Some Identities Relating Mock Theta Functions And Hikami Mock Theta Functions

International OPEN

Journal

ACCESS

Of Modern Engineering Research (IJMER)

Some Identities Relating Mock Theta Functions And Hikami Mock Theta Functions *M. M. Eltikali,N. U. Khan,*T. M. Elfrgani *A. Paul Department Of Applied Mathematics, Faculty Of Engineering And Technology Aligarh Muslim University, Aligarh, India. *Department Of Mathematics, Sam Higginbottom Institute Of Agriculture Technology And Sciences, Allahabad, India.

Abstract: In this paper, we established certain new representations of Mock theta functions and Hikami’s Mock theta functions by making use of identities due to Singh, S. P. and Mishra, B. P. [12].

Keywords: Mock theta functions of order two, six, eight and ten, Hikami’s Mock theta functions of order two, four and eight, identities.

I.

INTRODUCTION

Srinivasa Ramanujan submitted an interesting function defined by a q  series as a gift to the world of mathematics is known as Mock theta function. He gave a list of seventeen such functions classifying them as of order three, five and seven, but did not say what he meant by order. Furthermore after the discovery of Ramanujan's Lost Notebook more Mock theta functions of distinct orders studied by Andrews, G. E. [2], Andrews, G. E. and Hicerson, D. [3], Gordon, B. and McIntosh, R., J. [4,5] and Choi, Y. S. [6]. Hikami‟s [8, 9] discovered new Mock theta functions of order two, order four and order eight respectively. Recently, Eltikali, M., Paul, A. et al. [7] developed new representation among Hikami Mock theta functions related with many 

selected identities. If

F (q) :  f (q, n) is a Mock theta function, then the corresponding partial Mock theta n 0

m

function is denoted by the truncated series,

Fm (q) :  f (q, n) and the complete Mock theta function is n 0

defined by the bilateral series Fm (q) :

II.



 f ( q, n) .

n  

DEFINITIONS AND NOTATIONS

q  1 and use the notation

Throughout this article, we assume that

(a) 0 : (a; q) : 1,

(a; q k ) 0 : 1 n 1

(a) n : (a; q) n :  (1  aq r ),

(2.1)

n  1.

r 0



(a; q)  :  (1  aq ), r

r 0

(2.2)

n 1

(a; q ) n :  (1  aq k r ), k

r 0

(2.3)

where ( a; q ) n is q  shifted factorial. k

The second order Mock theta functions are: | IJMER | ISSN: 2249–6645

www.ijmer.com

| Vol. 7 | Iss. 5 | May. 2017 | 45 |

Some Identities Relating Mock Theta Functions And Hikami Mock Theta Functions 

q n 1 (q 2 ; q 2 ) n

n 0

(q; q 2 ) n 1

A(q) : 



q n (q; q 2 ) n

n 0

(q; q 2 ) n 1

B(q) : 

,

(2.4) 

(1) q (q; q ) n

n 0

(q 2 ; q 2 ) 2n

 (q) :  The last function

 (q)

n2

n

2

.

appears in Ramanujan's "Lost" Notebook.

The six order Mock theta functions are:

(1) n q ( n 1) (q; q 2 ) n  (q ) :  ; (q; q ) 2 n 1 n0

(1) n q n (q; q 2 ) n  (q) :  , (  q; q ) 2 n n0 2





 (q) : 

q

n ( n 1) 2

, (  q; q ) n 2

(q; q ) n 1

n 0

2





 (q) :  q

,

( n 1)( n  2 ) 2

n 0

(q; q ) n

2

(q; q ) n 1 (2.5)

(1) q (q; q ) n , (q; q ) n n0 

n

 (q) : 

n

(1) (q; q ) n , (  q; q ) n n 0 

2

n

 (q) : 

2

2



 (q) :  n0

q n ( q; q ) n (q 3 ; q 3 ) n

.

