On generalized eighth order mock theta functions Текст научной статьи по специальности «Математика»

URAL MATHEMATICAL JOURNAL, Vol. 6, No. 1, 2020, pp. 137-146

DOI: 10.15826/umj.2020.1.011

ON GENERALIZED EIGHTH ORDER MOCK THETA FUNCTIONS

Pramod Kumar Rawat

University of Lucknow, Lucknow 226007, India [email protected]

Abstract: In this paper we have generalized eighth order mock theta functions, recently introduced by Gordon and MacIntosh involving four independent variables. The idea of generalizing was to have four extra parameters, which on specializing give known functions and thus these results hold for those known functions. We have represented these generalized functions as q-integral. Thus on specializing we have the classical mock theta functions represented as q-integral. The same is true for the multibasic expansion given.

Keywords: q-Hypergeometric Series, Mock Theta functions, Continued Fractions, q-Integrals.

1. Introduction

The last gift to mathematics by Ramanujan was mock theta functions. In his last letter to Hardy [5], Ramanujan introduced 17 functions and called them mock theta functions as they were not theta functions and classified them as 4 functions of third order, 10 functions of fifth order and 3 functions of seventh order though Ramanujan did not say what he meant by "order" of mock theta function. Later Watson [12] introduced 3 more mock theta functions of third order. Gordon and Mcintosh [7] gave eight more mock theta functions and called them of eighth order. Andrews and Hickerson [3] said the "order" is connected with combinatorics interpretation. Andrews [1] generalized five third order mock theta functions. Srivastava [11] generalized eighth order mock theta function. Recently Choi [4] also generalized mock theta functions of third, fifth, sixth, seventh and tenth order.

Motivated by Andrews' generalization of five of seven third order mock theta functions and Choi's generalization, we have tried to generalize the eighth order mock theta functions by introducing four independent variables. The advantage is that by specializing the parameters we can have known functions.

In this paper we have represented these generalized functions as q-integral and we have also given the multibasic expansion. Thus we have on specializing the parameters, the classical mock theta functions representation as q-integral and the multibasic expansion for generalized functions reduced to classical mock theta function of eighth order.

2. Definitions and notations

The eighth order mock theta functions of Gordon and Mclntosh [7] are

= Sl(g) = f; """'+2>(-g-g2>'

^ (-q2; q2)n (-q2; q2)n

n=0 v ^ '^ "" n=0

^ (n+1)(n+

q(n+1)(n+2) ( q2; q2) ^ qn2+n( q2; q2)

To{d) = Y.q—(—t1^ TM = E ( I

n=0 (-q; q2)n+1 n=0 (-q; q2)n+1

qn2 ( q; q2) q(n+1)2 ( q2)

TT / \ V^ q ( —q q )n TT f N V^ q q /n

rw,)=r 1(,)=£ '

qn2 ( .... q2) q2n2 ( ...2. q4)

to ¿i

^ (q-,q2)n+i (<?;<?2)2n+2 '

n=0 ^ "+1 n=0

where

(a; qk)n ^(1 - aqk(j-1)), (a; qk= [](1 - aqk(j-1)), and (a; qk)o = 1.

j=i j=1 3. Generalized eighth order mock theta functions

The four variable generalization of the eighth order mock theta functions are

R , 1 t)nq"+nl3(-zq]q2)

rp /, O \ 1 ^ (¿)rag»3+3n+2-n+ra/3(_ 2/a 2)

n=0

«,(., a, ft t; 9) = ^ £--

tru a \ 1 i 1 v^ (t)nqn2 n+nl(-zq;q2)nan V0(t, a,f3,z;q) = -l + — ^-j—^-

ray \ ^HiH )n<-

n

n

(aq; q2)n

* = s ——

Tl(i'a'* = (ijl S (-«A^W^i

(i'* = mz S —H^tw—'

Vl(t, a, /J, z; q) = J— ^--

(t)~ ^ (aq; q )n+1

For t = 0, a = 1, ft = 1 and z = 1 these functions reduce to classical mock theta functions.

