THE RAMANUJAN JOURNAL 1, 299–308 (1997)c© 1997 Kluwer Academic Publishers. Manufactured in The Netherlands.

Some Cubic Summationand Transformation Formulas

MIZAN RAHMAN ∗ mrahman@math.carleton.caDept. of Mathematics and Statistics, Carleton University, Ottawa, Ont., Canada K1S 5B6

Received May 17, 1996; Accepted June 27, 1996

Abstract. q-Analogues of two cubic summation formulas that have recently caught the attention of Bill Gosperare found by first showing their connection with theq-binomial formula and then using some known transformationformulas. We also find aq-extension of a cubic transformation formula involving Gauss’ hypergeometric function,which turns out to be a relation between balanced and very-well-poised10φ9 series.

Key words: basic hypergeometric series, balanced, nearly-poised and very-well-poised series,q-binomial for-mula, quadratic and cubic transformation formulas, cubic summation formula

1991 Mathematics Subject Classification: Primary—33D15, Secondary—33C05

1. Introduction

The primary objective of this paper is to deriveq-analogues of the following summationformulas listed in [2, 2.9] and [1, 15.1]:

F

(−a,

1

2− a; 2a+ 3

2;−1

3

)=(

8

9

)2a0(4/3)0(2a+ 3/2)

0(3/2)0(2a+ 4/3),

2a+ 3/2 6= 0,−1,−2, . . . , (1.1)

F

(3a, 3a+ 1

2; 2a+ 5

6; 1

9

)=(

3

4

)3a 0(

12

)0(2a+ 5

6

)0(a+ 1

2

)0(a+ 5

6

) ,2a+ 5/6 6= 0,−1,−2, . . . . (1.2)

(Watch for a misprint in [2] on the second formula. The same misprint seems to havebeen trustingly reproduced in [1]). Another formula that is closely related to (1.1) but onewouldn’t probably suspect it at first glance is Watson’s summation formula [7]:

F

(a+ 1

3, 3a; 2a+ 2

3; e±iπ/3

)= 2π0

(2a+ 2

3

)3−(3a+1)/2

0(

23

)0(a+ 1

3

)0(a+ 2

3

)e±iπa/2,

2a+ 2

36= 0,−1,−2, . . . . (1.3)

∗This research was supported, in part, by an NSERC grant #A6197.

300 RAHMAN

It can be shown by using [2, 2.11(28)] and [2, 2.9(2)] that (1.3) reduces to (1.1) whenthe+ve sign is chosen. Formula (1.3) can also be seen as a special case of the cubictransformation formulas [2, 2.11(39)]

F

(3a

2,(3a− 1)

2;a+ 1

2;−z2

3

)= (1+ z)1−3aF

(a− 1

3,a; 2a; 2z(3+ z2)(1+ z)−3

).

(1.4)

Note that the argument−z2/3 on the l.h.s. appears as−z2/3 in the above reference whichis obviously a misprint. If one setsz= 1 in (1.4) then the argument of the hypergeometricfunction on the r.h.s. becomes 1 and so it can be summed by the Gauss formula [2, 2.8(46)].This provides an alternative proof of (1.1). It may be pointed out that Goursat’s [4] cubictransformation formulas [2, 2.11(41)] and [2, 2.11(43)] can also be derived from (1.4) byan application of the quadratic transformation formula [2, 2.11(28)] as well as a linearfractional transformation [2, 2.9(3)]. In this paper we shall derive aq-analogue of (1.4),rather the following analytic continuation of it:

F

(1

2− a/2, 1− a/2;a+ 1

2;−z2/3

)= (1+ z)1−3a(1+ z2/3)2a−10(2a)/0(a)

×{

0(

13

)0(a+ 1

3

) F

(a− 1

3,a; 2

3;(

1− z

1+ z

)3)

+ 0(− 1

3

)0(a− 1

3

) F

(a,a+ 1

3; 4

3;(

1− z

1+ z

)3)}.

