On certain Ramanujan's mock theta functions

Proe. Indian Acad. Sci. (Math. Sci.), Vol. 103, No. 3, December 1993, pp. 257-267. �9 Printed in India.

On certain Ramanujan's mock theta functions

ANJU G U P T A Department of Mathematics and Astronomy, University of Lucknow, Lucknow 226007, India MS received 1 November 1992; revised 1 March 1993

Abstract. Ramanujan's last gift to the mathematicians was his ingeneous discovery of the mock theta functions of order three, five and seven. Recently, Andrews and Hickerson found a set of seven more functions in Ramanujan's Lost Note Book and formally labelled them as mock theta functions of order six. In this paper the complete forms of these functions have been studied and connected with the bilateral basic hypergeometric series 21P2. Several other interesting properties and transformations have also been studied.

Keywords. Mock theta functions; basic hypergeometric series; bilateral series.

1. Introduction

Ramanujan's last gift to the mathematicians was his ingeneous discovery of the mock theta functions of order three, five and seven. They were first studied by Watson [23]. Andrews [5, 6], in 1966, proved Ramanujan's identities for the third and fifth order mock theta functions by means of certain general theorems. Agarwal [1] in 1968, observed that the results of Andrews are not isolated ones but belong to a more general class of basic hypergeometric identities. Later, Andrews and his coworkers in a series of remarkable papers [5-10, 12, 13] derived Ramanujan's results and conjectures by the use of hypergeometric transformations and generalized Lambert 's series. Agarwal [4], recently, suggested that it would be nice to study the mock theta functions in a unified manner through transformations of basic hypergeometric series and also try to prove the numerous identities of Ramanujan in the spirit "how Ramanujan might have proved them" instead of using techniques with which Ramanujan might not have been conversant. Andrews and Hickerson [13] in a recent paper in 1991 proved eleven other identities found in Ramanujan's 'Lost' Notebook and labelled the seven functions involved in these identities as the sixth order mock theta functions.

In w 3 of this paper, we study these seven sixth order mock theta functions further by defining their three 'complete' forms which throw more light on their structures and forms and subsequently derive interesting relations among these 'complete' sixth order mock theta functions and the sixth order functions.

We observe that all the mock theta functions of the third, fifth, seventh and the sixth order are simple limiting cases of 2~bl's, 3q~2's and 4~b3's basic hypergeometric series, on single bases (see for details Agarwal [4a]).

The definitions of sixth order functions as limiting cases of the a~b2 functions (with q as one of its numerator parameters) suggest the study of this function on the pattern

257

258 Anju Gupta

of the study made by Fine [15] for the Ramanujan's function - ~ ' (a),t" (b), " In w 4 we

derive three transformations connecting the sixth order mock theta functions, in a very straightforward manner, through the transformations of the special types of 2W 2 and 34~2 series studied earlier [18].

Finally, in w 5 the complete form of the ~(q) function, the last of the sixth order mock theta function has been studied. Andrews and Hickerson [13] have remarked that they could not find the pair for this function on the pattern that exists for the other six sixth order functions given by Ramanujan. We have suggested that the function x(q) may be justifiably called the missing pair of the y(q)-funetion. Some of the salient features of this paper were reported in the Nat. Acad. Sci. Letters, 15 (1992), 49-50.

2. N o t a t i o n s and resu l t s used

For Iql < 1, let

(a; q). = (1 - a)(1 -- aq)... (1 -- aq"- 1 );

(a; q)o -= 1, (a; q)~ = f i (I - aq~). j=O

n~>l.

We, then define a generalized basic hypergeometric series, for Iq] < 1, as:

al,a2,.. . ,a a -] (ax;q)....(aa;q),z ~ A~)A-1 bl,b2 . . . . b A _ l ; z J --- _ . = o (q; q) . (b i ; q) . "" (ba - 1 ; q)."