4. Relation between generalized eighth order mock theta functions

The differential operator Dq [8] is defined as

zDq,zF(z, a) = F(z, a) — F(zq, a).

By using the differential operator we shall connect the generalized eighth order mock theta functions.

n

Proposition 1. The following is true:

(i) Dq2 tS0(t, a, fl, z; q) = S1(t, a, fl, z; q),

(ii) q2D2,tT1(t,a,fl,z; q) = T)(t, z, a, fl, z; q).

Proof. Proof of (i):

+ n au a 1 ^ (t)nqn2-n+n^(-zq; q2)nan 1 ^ (tq)nqn2-n+n^(-zq; q2)nan

tDq,tS0(t, a, ft, Z] g)-— ^ — ^

1 1 A (t)nqn2-n+n^-zq-,q2)no^ n

(-aq2; q2)« (t)~ (-aq2; q2)

i Y^{t)nqn2+nß{-zq]q2)nan

(t)^ (-aq2; q2)n

Similarly

DqMt, a, ß, z- q) = — g--

1 (t) qn2+2ra-ra+raß( zq; q2)

1 V^ (t)nq (-zq q )na a „ „ a

=m S--=Sl(ti - A '

which proves (i). Proof of (ii):

1 ^ qra2+2ra-ra+raß ( q2 / q2) n T> (+ a \ 1 V^ q ( — q/a; q )™

n=0

and

D^fi, a, ft t;,) = ^ £ ■

n=0

1 ^ qn2+3n+2-n+n,3(-q2/a; q2)

= — = ft

which proves (ii). □

5. q-Integral representation for the generalized eighth order mock theta

functions

Thomae and Jackson [6, p. 19] defined q-integral

r 1 ^

/ f (t)dqt = (1 - q)£ f (qn)q 0

n=0

using limiting case of q-beta integral, we have

1 (1 - q)-1 Z1

f tx *(tq; t. o

(qx; q)~ (q; q)~ ./0

We now represent these generalized functions as q-integral. By specializing the parameters we have the integral representation for classical mock theta functions.

Theorem 1.

(i) So(qt, a, ft, z.

(ii) To(qt,a,ft,z

(iii) Uo(qt,a,ft,z;

(iv) V0(qt,a,ft,z

(v) S1(qt, a, ft, z.

(vi) T1(qt,a,ft,z

(vii) U1(qt, a, ft, z;

(viii) V1(qt,a,ft,z

(i - q)~

(q; q)« (i - q)"

(q; q)« (i - q)"

(q; q)«

(i - q)

(q; q)« (i - q)"

(q; q)«

(i - q)

(q; q)«

(i - q)

(q; q)«

(i-g)'

(q; q)

u (uq; q)«So(0,a,pu, z; q)dqu,

/ u* 1(uq; q)«To(0,a,pu,z; q)dqu, Jo

u (uq; q)«Uo(0,a,pu, z; q)dqu,

u 1(uq; q)«Vo(0,a,pu, z; q)dqu,

«o

/ u* :(uq; q)«Si(0,a,pu, z; q)dqu, Jo

/ ut-1(uq; q)«T1(0,a,pu, z; q)dqu, o

/ ut-1(uq; q)«U1(0,a,pu, z; q)dqu, o

/ ut-1(uq; q)«V1(0, a,pu, z; q)dqu. o

Proof. A detailed proof for So(ql,a,ft,z; q) is given.The proofs of the other functions are similar, so omitted.

Proof of (i): By definition

S0(t, a, ft, z; q) =

1 ^ (t)nqn2-n+n|(-zq; q2)raan

(t)

n=0

(-aq2; q2)r

Replacing t by q*, we have

(q )«

; q2)nan

n=0

(-aq2; q2)

;q2)n an

n=o

(-aq2; q2)n(qn+t; q)c

qn2-n+n^(-zq]q2)nan (1 -

(q-,q)

-1 1

n=o (-aq2; q2)n

« J0

/ u* 1(uq; q)«dqu, o

1

o

1

o

1

o

but

q (-zq; q )n

« qn2-n+ni3(-zq; q2)nan

So(0,a,f3,z-,q) = ,

^ (-aq2; q2)

n=0

putting qi = p, we have

n

x V- qn2 n{-zq]q2)nanpn

n=0

(1 - q)-1 f 1

S0{qt ,a,ft,z-,q) = —--- / ut~1(uq]q)ooS0(0,a,pu, Z]q)dqu,

(q; q)« JO

which proves (i).