The eight order Mock theta functions are: 2



q n (q; q) n

S 0 (q) : 

(q 3 ; q 3 ) n

n 0



T0 (q) : 



U 0 (q) :  n 0

S1 (q) : 

(q 3 ; q 3 ) n

(q 3 ; q 3 ) n 

q n (q; q 2 ) n

n 0

(q; q 2 ) n

V0 (q) : 1  

| IJMER | ISSN: 2249–6645

(q 3 ; q 3 ) n 2



T1 (q) :  n 0

2

q n (q; q) n

2

q n (q; q) n

n 0

2

q n (q; q) n

n 0



q n (q; q) n (q 3 ; q 3 ) n



U 1 (q) :  n 0

2

2

q n (q; q) n (q 3 ; q 3 ) n 

q ( n 1) (q; q 2 ) n

n 0

(q; q 2 ) n 1

V1 (q) : 

www.ijmer.com

2

(2.6)

| Vol. 7 | Iss. 5 | May. 2017 | 46 |

Some Identities Relating Mock Theta Functions And Hikami Mock Theta Functions The ten order Mock theta functions are:

q n ( n 1) / 2 , 2 n  0 ( q; q ) n 1

q ( n 1)(n  2) / 2 2 n  0 ( q; q ) n 1





 LC (q) : 

 LC (q) : 

(2.7)

(1) n q ( n 1) . n  0 (  q; q ) 2 n 1

(1) n q n , n  0 (  q; q ) 2 n 2



X LC (q) : 

2



 LC (q) : 

Hikami‟s Mock theta functions are: 

D5 (q) :  n 0

q n (q; q) n (q; q 2 ) n 1



q n (q 2 ; q 2 ) n

n 0

(q n 1 ; q) n 1

D6 (q) : 

,

, (2.8)



I 12 (q) : 

q 2 n (q; q 2 ) n

n 0

(q n 1 ; q) n 1



q n (q; q 2 ) n

n 0

(q n 1 ; q) n 1

I 13 (q) : 

,

.

We shall make use of following identities in our analysis:

aq, yq; q   a, y; qn q n n  aq, yq; qn  n :  r     q, ayq; q n0 n0 q, ayq; qn r 0 n0 q, ayq; qn 







n

 n1 .

(2.9)



n

    :              . n

n 0

r

r 0



n

n 0





r

r 0

n

n 0



n

r 0

r

n 0

n n

(2.10)



n

   :           .

n  

n

r 0

r

n

n 0

III.

r n

r

n  

n

r 0

r

n  

n n

(2.11)

MAIN RESULTS

1.We establish new representations involving partial Mock theta functions of order two, six, eight and ten. Also, partial Hikami‟s Mock theta functions of order two, four and eight are obtained.

aq, yq; q q, ayq; q

Where



A(q)   n 0

a, y; qn q n q, ayq; qn



An (q)   n 0

aq, yq; qn q, ayq; qn

(q; q 2 ) n  2

.

(3.1)

A(q) is a Mock theta function of order two.

  a, y; qn q n aq, yq; qn aq, yq; q  ( q)    n ( q)   q, ayq; q n 0 q, ayq; qn n 0 q, ayq; qn

Where

q n  2 (q 2 ; q 2 ) n 1

 (q)

q (n1)(n 2) / 2 (q; q) n1 2

(q; q ) n 2

.

(3.2)

is a Mock theta function of order six.

  a, y; qn q n aq, yq; qn aq, yq; q S 0 (q)   S 0, N (q)   q, ayq; q n  0 q, ayq; q n n 0 q, ayq; q n

q ( n1) (q; q 2 ) n1 2

(q 2 ; q 2 ) n1

.

(3.3)

Where S 0 (q) is a Mock theta function of order eight. | IJMER | ISSN: 2249–6645

www.ijmer.com

| Vol. 7 | Iss. 5 | May. 2017 | 47 |

Some Identities Relating Mock Theta Functions And Hikami Mock Theta Functions

aq, yq; q  (q)   a, y; qn q n  (q)   aq, yq; qn   LC ,n q, ayq; q LC n 0 q, ayq; q n n 0 q, ayq; q n Where

q ( n1)(n 2 ) / 2 . (q; q 2 ) n 2 (3.4)

 LC (q)

is a Mock theta function of order ten.

aq, yq; q q, ayq; q

a, y; qn q n q, ayq; qn



D5 (q)   n 0



D5,n (q)   n 0

aq, yq; qn q, ayq; qn

q ( n 1) (q; q) n 1 (q; q 2 ) n  2

.