(1.5)

This is obtained from (1.4) by applying [2, 2.9(2)] on the l.h.s. and [2, 2.10(1)] on ther.h.s. It is worth pointing out that (1.2) can be obtained from (1.5) by settingz = i /

√3

and using Kummer’s summation formula [2, 2.8(47)]. The same cannot be said about (1.4)because the argument of theF-function on the r.h.s. becomes 2 whenz= i /

√3 and so is

a divergent series.Using [2, 2.9(3)] in (1.3) and replacinga+ 1

3 by a we can rewrite the summation formulain the form

F(a, 1− a; 2a; e±iπ/3) = e±iπ(1−a)/6 2π0(2a)3−3a2

0(a)0(a+ 1

3

)0(

23

) . (1.6)

It is this formula for which the author found aq-analogue in [6], (note a misprint),

3φ2

[a,q/a,aω

a2,qω2;q,aω

]=√

3eπ i /6 (a,aω2;q)∞(a2, ω2;q)∞(q;q3)∞

, |a| < 1, (1.7)

whereω = e2π i /3. The complex conjugate of it is obtained by simply replacingω byω2 =1/ω in (1.7). The3φ2 symbol in (1.7) represents a special case of a basic hypergeometric

CUBIC SUMMATION AND TRANSFORMATION FORMULAS 301

functionr+1φr defined by

r+1φr

[a1, . . . ,ar+1

b1, . . . ,br;q, z

]=∞∑

n=0

(a1, . . .ar+1;q)n(q, b1, . . . ,br ;q)n zn, (1.8)

where

(a1, . . . ,ak;q)n =k∏

j=1

(aj ;q)n,

(a;q)n = (a;q)∞/(aqn;q)∞, (1.9)

(a;q)∞ =∞∏j=0

(1− aqj ), 0≤ |q| < 1.

The usual condition of convergence of the series (1.8) is|z| < 1. For further details see[3]. The seriesr+1φr in (1.8) is called balanced ifz= q andb1 · · ·br = qa1a2 · · ·ar+1. Itis called a nearly-poised series of the first kind ifqa1 6= a2b1 = a3b2 = · · · = ar+1br ; anearly-poised series of the second kind ifqa1 = a2b1 = a3b2 = · · · = ar br−1 6= ar+1br . Itis a well-poised series ifqa1 = a2b1 = a3b2 = · · · = ar br−1 = ar+1br , and, a very-well-poised series if, in addition,b1 = a1/2

1 , b2 = −a1/21 . These additional structures help us

transform ther+1φr series to other single series for certain values ofr . For a comprehensivetreatment of these transformation formulas see [3].

In [6] the author went on to derive a bilateral extension of (1.7) and analogous formulasassociated with complex fifth roots of unity, but did not attempt to work on theq-analoguesof the related formulas (1.1) and (1.2). An effort will be made in this work to complete thatunfinished job.

The motivation of this work was a recent series of e-mail communications with BillGosper who has found his own computer-aided proofs of (1.1) and (1.2) and many otheridentities, some of which are old but many are new.

2. Theq-binomial formula and more

We shall start by writing theq-binomial formula [3, II.3] in baseq3, instead of the usualq:

∞∑n=0

(q3/a3;q3)n

(q3;q3)ntn = (tq3/a3;q3)∞

(t;q3)∞, |t | < 1, (2.1)

since it is our intention to show that this is the formula that lies at the root of theq-analogues that we are seeking in this paper. It was already mentioned in [6], but forthe sake of completeness we shall more or less repeat the same arguments here. Since(a3;q3)n = (a,aω,aω2;q)n, the l.h.s. of (2.1) can be written as a3φ2 series in baseq:

3φ2

[q/a,qω/a,qω2/a

qω2,qω;q, t

], (2.2)

which is not a big deal except that it is well-poised, so it has a nice structure that can beexploited. In [3, (3.4.1)] there is a transformation formula for a series precisely like this,

302 RAHMAN

so we can write

(tq3/a3;q3)

(t;q3)∞= (qt/a2;q)∞

(t/a;q)∞ 5φ4

[(q/a)1/2,−(q/a)1/2,q/a1/2,−q/a1/2,a

qω,qω2,qt/a2,aqt;q,q

]+ (a,q/a,qωt/a,qω2t/a;q)∞

(t,qω,qω2,a/t;q)∞

× 5φ4

[t, tq1/2/a3/2,−tq1/2/a3/2, tq/a3/2,−tq/a3/2

qωt/a,qω2t/a,qt2/a,qt/a;q,q

], (2.3)

which looks like an impressive summation formula that we got almost free of cost, exceptthat it is not a very useful formula.

