q3,p)
n
When b=a this is the elliptic analogue of [16, Eq. (5.22), c→q−3n] of Gasper.
Theorem 4.3 also leads to a summation formula if we choose d = a. Indeed,
ω (af/c;a/c,fq3/c,bef/a,a,aq,f,q−3n;q3,p) with f = a/b, bc = q and e =
10 9
aq3n+1 becomes ω (a2/q;a/b,ab/q,aq3n+1,q−3n;q3,p) which, by (2.11), evaluates
7 6
to
(q3,a2q2,bq,q2/b;q3,p)
n
.
(aq,q2/a,aq3/b,abq2;q3,p)
n
10 S.O. WARNAAR
We therefore conclude the following result, which for p = 0 corresponds to [18,
Eq. (3.7)].
Corollary 4.6. For bc=q and e=aq3n+1,
n E(aq4k) (b,c;q,p) (a;q,p) (e,q−3n;q3,p)
k 2k k qk
E(a) (aq3/b,aq3/c;q3,p) (q;q,p) (aq/e,aq3n+1;q,p)
k 2k k
k=0
X
(aq2,aq3,bq,cq;q3,p)
n
= .
(q,q2,aq3/b,aq3/c;q3,p)
n
The next three results, stated as separate theorems, are somewhat less trivial
as their proof deviates from the standard polynomial argument applicable in the
p=0 case.
Theorem 4.7. For bcd=a2q and ef =aqn+1,
n E(aq3k) (b,c,d;q2,p) (e,f,q−n;q,p)
k k qk
E(a) (aq/b,aq/c,aq/d;q,p) (aq2/e,aq2/f,aqn+2;q2,p)
k k
k=0
X
(aq,aq/bc;q,p) (aq1−n/b,aq1−n/c;q2,p)
n n
(aq/b,aq/c;q,p) (aq1−n,aq1−n/bc;q2,p)
n n
× ω (a2/ef;b,c,d,a/e,a/f,q1−n,q−n;q2,p).
10 9
For p=0 this is [18, Eq. (5.15)] (with corrected misprint).
Theorem 4.8. For bc=a2qn+1 and de=aqn+1,
⌊n/2⌋E(aq4k) (b,c;q3,p) (q−n;q,p) (d,e;q,p)
k 2k k qk
E(a) (aq/b,aq/c;q,p) (aqn+1;q,p) (aq3/d,aq3/e;q3,p)
k 2k k
k=0
X
(aq;q,p) (aq2−n/b;q3,p)
= n n ω (a2/de;b,c,a/d,a/e,q2−n,q1−n,q−n;q3,p).
(aq/b;q,p) (aq2−n;q3,p) 10 9
n n
When p=0 this can be recognized as [18, Eq. (3.19), c→bq−n−1].
Theorem 4.9. For bcd=a2q and de=aqn+1,
n E(aq4k) (b,c;q3,p) (d;q,p) (e,q−n;q,p)
k 2k k qk
E(a) (aq/b,aq/c;q,p) (aq/d;q,p) (aq3/e,aqn+3;q3,p)
k 2k k
k=0
X
f ω (a2/deq;a/dq,a/e,b,c,d,q1−n,q−n;q3,p) n6≡2 (mod 3)
n 10 9
= g ω (a2/de;a/d,a/e,b,c,dq,q2−n,q−n;q3,p) n6≡1 (mod 3)
 n 10 9
hn 10ω9(a2q/de;aq/d,a/e,b,c,dq2,q2−n,q1−n;q3,p) n6≡0 (mod 3),
with 
(aq3−σ,aq3−σ/bc,aq3−σ/bd,aq3−σ/cd;q3,p)
(n+σ)/3
f =
n (aq3−σ/b,aq3−σ/c,aq3−σ/d,aq3−σ/bcd;q3,p)
(n+σ)/3
(aq3−σ,aq3−σ/bc,aq2−σ/bd,aq2−σ/cd;q3,p)
(n+σ)/3
g =
n (aq3−σ/b,aq3−σ/c,aq2−σ/d,aq2−σ/bcd;q3,p)
(n+σ)/3
E(aqσ)E(aqσ/bc)E(aq/bd)E(aq/cd)
h =
n E(aqσ/b)E(aqσ/c)E(aq/d)E(aq/bcd)
(aq3−σ,aq3−σ/bc,aq1−σ/bd,aq1−σ/cd;q3,p)
(n+σ)/3
×
(aq3−σ/b,aq3−σ/c,aq1−σ/d,aq1−σ/bcd;q3,p)
(n+σ)/3