(3.5)

Where D5 ( q ) is Hikami‟s Mock theta function of order two.

2.In this section we shall make use of identity (2.10) to establish product formulae for any two Mock theta functions. 

q n 1 (q; q 2 ) n

n 0

(q; q 2 ) n 1

A(q) B(q)  



q n 1 (q 2 ; q 2 ) n

n 0

(q; q 2 ) n 1

An (q)  



q 2 ( n 1) (q 2 ; q 2 ) n (q; q 2 ) n

n 0

(q; q 2 ) 2n 1

B n (q)  

(3.6) This is a product formula for two Mock theta functions of order two. 

q n ( n 1) / 2 (q; q) n

n 0

(q; q 2 ) n 1

A(q)  (q)  



q n 1 (q 2 ; q 2 ) n

n 0

(q; q 2 ) n 1

An (q)  



q ( n 1)(n  2) / 2 (q; q) n (q 2 ; q 2 ) n

n 0

(q; q 2 ) 2n 1

 n (q)  

(3.7) This is a product formula for Mock theta functions of order two and six. 

q n (q; q 2 ) n

n 0

(q 2 ; q 2 ) n 1



q n 1 (q 2 ; q 2 ) n

n 0

(q; q 2 ) n 1

2

A(q) S 0 (q)  

An (q)  



q n ( n 1) 1 (q; q 2 ) n

n 0

(q; q 2 ) n 1

S 0, n (q )  

.

(3.8)

This is a product formula for Mock theta functions of order two and eight.   q n 1 (q 2 ; q 2 ) n q ( n 1)(n  2) / 2 (q 2 ; q 2 ) n q n ( n 1) / 2 A ( q )   ( q )    n LC , n 2 (q; q 2 ) n 1 (q; q 2 ) 2n 1 n  0 (q; q ) n 1 n 0 n 0 

A(q) LC (q)  

(3.9) This is a product formula for Mock theta functions of order two and ten. 

q n (q; q) n

n 0

(q; q 2 ) n 1

A(q) D5 (q)  



q n 1 (q 2 ; q 2 ) n

n 0

(q; q 2 ) n 1

An (q)  



q 2 n 1 (q; q) n (q 2 ; q 2 ) n

n 0

(q; q 2 ) 2n 1

D5, n (q)  

.

(3.10) This is a product formula for Mock theta functions of order two and Hikami‟s Mock theta function of ordero.   () n q ( n 1) (q; q 2 ) n () n q n (q; q 2 ) n () 2 n q 2 n ( n 1) 1 (q; q 2 ) 2n  (q) (q)    n (q)    n (q)   . (q; q) 2 n 1 (q; q) 2 n n 0 n 0 n  0 (q; q) 2 n (q; q) 2 n 1 2



2

(3.11) This is a product formula for two Mock theta functions of order six. 

q n (q; q 2 ) n

n 0

( q 2 ; q 2 ) n

 (q) S 0 (q)  

2

| IJMER | ISSN: 2249–6645

 () n q n (q; q 2 ) n () n q 2 n (q; q 2 ) n  n (q)   S 0, n (q)   . (q; q) 2 n ( q 2 ; q 2 ) n n 0 n 0 (3.12) 

2

www.ijmer.com

2

| Vol. 7 | Iss. 5 | May. 2017 | 48 |

Some Identities Relating Mock Theta Functions And Hikami Mock Theta Functions This is a product formula for Mock theta functions of order six and eight.   () n q n (q; q 2 ) n () n q n (3n 1) / 2 (q; q 2 ) n q n ( n 1) / 2  (q) LC (q)    n (q)    LC , n (q)   .

2 2 (q; q) 2 n n  0 (q; q ) n 1 n 0 n  0 (q; q ) n 1 (q; q ) 2 n 2



(3.13) This is a product formula for Mock theta functions of order six and ten. 

q n (q; q) n

n 0

(q; q 2 ) n 1

 (q) D5 (q)  

 () n q n (q; q 2 ) n () n q n ( n 1) (q; q 2 ) n (q; q) n  n (q)   D5, n (q)   . (q; q) 2 n (q; q) 2 n (q; q 2 ) n 1 n 0 n 0 2



(3.14) This is a product formula for Mock theta functions of order six and Hikami‟s Mock theta function of order two. 

q n ( n  2) (q; q 2 ) n

n 0

( q 2 ; q 2 ) n

S 0 (q) S1 (q)  



q n (q; q 2 ) n

n 0

( q 2 ; q 2 ) n



q 2 n ( n 1) (q; q 2 ) 2n

n 0

(q 2 ; q 2 ) 2n

2

S 0, n (q)  

S1, n (q)  

.