As long as 0< |q| < 1, which we shall always assume to be true (in fact, we shall assumeq to be real and 0< q < 1), both5φ4 series in (2.3) are balanced and convergent, butbecomes divergent when we letq→ 1. The only hope of salvaging something meaningfulfrom the r.h.s. is to seek an analytic continuation that does not have this divergence problem.Unfortunately, as long as these are5φ4 series, no further transformation into single seriesseems possible, unless they become mutually matched4φ3 series which happens whent = a3/q. In this case the3φ2 series in (2.2) becomes what is called in [3] a series ofType II, and the two4φ3 series in (2.3) combine nicely to become a very-well-poised8φ7

series via [3, III.36] (it is in this sense that we just described the two4φ3 series as “mutuallymatched”). This gives

(q2;q3)∞(a3/q;q3)∞

= (aω,a2;q)∞(qω,a3/q;q)∞ 3φ2

[a,q/a,aω

a2,qω2; q,aω

]= (a2,−a3/2;q)∞(a3/q,−qa1/2;q)∞ 8W7(−a1/2; (q/a)1/2,

−(q/a)1/2,q/a1/2,−ωa1/2,−ω2a1/2;q,−a3/2). (2.4)

Note that we used [3, III.9] to transform the3φ2 series in (2.2) and have used theW-notationof [3] for very-well-poised series:

r+1Wr (a;a1,a2, . . . ,ar−2;q, x) := r+1φr

[a,q√

a,−q√

a,a1,a2, . . . ,ar−2√a,−√a,qa/a1,qa/a2, . . . ,qa/ar−2

;q, x

].

(2.5)

Using the identities [3, (I.26)] and [3, (I.29)] we can derive (1.7) from the first line of (2.4).The second line contains aq-analogue of (1.1). To see this we first replacea by q2a+1 sothat it can be written as

8φ7

[−qa+1/2, iqa/2+5/4,−iqa/2+5/4,q−a,q1/2−a,−q−a,−ωqa+1/2,−ω2qa+1/2

iqa/2+1/4,−iqa/2+1/4,−q3/2+2a,−q1+2a,q3/2+2a, ω2q, ωq;q,−q3a+3/2

]

=(q6a+2,−qa+3/2;q)∞(q2;q3

)∞(

q4a+2,−q3a+3/2;q)∞(q6a+2;q3)∞

= 0q3

(43

)0q2

(2a+ 3

2

)0q3

(2a+ 4

3

)0q2

(32

)( 1+ q

1+ q + q2

)4a

(−qa+3/2;q)2a, (2.6)

CUBIC SUMMATION AND TRANSFORMATION FORMULAS 303

where the r.h.s. has been obtained by repeated application of theq-gamma function

0q(x) = (q;q)∞(qx;q)∞ (1− q)1−x, |q| < 1, x 6= 0,−1,−2, . . . , (2.7)

and its multiplication and limit properties [3, (I.36)–(I.38)]. Using the fact that

limq→1−

(x;q)b = limq→1−

(x;q)∞(xqb;q)∞ = (1− x)b (2.8)

and that

limq→1−

(ωq, ω2q;q) j = 3 j , limq→1−

(−ωqa,−ω2qa;q) j = 1 (2.9)

it is easy to see that the limitq→ 1− of (2.6) is (1.1).

3. A q-analogue of the cubic transformation formula (1.5)

Clearly, we need an extension of (2.2) which is still transformable. A strong candidate is

4φ3

[q/a, ωq/a, ω2q/a,q/x

qω2,qω,q3/xa3;q,q

]which reduces to (2.2) whenx → 0 but otherwise a balanced and nearly-poised series ofthe second kind. However, it cannot be transformed into other single series unless the seriesterminates which happens only ifa or x is of the typeqn+1, n a nonnegative integer. Forgeneral values of these parameters one needs its companion4φ3 series so that the two ofthem together combine to make an8φ7 series via Bailey’s formula [3, (2.10.10)]. Also,because of the nearly-poised structure of the4φ3 series, it admits a second transformationwhich is a special case of a formula due to Jain and Verma [6], see also [3, Ex. 2.25]:

4φ3

[q/a,qω/a,qω2/a,q/x

qω2,qω,q3/xa3;q,q

]+ (q/a,qω/a,qω

2/a,q/x, xa3/q2, ωxa3/q, ω2xa3/q;q)∞(qω,qω2,q2/xa3,a3/q, xa2/q, ωxa2/q, ω2xa2/q;q)∞