The series converges for ]z] < 1. We denote (a; q). simply by (a). whenever there is no chance of any confusion with any other base. The symbol a(1)~ ~) denotes the generalized basic hypergeometric series on base q2.

A generalized basic bilateral hypergeometric series is defined by

n•nral ..... an;z]= ~ (at;-q)"'"(as;q)"z"

where ]blb2...be/ala2...aaz I < 1, Izl< 1, for convergence and

(_).q.(.+1)/2 (a;q)_.=

a"(q/a;q).

Andrews and Hickerson [13] have defined the following seven sixth order mock theta functions as found in the 'Lost' Notebook of Ramanujan:

~b(q)= ~ (_),,q,,*(q;q2),, ,=o (-- q)2, (1)

�9 (q)= ~ (-)"q("+ t)2(q;q2 )" (2)

. = o (-q)2.+I '

Ramanujan's mock theta functions 259

P(q)= ~ ql/2.~.+l)(_q). n=O (q;q2)n+ 1

( - q/n ql/2tn+ l)(n+ 2) _'~ tY(q) .~-.

.=o Z.., (q; q2).+ 1

).(q) = (-- ).q.(q;q2).

.=o (-- q). '

P(q)= ~ (_) . (q;q2) . ,

.=o ( - q ) . q"'(q).

~(q) = .~=o ( ~ . "

(3)

(4)

(5)

(6)

(7)

It may be noted that #(q) in (6) represents a divergent series. As remarked by Andrews [13] one gives it a meaning as "the sequence of even partial sums converge, as does the sequences of odd partial sums; we define g(q) as the average of these two values."

Indeed one could see this fact more clearly if one uses the t ransformation

(ct; q2)n(fl).z* = (fl)oo(0tz; q2)o o ~ (~/fl)2,n(Z; q2),.f12. n=o (q2;q2)n()') ~ ()')~(T;qZ)~ m=o (q)z,n(txz;q2)m

_~ (fl)oo (Os q2)oo ~ (~/fl)2m +1 (T,q', q2)m fl2m + 1

(?)~ ('rq; q2)~ m=O (q)2t~+t(Ot'cq;q2)m '

+

obtained from Theorem A 1 of Andrews [5; p. 671. Taking ~t = q, fl = q and y = 0, z = - 1, the LHS of the above eqn becomes #(q)

and the RHS becomes a sum of two convergent series with even and odd powers of ~.

3. The 'complete' sixth order mock theta functions

(i) Let us consider the bilateral series

Fc(ct)= ~ (_).q.~a2.(q;q2). - oo ( - otq)2n

(-- )nqS2ot2n(q; q2)n oo oo ( - )" q"2a- 2"(q; q2)_. = Y~ ~ Y~

o ( -- 0tq)2. 1 ( -- 0tq)_ 2.

~(_).q.2~tZ.(q; q2). q(1 + 1/00(1 + q/ot)~ q"(-- q2/cz; q)2.

: o i - q :o "

In (8) taking ~t = 1 and q, respectively, we get

2q(l_+_q) a,~2)F-q2,-q3,q 2 ] Fc(1) = q~r = dp(q)-~ I - q 3wa [_ q3 ; q.j'

(8)

(9)

260 Anju Gupta

where ~br denotes the 'complete' or 'bilateral' ~(q) function (1) and

2q .~(q,) [ -- q, -- q2, q2, q 1 l + q q Fc(q)- V,(q)= tIJ(q)+-~_qaW. 1 qa �9 J (10)

where We(q) denotes the 'complete' or 'bilateral' W(q) function (2). (ii) Next, consider the function Rc(~t ) =

q.(.+ l)/2(__q).~. _

1 co (--)"q"ct-"(q;q2). ~. ~ qn(n+ l)/2(__q)nO~n.+ ~ ~0 (11) o (q;q2).+, ( _ q;q),

In (11), taking 0t = 1, we get from (3) and (5),

Re(l) = Pc(q) = P(q) + �89 (12)

In (11), taking ~t = q, we get from (4) and (6)

qRc(q) = ac(q) = a(q) + �89 (13)

where Pc(q) and at(q) are, respectively, 'complete' forms for p(q) and a(q). (iii) Again, consider the function Mc(ct)=

(-)"~

-o~ (-- q).