The proof of all the other functions is similar. Taking a = 1, ft = 1 and z = 1 we have the integral representation of the classical eighth order mock theta functions. □

6. Multibasic expansion of generalized eighth order mock theta functions

The following bibasic expansion will be used to give multibasic expansion for the generalized functions.

Theorem 2. The following is true :

(1 — apkqk){l — bpkq~k){a, b]p)k{c, a/bc; q)kqk

ÔI ^ "m+fc

(6.1)

\ - yi - up q - up g )(u,,u,p)k\c,u,/uc,q)kq \ -

¿J (1 - a)(l - b)(q, aq/b; q)k(ap/c, bcp;p)k "m+fc

_ y^ (ap, bp', p)m(cq, aq/bc; q)mqm (q,aq/b]q)m(ap/c,bcp]p)m

P r o o f. Using the summation formula [6, (3.6.7), p. 71] we have

apkqk)( 1 - bpkq~k) (a, b;p)k(c, a/bc; q)k k ^ (1 - a)( 1 - b) (q, aq/b; q)k(ap/c, bcp;p)k

_ (ap,bp;p)n(cq, aq/bc; q)n (q, aq/b; q)n(a,p/c, bcp;p)n

and [9, Lemma 10, p. 57],

EE5 (k,n) = EE b (k,n+k),

n=0fc=0 n=0fc=0

therefore we get the statement of the theorem. □

x n

X X

We will consider the following case of Theorem 2. Case I. Letting q ^ q3 and c ^ to in Theorem 1, we have

x

E-

fc=Q

(1 - apkq3k)( 1 - />//'</ 3fc)(a, b-,p)kq(-3k'2+3k^2 (1 - a)(l - 6)(q3, ag3/6; q3)fc6fc;p(fc2+fc)/2

am+fc

m=Q

E

m=Q

(ap,bp-,p)mq(-3m2+3m)/'2 (q3, aq3/b; q3)m&"Vm2+m)/2

(6.2)

am

Theorem 3. The multibasic hypergeometric expansion of these generalized functions are :

(i) So (t,a, 1, z; q) =

1 ^ (1 - tq4k-1)(1 - k-2fc+2)(i; q)fc-i(-zq; q2)fcqfc2ak

(t)c

E

fc=Q

(1 - qk+2)(-aq2; q2)fc

x 0

(ii) To(t, a, 1, z; q) =

1

q;-zq2fc+1;iq3fc ;q3k+3 qk+3;-aq2k+2:Q ; x ^ . 4fc-l

2 3

q, q2, q3; qa

E

(1 - tq4k-1)(1 - k-2k+2)(t; q)fc-i(-q2/a; q2)fcqk2+3k+2ak

x 0

(t)x k=Q (1 - qk+2)(-q/z; q2)fc+izk+1

q;-q2fc+2/a;iq3fc ;q3fc+3_ qk+3;-q2k+2/z:Q ; x

; q, q2, q3; q4a

(iii) UQ (t,a, 1,z; q) =

1 ^ (1 - tq4k-1)(1 - k-2k+2)(t; q)fc-i(-zq; q2)fcqfc2akz2k

(t)c

E

fc=Q

(1 - qk+2)(-aq4; q4)fc

x 0

q;-zq2fc+1;iq3fc ,q3fc+3;Q 2 3 4 2

qk+3:Q;Q;-«q4fc+4 ; ;q,q ,q ,q ;z qa

v = + --

X $

q-_zq2k + 1-tq3k q3k + 3 2 3 2

qk+3^_aq5!fc+i:o ; q,q ,q; qaz

1 ^ (1 -¿^-^(l - fc-2fe+2)(f;g)fe-i(-^;g2)fegfe3+2feafe (V) Sl(i,a,Mi?)=(^i-(1 — gk+2)(—ag2] g2)k-