× 4φ3

[xa2/q, xωa2/q, xω2a2/q,a3/q

xa3/q, xωa3/q, xω2a3/q;q,q

]= (a2, xa3/q2,−a3/2,−xa1/2;q)∞(a3/q, xa2/q,−qa1/2,−xa3/2/q;q)∞× 10W9(−a1/2; (q/a)1/2,−(q/a)1/2,q/a1/2,−ωa1/2,−ω2a1/2,q/x, xa2/q;q,q)

+ (aq/a,q/x,ax, x2a5/q2, xa3/q2,−ωa1/2,−ω2a1/2;q)∞(a3/q,qω,qω2, xa2/q, ωxa2/q, ω2xa2/q,−q/a1/2,−xa5/2/q;q)∞

× (−xωa3/2,−xω2a3/2;q)∞(−q/xa3/2,−x2a7/2/q;q)∞ 10W9(−x2a7/2/q2;

−ax/√

q,ax/√

q,−ax,−a3/2, xa2/q, ωxa2/q, ω2xa2/q;q,q). (3.1)

304 RAHMAN

Use of [3, (2.10.10)] then gives the transformation formula

8W7(x;ωx, ω2x,q/a, ωq/a, ω2q/a;q,a3/q)

= (a2,qx, ωxa2/q, xω2a2/q,−a3/2,−xa1/2;q)∞(a3/q,ax,axω,axω2,−qa1/2,−xa3/2/q;q)∞× 10W9(−a1/2; (q/a)1/2,−(q/a)1/2,q/a1/2,−ωa1/2,−ω2a1/2,q/x, xa2/q;q,q)

+ (a,q/a,qx,q/x, x2a5/q2,−ωa1/2,−ω2a1/2,−xωa3/2,−xω2a3/2;q)∞(a3/q,qω,qω2,axω,axω2,−q/a1/2,−xa5/2/q,−q/xa3/2,−x2a7/2/q;q)∞× 10W9(−x2a7/2/q2;−ax/

√q,ax/

√q,−ax,−a3/2, xa2/q, ωxa2/q, xω2a2/q;q,q),

(3.2)

where, it is assumed that|a3/q| < 1. The l.h.s. of (3.2) is

∞∑n=0

(x3,q3/a3;q3)n

(q3,a3x3;q3)n

1− xq2n

1− x(a3/q)n

= (1− x)−12φ1

[q3/a3, x3

a3x3;q3,a3/q

]− x(1− x)−1

2φ1

[q3/a3, x3

a3x3;q3,qa3

]= (1− x)−1 (x3;q3)∞

(a3x3;q3)∞

{(q2;q3)∞(a3/q;q3)∞

2φ1

[a3/q,a3

q4;q3, x3

]− x

(q4;q3)∞(qa3;q3)∞

2φ1

[qa3,a3

q4;q3, x3

]}, (3.3)

where we have used [3, III.1] to derive the last line from the second last on the r.h.s.If q/a = q−n, n a nonnegative integer then it is clear that (3.2) and (3.3) together give a

q-analogue of (1.5), but for other values ofa, subject to the condition|a3/q| < 1, this isnot at all obvious since this would mean there would be no contribution from the secondterm on the r.h.s. of (3.2) when one takes the limitq→ 1− after replacinga by qa. So, by(1.9) and (2.7) one can rewrite (3.2) and (3.3) in the form

(1− x)−1(x3;q3)a

{0q3

(a− 1

3

)0q3

(23

) 2φ1

[q3a−1,q3a

q2;q3, x3

]

− x0q3

(a+ 1

3

)0q3

(43

) 2φ1

[q3a+1,q3a

q4;q3, x3

]}

= (1+ q + q2)1−a0q(3a− 1)

0q(2a)

(qx;q)2a−2(−xqa/2;q)a−1

(x3q3a;q3)a−1(−qa/2+1;q)a−1

× 10φ9

[−qa/2,q√·,−q

√·,q 1−a2 ,−q

1−a2 ,q1− a

2 ,−ωqa2 ,−ω2q

a2 ,q/x, xq2a−1

√·,−√·,−qa+ 12 ,qa+ 1

2 ,−qa, ω2q, ωq,−xqa/2,−q2− 3a2 /x

;q,q]

+ (1− x)−1(1+ q + q2)1−a(1− q)2a 0q(3a− 1)

0q(a)0q(1− a)