_2 ~ (14) (--)"ct"(q; q2). ~ C t - " ( - - q;q).q("+ l)(n+ 2)/2

+ /. o (--q). o~ o (q;q2).+,

For ~ = 1, q, respectively in (14), we get from (4) and (6)

Me(l) -=/zc(q) =/z(q) + 2a(q); (15)

Mc(q) - At(q) = 2(q) + 2p(q); (16)

where/zr and 2c(q), respectively, are the 'complete'/~(q) and 2(q) functions. From (12) and (16), we get, the surprisingly curious equivalence

At(q) = 2pc(q). (17)

Also, from (13) and (15), we get

lac(q) = 2ac(q). (18)

(17) and (18) can also be obtained from the following transformation of Bailey [141;

W [a,b ] = (az, bz, cq/abz, dq/abz;q)o~ [abz/c, abz/d, c d 7 2 2Lc, d ,z (q/a,q/b,c,d;q)~ 2W2Laz, bz 'abz]"

(19)

In (19) taking a = ~t, b = - q, c = q 3 / 2 , d = - q3/2, g = - - q / c t and then letting ct~ oo, we again obtain (17).


Next, in (19) taking a = ~t, b = - q, c = q3/2, d = - q3/2, z = - q2/~ and then letting ~ - , ~ , we again obta in (18).

Again in (19) taking q = q2, a = ~t, b = q, c = - q2, d = - q, z = q/Qt and then letting ~t ~ ~ , we obta in (9).

Finally, in (19) taking q = q2, a = ~t, b = q, c = q2, d = - q3,z = q3/~t and then letting ~t~ o0, we obta in (10).

The above analysis also shows that the sixth order mock theta functions can be conveniently studied through the known t rans format ion theory of 2~F2 functions.

If one uses the k n o w n identities (0.18)g - (0.21)R of Ramanu jan , p roved by Andrews and Hickerson [13], it can be easily seen tha t the var ious ' comple te ' mock theta functions of order six can be reexpressed as

dpc(q) + 2a(ql/2) = 2q(14-_ q) ./,(a2)F -- q2, _ q3, q2 "] 1- -q awl L q3 ;q J +

+ ( _ q 1/2; q)~j( _ q3/2; q3). (20)

2q(1 + q) i - - -q [ - - q 2 ' q 3 ' q 2 ; q 1 __ ~c(q)--/t(ql/2) = 3r (q 2 ) ~ +

+ 1 (ql/2 ; ,,uq2 i(rt3/2 ~J~., ~ q3). (21)

2q .4,(,12) -- + ~tJc(q) q-ql/2p(ql/2)=l--q 3wl I q'--q2'q2;qTq3 _]

+ ql/2(_ ql/2; q)2 j (_ ql/2; qa). (22)

ql/2 l ~ q a~b(lq2)I- q ' _q2, q2;q]+ ~c(q) + --~--2(-- ql/2) = q3

ql/2 + 2 (_ql/2;q)2j(_ql/2;q3). (23)

pc(q)= �89 1 . 2 2 �9 =-~(q,q )~oJ(q;q6)+ + ( _ q; q 2 ) 2 j ( _ q; q6). (24)

Pc(q) = �89 + 2(-- q) -- q-1 tiJ(q2). (25)

1 2 2 ' 3 trctq) = �89 ) = �89 q; q 2 ) 2 j ( _ q3; q6) _ ](q; q )ooJ(q ;q6).