x $

q-_zq2fc+1-tq3fc ,q3k+3 _ ^ ^2 ^,3.^,3 q

fc+3:_aq2fc+2:o ; q,q ,q; q a

rvn T d ™ \ ^-tciik-l)ii-k-2k+2){t]q)k.l{-q2/a]g2)kgk2+kak

(t)« ^ (1 - qk+2)(-q/z; q2)fc+1zk+1

x $

q-_q2fc+2/a-tq3fc „3fc+3 2 3 2 _1

qfc+3-_q2k+3/z:o ;q,q ,q ;qz a

t .. , 1 ^ (1 -tg^jl -k-2k+2)(t-,q)k.1(-zq-,q2Wk+^akz2k (VU) -(1 -g^-ag2^-

X $

q-_zq2k+1-tq3k ,q3fc+3 2 3 4 3 2 qfc+3-o-o-_aq4k+6:o ; q,q ,q ,q ; q z a

(V111) -(1 - gk+2)(ag] g2)fc+i-

X$

q-_zq2k+1-tq3k,q3k+3; q q2 q3; 3 2a

qfc+3-aq2fc+3:o ; q,q ,q ; q z a

Proof. We shall give the proof of (i) only, for others we will state the value of parameters. Proof of (i): Taking a = t/q, b = q2, p = q and

(g3; g3)m(t; g3)m(~zg] g2)mamgm . am =-———-^—- m (6.2),

f; q)m(-aq2; q2)m

we have

^ (1 - tg^-^j 1 - g-2k+2)(t/g, g2; g)fcgfc3+fc

(! - t/g)^ - g2)(g3,t; gi)kg2k (t; q3)m+fc(q3; q3 Wfc(-zq; q2)m+fcam+kqm+k

oo

¿J, (gi-,g)m+k{-ag2]g2)m+k

« (t, q3; q)mqm2+m (q3; q3)m(t; q3)m(-zq; q2)mamqm

m=o (q3,t; q3)mq2m (q3; q)m(-aq2; q2)r

The right hand side is equal to

E

(t, q3; q)mqm (q3; q3)m(t; q3)m(-zq; q2)mamqm

m=o (q3,t; q3)mq2m (q3; q)m(-aq2; q2)

y^ jt] g)m{-zg] g2)mgm2am n ( cxg2] g2)m

m=o

= (t)«So(t,a,ft,z; q).

The left hand side of (6.3) is equal to

^ (1 - ^-^(l - q-2k+2)(t/q, q2; q)kqk2+k (l-i/9)(l"92)(93>i;93)fc92fe

A (t; q3)k(tq3; q3)m(q3; q3)k(q3k+3] q3)m(~zq; q2)k(-zq2k+l] q2)mam+kqm+k ¿0 (q3; q)k(qk+3; q)m(~aq2; q2)k(-aq2k+2; q2)m

= ~ (1 -¿^-^(l - q-2k+2)(t]q)k.1(-zq]q2)kqk2ak (l-9fe+2)(-aç2;ç2)fc

~ (tq3; q3)m(q3k+3; q3)m(-zq2k+1; q2)mamqm

E

(qk+3; q)m(—aq2k+2; q2)

m=0 <x

1 g (1 -^-^(l -k-^rnqh^-zqiq2)^^

(t)x k=0 (1 - qk+2)(-aq2; q2)fc

q.-zq2fc+l.tq3fc ;q3fc+3 2 3

qfc+3;-aq2fc+2:0 ; q,q ,q ; qa

which proves (i).