(3.15) This is a product formula for two Mock theta functions of order eight. 

q n ( n 1) / 2 (q; q 2 ) n

n 0

(q; q 2 ) n 1

S 0 (q) LC (q)  



q n (q; q 2 ) n

n 0

( q 2 ; q 2 ) n

S 0, n (q)  

2



q n (3n 1) / 2 (q; q 2 ) n

n 0

(q; q 2 ) n 1 (q 2 ; q 2 ) n

 LC , n (q)   (3.16)

This is a product formula for Mock theta functions of order eight and ten. 

q n (q; q) n

n 0

(q; q 2 ) n 1

S 0 (q) D5 (q)  



q n (q; q 2 ) n

n 0

( q 2 ; q 2 ) n

2

S 0, n (q)  



q n ( n 1) (q; q 2 ) n (q; q) n

n 0

(q 2 ; q 2 ) n (q; q 2 ) n 1

D5 , n ( q )  

.

(3.17) This is a product formula for Mock theta functions of order eight and Hikami‟s Mock theta function of order two. 

q ( n 1)(n  2) / 2 (q; q 2 ) n

n 0

(q; q 2 ) n 1

 LC (q) LC (q)  

 q n ( n 1) / 2 q n ( n  2) 1  LC , n (q)    LC , n (q)   .

2 2 2 n  0 (q; q ) n 1 n  0 (q; q ) n 1 

(3.18) This is a product formula for two Mock theta functions of order ten. 

q n (q; q) n

n 0

(q; q 2 ) n 1

 LC (q) D5 (q)  

 q n ( n 3) / 2 (q; q) n q n ( n 1) / 2 D ( q )  .  5, n 2 (q; q 2 ) 2n 1 n  0 (q; q ) n 1 n 0 

 LC , n (q)  

(3.19)

This is a product formula for Mock theta functions of order ten and Hikami‟s Mock theta function of order two. 

q n ( q 2 ; q 2 ) n

n 0

(q n 1 ; q) n 1

D5 (q) D6 (q)  



q n (q; q) n

n 0

(q; q 2 ) n 1

D5, n (q)  



q 2 n (q; q) n (q 2 ; q 2 )

n 0

(q; q 2 ) n 1 (q n 1 ; q) n 1

D6 , n ( q )  

.

(3.20) This is a product formula for two Hikami‟s Mock theta function of order two and order four.

3.Finally, we establish product formulae among complete Mock theta function of order two and Mock theta function of distinct orders by making use of identity (2.11).

| IJMER | ISSN: 2249–6645

www.ijmer.com

| Vol. 7 | Iss. 5 | May. 2017 | 49 |

.

Some Identities Relating Mock Theta Functions And Hikami Mock Theta Functions 

q n 1 (q 2 ; q 2 ) n

n 0

(q; q 2 ) n 1

AC (q) B(q)  



q n (q; q 2 ) n

n 0

(q; q 2 ) n 1

where AC (q )

is complete Mock theta function of order two and

A(  n, n ) (q)  

B n (q) 

q 2 n 1  2 2 . n   (q; q ) n 1 

(3.21)

B(q) is another Mock theta function of

order two. 

() n q n (q; q 2 ) n

n 0

(q; q 2 ) 2 n

AC (q) (q)  

Where

 (q)



q n 1 (q 2 ; q 2 ) n

n 0

(q; q 2 ) n 1

2

A(  n, n ) (q)  

 n (q) 



() n q n ( n 1) (q; q 2 ) n (q 2 ; q 2 ) n

n  

(q; q) 2 n (q; q 2 ) n 1



(3.22) is Mock theta function of order six. 

q n (q; q 2 ) n

n 0

( q 2 ; q 2 ) n

AC (q) S 0 (q)  



q n 1 (q 2 ; q 2 ) n

n 0

(q; q 2 ) n 1

2

A(  n, n ) (q)  

S 0, n (q) 



q n ( n 1) 1 (q; q 2 ) n

n  

(q; q 2 ) n 1



.