(26)

trc(q) = �89 _ / ~ ( _ q) + �89 (27)

where j(~t; qk) = (~t, qk/ct, qk; qk)=. Next let us consider another bilateral t r ans format ion of Bailey [14], namely

Fa, b;z ] (az, d/a, c/b, dq/abz; q)oo 2~ 2 ~a, abz/d;; d/a ~. (28) 2~I)2 L c, d = (z, d, q/b, cd/abz; q)~ Laz, c d

In (28), changing the base from q to q2 and put t ing a = t, b = q, c = - q, d . . . . . . .

262 Anju Gupta

z = q/t and then letting t ~ 0o, we get

(-- 1; q2)oo

(q; q 2 ) ~ q.(.+ lj(_ 1;q2)~ 4,4q) = z . '

-oo, q; q ),, (29)

which shows that (q; q2)~176 ~bc(q) is an even function of q. ( - 1;q2)oo

In (28), also changing the base from q to q2 and putting a = a, b = q, c = -q2 , d = - q3, z = q3/o~ and then letting ~ --* o0, we get

Wc(--q)= (q)oo tp(q)+ 2q _ ( , 2 ~ [ - q , - q 2 , q 2 q 0 o )

Comparing (30) with (10) we obtain that (q)|162 is an odd function of q. It may be interesting to compare this with a similar phenomenon in the case of

the 'complete' series for the fo(q), Wo(q), fl(q), and ~k I (q), the four fifth order mock theta functions, as remarked by Watson [23; section VII, p. 296].

4. The sixth order mock theta functions and three families of generalized functions associated with them

It may be observed that the third, fifth and seventh order mock theta functions are expressible in terms ofa 2~bl, 3~b2 and a ,~a series, respectively. This seemingly simple observation has far reaching ramifications and has in it the germs of the basis of classification of the mock theta functions of Ramanujan, as of order three, five and seven, a fact which neither Ramanujan nor anybody else during the past seven decades has, thus far, implicity or explicitly defined. (For details see [4a] and [18a]).

In particular, the seven "sixth" order mock theta functions are easily seen to be exvressible as follows;

q Lt ,~tq~)rl/~ q2 ] qJ(q) - ( l+q)~ -.o3w2 L _q2,_q3;q~a ; (32)

1. Lt 3dd2[1/ct'q'-q �9 ] P(q) = 1 - q=-*o qa/2, _ qa/2' - q~t ; (33)

o,q,-- 4,0 [ _ _ Fat/2 ]

)'(q)=3t~2L-" , --qtl2, q ; --q,O --q (35)

iz(q)=a~2[ql/2, _--ql/2'q;q,O - -1] ; (36)


I1/~, 1/fl, q . o-1 ~'(q) = ~,a~oLt 3dP2 qog, qo) 2'qct# l'd (37)

= (1 -- co 2) 2 ~b t [0, q; toq; (.02 ],

on using a well-known transformation due to Sears [20], and where 02 is a primitive cube root of unity.

The above forms of definitions of the 'sixth' order functions suggest the study of transformations of the function 3qb2(q, a, b; c, d; t) and its limiting cases.

We now proceed to prove that for ct = qr, r = 0, 1, 2 , . . . the following three functions A(ct, q), B(ct, q) and C(~,q) form a family of functions closely associated with mock theta functions of order six. The three functions are defined as

A(~;q) = ~. (-)"qn2ct2"(q;q2)"', (38)

n = o ( - ~q; q)2n

B(~;q) = ~ q"("+l)/2ct"(- q)"" (39) .=o (q;q2).+ 1 '

(-- ).a.(q; q2). C0t;q) = L (40)

.=o ( - q ) .

To prove that (38) gives a family of functions, associated with the mock theta functions ~b(q) and W(q), take q - q2, a = ~, b = q, c = - q2, d = - q, z = q/ct and let ct--,0, in (19), to obtain, after some simplification,

~b(q) + W(q) - (1 + q)q2 ~ (_) .q .~+4.(q;q2) .

. = o ( - q; q ) 2 . + 2 - 1 = 0 .