(q, x,q/x;q)∞(qω,qω2,qaxω;q)∞

CUBIC SUMMATION AND TRANSFORMATION FORMULAS 305

× (x2q5a−2,−ωqa/2,−ω2qa/2,−ωxq3a/2,−ω2xq3a/2;q)∞(qaxω2,−q1−a/2,−xq5a/2−1,−q1−3a/2/x,−x2q7a/2−1;q)∞

× 10φ9

[−x2q7a2 −2,q

√·,−q√·, xqa− 1

2 ,−xqa− 12 ,−xqa,−q

3a2 , xq2a−1,

√·,−√·,−xq5a2 − 1

2 , xq5a2 − 1

2 , xq5a2 −1, x2q2a−1,−xq

3a2 ,

ωxq2a−1, ω2xq2a−1

−ω2xq3a2 ,−ωxq

3a2;q,q

], (3.4)

where the expressions under the√· sign in both10φ9 series are the top left hand corner

parameters. So it is√−qa/2 in the first and

√−x2q

7a2 −1 in the second. Our claim is that

(3.4) is aq-analogue of (1.5). Indeed, taking the limitq→ 1− one finds that

(1− x3)a{0(a− 1

3

)0(

23

) 2F1

[a− 1

3,a23

; x3

]− x

0(a+ 1

3

)0(

43

) 2F1

[a+ 1

3,a43

; x3

]}

= 61−a0(3a− 1)

0(2a)

(1+ x)a−1

(1+ x + x2)a−1 2F1

[1−a

2 , 1− a2

a+ 12

;−1

3

(1− x

1+ x

)2]

+ 31−a 0(3a− 1)

0(a)0(1− a)lim

q→1−

{(1− q)2a (q, x,q/x, x2q5a−2;q)∞

(qω,qω2, xωqa, xω2qa;q)∞

× (−ωqa/2,−ω2qa/2,−ωxq3a/2,−ω2xq3a/2;q)∞(− q1−a/2,−xq5a2 −1,−q1− 3a

2 /x,−x2q7a2 −1;q)∞

× 10W9(− x2q

7a2 −2; xqa− 1

2,−xqa− 12,−xqa,−q

3a2 , xq2a−1, ωxq2a−1, ω2xq2a−1;q,q)}.

(3.5)

We will show in the Appendix that the limit on the r.h.s. is zero. Formula (1.5) thenfollows from (3.5) by use of the multiplication formula [2, 1.2 (11)]. Whenx = −ωq1−a or−ω2q1−a, both2φ1 series on the l.h.s. of (3.4) can be summed by theq-Kummer’s formula[3, II.9], provided|q1−a| < 1. In this special case, however, it is easier to go back to the8φ7 series in (3.2) because it now takes the form

8φ7

[−ωq/a,q√·,−q

√·, ωq/a,q/a,−q/a, ω2q/a,−ω2q/a√·,−√·,−q,−ωq, ωq,−ω2q, ω2q;q,a3/q

]= 4φ3

[ω2q2/a2,−q2√·,q2/a2, ωq2/a2

−√·,q2ω2, ωq2 ;q2,a3/q

]= (ω2q4/a2,aq, ωaq, ωa2;q2)∞(ωq2, ω2q2, ωq3/a,a3/q;q2)∞

, (3.6)

by theq-Dixon formula [3, II.13]. This leads us to the summation formula

(a2,−ωq2/a,−ω2a,−a,−a3/2, ωq/a1/2;q)∞(a3/q,−q,−qω,−qω2,−qa1/2, ωa1/2;q)∞× 10φ9

[−a1/2,q√·,−q

√·, (q/a)1/2,−(q/a)1/2,q/a1/2,−ωa1/2,−ω2a1/2,−ωa,−ω2a√·,−√·,−aq1/2,aq1/2,−a,qω2,qω,qω2/a1/2,qω/a1/2 ;q,q]

306 RAHMAN

+ (a,q/a,−ωq2/a,−ω2a, ω2a3,−ωa1/2,−ω2a1/2,qa1/2, ωqa1/2;q)∞(a3/q2,qω,qω2,−q,−qω2,−q/a1/2, ωa3/2, ω2/a1/2,−ω2qa3/2;q)∞

× 10φ9

[ −ω2a3/2,q√·,−q

√·, ω√q,−ω√q, ωq,−a3/2,−ωa, ω2a,−a√·,−√·,−ωq1/2a3/2, ωq1/2a3/2,−ωa3/2,qω2, ωqa1/2,qa1/2, ω2qa1/2;q,q