Proof of (ii): Take a = t/q, b = q2, p = q and

_ (q3; q3)m(t; q3)m(-q2/a; q2)mq4m+2«m . (q3; q)m{-q/z-, q2)m+izm+1 111

Proof of (iii): Take a = t/q, b = q2, p = q and

„ _ (q3; q3)m(t; q3)m(-zq; q2)mqmz2mam

am — /„z.„\ < „„A.„A\ 111

(q3;q)m(-aq4;q4)m Proof of (iv): Take a = t/q, b = q2, p = q and

„ _ (q3; q3)m(t; q3)m(-zq; q2)mqmz2mam

OLm — / o N , o\

(q3; q)m(aq; q2) Proof of (v): Take a = t/q, b = q2, p = q and

(q3; q3)m(t; q3)m(—zq] q2)mq3mam . am =--—^——-^—- m (6.2).

(q3;q)m(-aq2;q2)m Proof of (vi): Take a = t/q, b = q2, p = q and

_ (q3j q3)m{t] q3)m{ — q2/Q; q2)mq2mQm .

(q3; q)m{—q/z] q2)m+izm+l m (b-2j-

Proof of (vii): Take a = t/q, b = q2, p = q and

= (q3; q3)m(i; q3)m(-zq; q2)mq3m+VmQ:m ^ (q3;q)m(-aq2;q4)m+i

Proof of (viii): Take a = t/q, b = q2, p = q and

(q3; q3)m(t; q3)m(-zq; q2)mq3m+1z2mam .

«m =-r-î—-- m (6.2).

(q3; q)m(aq; q2)m+i

By taking a = 1, fi = 1 and z = 1 we have multibasic expansion of classical eighth order mock theta functions.

m

m

7. Special cases and Ramanujan's cubic continued fraction Proposition 2. We have the following special cases

f (—q, —q)

(i) Uo(0, -1,1,1; q) =

(ii) Uo(0,-1,1,1; —q) =

(iii) Uo(0, —1, 3, —1; —q) =

(iv) Uo(0, —1,1, —1; —q) =

q)

f(-q2,-q2) q)

f (—q, —q5)

f (—q3, —q3)

$(-q)

P r o o f. Proof of (i): By definition we have

-t^MZ-' <">

put t = 0, a = -1, ft = 1 and z = 1, therefore we have

=±{-lr^f)'\ (7.2)

n=0

from [10, eq. (A.13), p. 171], we have

M-q) to '

by (7.2) and (7.3), we get which proves (i).

Proof of (ii): Put t = 0, a = -1, ft = 1, z = 1 and replace q = -q in (7.1), we have

n=o (q ; q )n

from [10, eq. (A. 23), p. 172], we have

f{-q2,-q2) _ y^

q) n=o (q4; q4)n

by (7.4) and (7.5), we get which proves (ii).

Proof of (iii): Put t = 0, a = —1, ft = 3, z = —1 and replace q = — q in (7.1), we have

n=0

from [10, eq. (A. 52), p. 175], we have

/(-<?, -q5) v g"3+2ra(-g;g2)n r7 ~

by (7.6) and (7.7), we get

4>(_q) n=0 (q4; q4)n

C/o(0,-l,3,-l;-g) = /(^g)g5), (7.8)

which proves (iii).

Proof of (iv): Put t = 0, a = -1, ft = 1, z = -1 and replace q = -q in (7.1), we have

n=o (q ; q )n

from [10, eq. (A. 53), p. 175], we have

f (-q3, -q3) qn2 (-q; q2)n

^(_q) n=0 (q4; q4)n '

(7.10)

by (7.9) and (7.10), we get

Uo(0,-1,1,-1 -,-Q) = f{/{:/\ (7.11)

which proves (iv). □

Remark 1. Dividing (7.8) by (7.11), we have

Up(0, —1, 3, —1; —g) = f(-q,-q5) = q + q2 q2 + qA q3 + q6 Uo(0, —1,1, —1; —q) f(-q3,~q3) 1+1 + 1 +"'

which is Ramanujan's cubic continued fraction [2, (3.1.6), p. 86].

8. Conclusion

The advantage of the generalization presented in the paper is that by specializing the parameters we can obtain known functions which connects mock theta functions with continued fractions. So the results obtained for mock theta functions are reduced to continued fractions.

Acknowledgement

The author thanks the reviewers for their useful comments and Prof. Bhaskar Srivastava for his guidance.

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