In terms of A(~,q), we have

q2 q A ( q ; q ) - ~ A ( q 2 ; q ) - 1 = 0 . A(1,q) +

1 + q + q ' l (41)

Since A(1;q) and A(q;q) are known to be mock theta functions it follows on repeating the argument , that A(q'; q), r = 0, 1, 2 , . . . is an infinite family of functions connected with q~(q) and ~F(q).

Next, consider a t ransformat ion between three 302's derived by me in an earlier paper [.18], namely

q2(1 - - t ) 3(I) 2 I d , b, q; c, d; t] + ]-(a + b)tq 2 - (c + d)q] 302 [a, b, q; c, d; tq] +

+ [cd - abtq 2 ] 302 [a, b, q; c, d; tq 2 ] + (c + d - q)q - cd = O. (42)

In (42), taking a = ~, b = - q , c = q3/2, d = -qS/2 , t = - q/ct, letting o t~ oo and using (33) and (34), we get

oo r~n2/2 + Sn/2 { r[I

p ( q ) - a(q) -- q(1 + q)~o (q~q2)---~ 1

264 Anju Gupta

In terms of the B(ct; q) functions, we have

B(1; q) - qB(q; q) - q(1 + q)B(q2; q) - 1 = O. (43)

As in (41) it follows that B(q'; q), r = 0, 1, 2 . . . . defines an infinite family of functions associated with p(q) and o(q). Lastly, in (42) taking a = ql/2, b = - q~/2, c = - q, d = O, t = - 1 and using (35) and (36), we get

oo

2u(q) + 2(q) -- q ~ ( _ )n qZ.(q; q2). _ 2 = O. o (-- q),,

In terms of C(~, q) this can be rewritten as

2C(1; q) + C(q; q) - qC(q2; q) - 2 = O. (44)

Therefore it follows that C(qr; q) defines an infinite family of functions associated with/~(q) and 2(q).

The functions A(~; q), B(~t; q) and C(~t; q) are not without interest. As for instance, for ct = - 1, we have

(q2, q4, q6; qO)~ (q; q2)o ~ A ( - 1; q) = , (45) (q2 ; q2)~

on using an identity due to Slater [21;(23)]. An identity of Ramanujan found in the 'Lost' notebook [11;(3.17)R], namely,

q + ~ aq).(-- bq). = c/ab ~ q"~"+ l~/2a"bnc-"(- l/c).

x ( - cq). 1 (aq/c). (bq/c).

- c/ab ( - aqLo ( - bq)~ | q"~a"b"c- 2. ( - cq)~o ~ ( ~ , (46)

for a = -q-1 /2 , b = q-1/2, c = 1 connects C ( - q ; q) with B ( - 1; q) in the form

q C ( - q; q) = 2qB(- 1; q) + q(q; q2)~ (__).qn~.- 1)

( - q ) ~ x (q; q2). (47)

5. The 'complete' mock theta function u ?(q) and its missing pair function

While searching for the pair function corresponding to the last of the 'sixth' order mock theta function ?(q), we first derive some simple alternative forms for the ?(q) function. We have

~(q)= ~ qn~(q).

.=o (q3; q3) n

= (1 - (D2) 2(~1 [0, q; ~q; ~2],

where ~o is a primitive cube root of unity.

(48)

Ramanujan's mock theta functions

Using the following 3(I)2 transformation of Sears [20], namely,

(e),,(f),,(qbc/ef)" = (qbc/f)=(qc/e)oo x ,, "-'= o (qb),, (qc),, (qc)= (qbc/ef)=

x ~. (b).(e).(qb/f).(qc/e)" .=o (q).(qbc/f).(qb).