]= (ω2q4/a2,aq,aωq, ωa2;q2)∞(ωq2, ω2q2, ωq3/a,a3/q;q2)∞

. (3.7)

This leads to aq-analogue of (1.2). To see this we first replacea by qa and observe thatthe limit of the first10φ9 series on the l.h.s. of (3.7) is

F

[1−a

2 , 1− a2

a+ 12

; 1

9

]

which, by [2, 2.9(2)], transforms to(8/9)2a−1F( 3a−12 , 3a

2 ;a+ 12; 1/9). This F-function is

the same as the one in (1.2) if we replacea by 2a+ 13. Indeed, ifa = qn+1, n = 0, 1, 2, . . .,

then the second term on the l.h.s. of (3.7) automatically drops out because of the factor(q/a;q)∞ so that the summation formula can simply be expressed in the form

10φ9

[−qn+1

2 ,q√·,−q

√·,q− n2,q

12− n

2,−q−n2,−ωq

n+12 ,−ω2q

n+12 ,−ωqn+1,−ω2qn+1

√·,−√·,−qn+ 32 ,−qn+1,qn+ 3

2 , ω2q, ωq, ω2q1−n

2 , ωq1−n

2;q,q

]

=(q3n+2,−q,−qω,−qω2,−q

n+32 , ωq

n+12 ;q)∞(

q2n+2,−ωq1−n,−ω2qn+1,−qn+1,−q3n+3

2 ,−ωq1−n

2 ;q)∞× (q

n+2, ωqn+2, ωq2n+2, ω2q2−2n;q2)∞(ωq2, ω2q2, ωq2−n,q3n+2;q2)∞

. (3.8)

In the limit q→ 1− this simplifies to

F

(−n

2,

1− n

2; n+ 3

2; 1

9

)=(

4

3

)n0(n+ 1)0(n+ 3

2

)0(

n+22

)0(

3n+32

) = 23n

3n−1

0(n+ 3

2

)0(

n+32

)0(

3n+52

)0(

12

) . (3.9)

In fact, by using the formulas (1.9), (2.7) and [3, I.29] one can rewrite (3.8) in the moresuggestive form:

10φ9

[−qn+1

2 ,q√·,−q

√·,q− n2,q

1−n2 ,−q−

n2,−ωq

n+12 ,−ω2q

n+12 ,−ωqn+1,−ω2qn+1

√·,−√·,−qn+ 32 ,−qn+1,qn+ 3

2 , ω2q, ωq, ω2q1−n

2 , ωq1−n

2;q,q

]

= (−1;q)n(1+ q)2n+2

2(1+ q

n+12) (

1+ qn+1

2 + qn+1)

(1+ q + q2)n

(ω2q1−n;q2)n(ω2q

1−n2 ;q)n

× 0q(n+ 1)0q2(n+ 2)

0q3(n+ 1)0q(n+ 2)· 0q2

(n+ 3

2

)0q(n+ 3

2

)0q(

3n+52

)0q2

(12

) . (3.10)

CUBIC SUMMATION AND TRANSFORMATION FORMULAS 307

Appendix

By the transformation formula [3, (2.12.9)],

10W9(−x2a7/2/q2; xa2/q,ax/√

q,−ax/√

q,−ax,−a3/2, ωxa2/q, ω2xa2/q;q,q)

= − (−x2a7/2/q,−q/xa3/2,ax/√

q,−ax/√

q,−ax,−a3/2;q)∞(−qa1/2,−xa3/2/q,−xa5/2/

√q, xa5/2/

√q, xa5/2/q, x2a2/q;q)∞

× (ωxa2/q, ω2xa2/q,aq1/2,−aq1/2,−a,−xa1/2,qω,qω2;q)∞(−ωxa3/2,−ω2xa3/2,q/x, (q/a)1/2,−(q/a)1/2,q/a1/2,−ωa1/2,−ω2a1/2;q)∞× 10W9(−a1/2; xa2/q,−(q/a)1/2, (q/a)1/2,q/a1/2,q/x,−ωa1/2,−ω2a1/2;q,q)