In (49) taking e = 1/~t, f = 1/fl, b = co, c = o92 and letting 0t and fl~O,-we get

~(q)= 1 ~ ( -- )n(CO)nco2" qn(n + 1)/2

(qco2)~ , ,=o (q)n(qco)n "

(49)

where co is a primitive cube root of unity. Again, use another transformation of Sears [20], namely

(qb/e),,(qb/f),,(q/a),,c" ,=o (q),,(qb/a),,(q2bc/ef),,

(qc/a)oo(qabc/ef)~ ~ (e/a),,(f /a),,(q/a),(qabc/ef)" (50)

(q2 bc/ef)oo (c)= ,/'= o (q),(qb/a),,(qc/a),,

In (50) first putting a = 1, then taking e = 1/0t, f = 1/fl, b = co, c = co2 and letting ~t and fl ~ 0, we get

oo co2n y(q) ---- (1 - - co2)nE-:-O ( coq) , ' (51)

where co is a primitive cube root of unity. Also, one can easily obtain from the result (16.4) of Fine [15] another representation

for ~(q) given by Andrews and Hickerson [13, (2.25)], without going through the various steps adopted by them. In (16.4) of Fine [15], taking b = co and t = co2, we get

or

Y (q) = (1 - (o2) 2(I) 1 (0 , q; coq; 0) 2 )

_ 1 { 1 + 3 ~ ( - )"(1 +qn)q (3nz+n)/2 } (q)oo . = 1 1 "b qn + q2n

,~, 3(--1)nq n(3n+1)/2 (q)~(q) =

- ~ ~ 1 + q" + q2. (52)

which is the result of Andrews and Hickerson [13, (2.25)]. From (48), we observe that the definition of ),(q) suggests a similar form for )'l(q),

the pair function of y(q), i.e.,

at)

Yl(q) = ,~o q,,2(_q),, = ( - - q 3 ; q 3 ) n

= (1 + c02)2~b1(0, q; -- coq; -- (o2). (53)

One easily sees that y~(q) is the third order mock theta function z(q).

266 Anju Gupta

We, now, define a 'complete' mock theta function Gc(~ ) by the bilateral sereis

n 2 q (~tq)n -o0 (e3q3;q3) "

n 2 q (otq)n - oo (Gt3q 3, q3)n

_ qn'(otq)n

We have

~3q(1 - 1/~ a) -~'~'-l/-~ ~0 (ot2q)n(q3/Ot3'q3)n(q/ct; q). (54)

In (54), taking ~t = 1, we get

Gr = L(q) = )'(q) + 3q x~ q,(q3;q3), o (q; q),

(55)

where y,(q) denotes the 'complete' y(q) function. Again, in (54) taking ~t = - 1, we get

oo q~,(_ q3; q3)n G,(- 1)= gc(q)= x(q) + qZ ,

( - q;q). (56)

where xc(q) denotes the 'complete' x(q) function. In the result of Fine [15, (16.4)] taking t = - c o 2, we get

x(q) = (1 + ( . D 2 ) 2 ~ 1 ( 0 , q; - - (_Dq; - - ( / )2)

: l {1 "~- ~ (--)n(-l-Jt-qn)q'3n2+n)/2~ (q)~ . = 1 1 -- q" + q2n j

o r oo ( )nqnI3n+l)/2

(57)

In the 'Lost' Note book on p. 17, the last but one formula mentions a relation between 3'(-q), ~ ( -q ) , the third order mock theta function. This indicates that probably ),(q) belongs to the class of the third order mock theta functions.

6. C o n c l u s i o n .

The above analysis in w167 3, 4 and 5 and the works of earlier authors makes it very clear that for a better understanding of the structure and transformation theory of the various mock theta functions, perhaps, the best tool is the corresponding basic bilateral hypergeometric series or their 'complete' forms (See Agarwal [4a] and Gupta [18a]).

I think it may also be possible to deduce some of the relations derived in this paper by using series manipulation of the Lambert series representations of the functions of Andrews and Hickerson [13].


Acknowledgements

I am grateful to the University Grants Commission, New Delhi for providing me with Senior Research Fellowship. I am also grateful to Prof. R P Agarwal for his guidance.

References

[1]

[2]

[3] [4]

[4a]

[5]

[6]

[7]

[8] [9]

[lO]

[11] [12]

[13]

[14] [15]

[16]

[17] [18]

[18a]

[19]

[2o]

[21]

[22] [23]

[24]

Agarwal R P, Certain basic hypergeometric identities associated with mock theta functions, Q. J. Math. (Oxford) 20 (1968) 121-128 Agarwal R P, A 'Lost' notebook of Ramanujan of G E Andrews, Adv. Math. (M.I.T.) 53 (1984) 291-300 Agarwal R P, Ramanujan's Last Gift, Math. Stud. 58 (1991) 121-150 Agarwal R P, Lambert series and Ramanujan, Platinum Jubilee Lecture delivered at the Indian Science Congress session (1992) at Baroda. (under publication) Agarwal R P, Mock tbeta functions - An analytical point of view, Nat. Acad. Sci. Letters. 15 (1992) 51-52 Detailed paper under publication in Proc. Nat. Acad. Sci. India (1993) Andrews G E, On basic hypergeometric series, mock theta functions and partitions - I, Q. J. Math. (Oxford) 17 (1966) 64-80 Andrews G E, On basic hypergeometric series, mock theta functions and partitions - II, Q. J. Math. (Oxford) 17 (1966) pp. 132-143 Andrews G E, On the theorems of Watson and Dragonette for Ramanujan's mock theta functions, Am. J. Math. 88 (1966) 454-90 Andrews G E, An introduction to Ramanujan's 'Lost' Notebook, Am. Math. Mon. 86 (1979) 89-108 Andrews G E, The fifth and seventh order mock theta functions, Trans. Am. Math. Soc. 293 (1986) 113-134 Andrews G E, Ramanujan's fifth order mock theta functions, Ramanujan revisited. Acad. Press, N. Y. (1988) 47-56 Andrews G E, Ramanujan's 'Lost' n otebook-I Partial 0-functions, Adv. Math. 2, 41 (198 I) 137-172 Andrews G E and Garvan F G, Ramanujan's 'Lost' Notebook-VI - The mock theta conjectures, Adv. Math. 73 (1989) 242-255 Andrews G E and Hickerson Dick, Ramanujan's 'Lost' Notebook VII: The sixth order mock theta functions, Adv. Math. 89 (1991) 60-104 Bailey W N, On the basic bilateral hypergeometric sereis 2u12, Q. J. Math. (Oxford) 2 (1950) 194-198 Fine N J, Basic hypergeometric series and Applications (1988) Mathematical Surverys and Monographs, Vol 27, Amer. Math. Soc. Providence, RI Gupta Mukul, Certain basic hypergeometric transformations, mock theta functions and related identities - I, Proc. Natl. Acad. Sci., India. 59 (1989) 573-586 ibid II, 60 (1990) 297-305 Gupta A, A study of transformations for a special type of 302 series and Mock theta functions (under publication in Ganita 44 (1993)) Gupta A, A study of certain Basic Hypergeometric Functions and Transformations Associated with Ramanujan, (Doctoral thesis approved by the University of Lucknow) (1992) Ramanujan S, The "Lost" Notebook and other unpulished papers. (1988) Narosa Publishing House, New Delhi Sears D B, On the transformation theory of Basic Hypergeometric function, Proc. London Math. Soc. 53 (1951) 158-179 Slater L J, Further identities of the Rogers-Ramanujan type, Proc. Lond. Math. Soc., (2) 54 (1952) 147-167 Slater L J Generalized Hypergeometric series. (1966) Camb. Uni. Press Watson G N, The final problem: an account of the mock theta functions, J. London Math. Soc. 11 (1936) 55-80 Watson G N, The mock theta functions (2), Proc. Lond. Math. Soc. 42 (1937) 274-304

Documents

On certain Ramanujan's mock theta functions