+ (−x2a7/2/q,−q/xa3/2,−xa5/2/q,−q/a1/2, ωx, ω2x, ωa2, ω2a2;q)∞(q/a2,q/x, xa4/q, x2a2/q,−ωa1/2,−ω2a1/2,−ωxa3/2, ω2xa3/2;q)∞× 10W9(xa4/q2; xa2/q,−a3/2/q1/2,a3/2/q1/2,a3/2,−a3/2, ωxa2/q, ω2xa2/q;q,q)

+ (−x2a7/2/q,−q/xa3/2,−a3/2/q1/2,a3/2/q1/2,a3/2,−a3/2, xa1/2,−xa1/2;q)∞(qx,q/x,a2/q, (q/a)1/2,−(q/a)1/2,q/a1/2, x2a2/q;q)∞

× (x(aq)1/2,−x(aq)1/2;q)∞(xa5/2/q1/2,−xa5/2/q1/2;q)∞

(qω,qω2, ωxa2/q, ω2xa2/q;q)∞(xa5/2/q,−ωa1/2,−ω2a1/2,−ωxa3/2,−ω2xa3/2;q)∞

× 10W9(x; xa2/q,−(q/a)1/2, (q/a)1/2,q/a1/2,−q/a1/2, ωx, ω2x;q,q). (4.1)

So the expression in{ } on the r.h.s. of (3.5) is

−0q(a)0q(1− a)

0q(2a)

(x;q)a(−xqa/2;q)a−1

(−q1+a/2;q)a−1(ωxqa, ω2xqa;q)a−1

× 10W9(−qa/2; xq2a−1,−q

1−a2 ,q

1−a2 ,q1− a

2 ,q/x,−ωqa/2,−ω2qa/2;q,q)+0q(1− 2a)

(x;q)4a−1(ωx, ω2x;q)a(x2q2a−1;q)3a−1(qω,qω2;q)2a−1

× 10W9(xq4a−2; xq2a−1,q

3a2 ,−q

3a−12 ,q

3a−12 ,−q

3a2 , ωxq2a−1, ω2xq2a−1;q,q)

+ 0q(1− a)0q(2a− 1)

0q(3a− 1)(1− x)

(x2qa;q)a−1

(ωxqa, ω2xqa;q)a−1

× 10W9(x; xq2a−1,−q

1−a2 ,q

1−a2 ,q1− a

2 ,−q1− a2 , ωx, ω2x;q,q). (4.2)

The limit of this expression asq→ 1− is

(1− x)a(1+ x)a−1

(1+ x + x2)a−1

0(a)0(1− a)

0(2a)2a−1

{− F

(1− a

2, 1− a

2;a+ 1

2;−z2/3

)+ 0(2a)0(2a− 1)

0(a)0(3a− 1)2a−1F

(1− a

2, 1− a

2; 2− 2a; 1+ z2/3

)+ 0(2a)0(1− 2a)

0(a)0(1− a)

1

23a−1(1+ z2/3)2a−1F

(3a− 1

2,

3a

2; 2a; 1+ z2/3

)}(4.3)

308 RAHMAN

wherez= (1− x)/(1+ x). By use of [2, 2.10(1), 2.10(5)] and [2, 1.2(11)] one can showthat the above expression vanishes. One has to take care, however, that this identity is validin some region of the complexz-plane. It is clear that for real values ofz the last twoF-series in (4.3) diverge unlessz= 0 anda < 1

2. If z is purely imaginary, which is indeedthe case for (1.2), then all theF-series above are convergent for|z| < √3.

References

1. M. Abramowitz and J.A. Stegun,Handbook of Mathematical Functions, Dover, New York, 1965.2. A. Erdelyi, ed.,Higher Transcendental Functions, McGraw-Hill, New York, 1953, vol. I.3. G. Gasper and M. Rahman, “Basic Hypergeometric Series,”Encyclopedia of Mathematics and Its Applications

35, Cambridge University Press, Cambridge, 1990.4. E. Goursat, “Sur l’equation differentielle lin´eaire,...,”Ann. Sci.Ecole Norm. Sup.10(2) (1881), 3–142.5. V.K. Jain and A. Verma, “Transformations of nonterminating basic hypergeometric series, their contour integrals

and applications to Rogers-Ramanujan identities,”J. Math. Anal. Appl.87 (1982), 9–44.6. M. Rahman, “A cubic and a quintic summation formula,”Ganita43 (1992), 45–61.7. G.N. Watson, “The cubic transformation of the hypergeometric function,”Quart. J. Math.41 (1909), 70–79.