(3.23) Where S 0 (q) is Mock theta function of order eight.   q n 1 (q 2 ; q 2 ) n q ( n 1)(n  2) / 2 (q 2 ; q 2 ) n q n ( n 1) / 2 A ( q )   ( q )    (  n, n ) LC , n 2 (q; q 2 ) n 1 (q; q 2 ) 2n 1 n  0 (q; q ) n 1 n 0 n   

AC (q) LC (q)  

Where

 LC (q)

(3.24) is Mock theta function of order ten. 

q n (q; q) n

n 0

(q; q 2 ) n 1

AC (q) D5 (q)  



q n 1 (q 2 ; q 2 ) n

n 0

(q; q 2 ) n 1

A(  n, n ) (q)  

D5, n (q) 



q n ( n 1) (q; q) n (q 2 ; q 2 ) n

n  

(q; q 2 ) 2n 1



(3.25) Where D5 ( q ) is Hikami‟s Mock theta function of order two.

IV.

PROOF OF MAIN RESULTS

1.0As an illustraction, we shall prove the results (3.1) to (3.5). In order to prove the result (3.1), taking

n 

q n 1 (q 2 ; q 2 ) n

in the identity (2.9) and then after simplification, we obtain the result (3.1).

(q; q 2 ) n 1

Similarly, suitable selections of

n

and making use of respective identity (2.9) one can establish the results

(3.2)-(3.5). Other Mock theta functions of order two, six eight, ten and Hikami's Mock theta functions can be expressed similarly by proper choice of  n in identity (2.9).

2.As

n 

an

illustration,

q n 1 (q; q 2 ) n (q; q 2 ) n 1

properly the values of

we

shall

prove

the

result

(3.6).

n 

Taking

q n 1 (q 2 ; q 2 ) n (q; q 2 ) n 1

and

in identity (2.10) and then after simplification, we get the result (3.6). Choosing

 n and  n and proceeding as above, one can obtain the results (3.7) to (3.20).

3.finally by making use of the identity (2.11) in order to prove the product formulae for Mock theta functions.

As an illustration, we prove the result (3.21).

n 

q n (  q; q 2 ) n

value of

(q; q 2 ) n 1

n

and

n

Taking

n 

q n 1 (q 2 ; q 2 ) n (q; q 2 ) n 1

and

in identity (2.11) and then after simplification, we get the result (3.6).\\By choosing the and proceeding as above, one can obtain the results (3.22) to (3.25).

| IJMER | ISSN: 2249–6645

www.ijmer.com

| Vol. 7 | Iss. 5 | May. 2017 | 50 |

.

Some Identities Relating Mock Theta Functions And Hikami Mock Theta Functions

Choosing

a  y  1 in (3.1), we get

 q; q2 q; q2

Choosing

 1; q2n q n q; q2n

 1; q2n q n  (q)   q; q2n n 0 

a  y  1 in (3.3), we get

 q; q2 q; q2 Choosing

n 0

 1; q2n q n S 0 (q)   q; q2n n 0 



An (q)   n 0

a  y  1 in (3.5), we get

 q; q2 q; q2



D5 ( q )   n 0

 q; q2n q; q2n

q n  2 (q 2 ; q 2 ) n 1 (q; q 2 ) n  2

.

(5.1)

 q; q2n  n (q)   q; q2n n 0

q ( n 1)(n  2) / 2 (q; q) n 1

 q; q2n S 0, n (q)   q; q2n n 0

q ( n 1) (q; q 2 ) n 1





a  y  1 in (3.4), we get

  1; q2n q n  q; q2  ( q )   q; q2 LC q; q2n n 0

Choosing

APPLICATIONS

a  y  1 in (3.2), we get

 q; q2 q; q2 Choosing



A(q)  

V.



 LC , n (q)   n 0

 1; q2n q n q; q2n

 q; q2n q; q2n

(q; q 2 ) n 1

.

(5.2)

2

(q 2 ; q 2 ) n 1

.

q ( n 1)(n  2) / 2 (q; q 2 ) n 1 (q; q 2 ) n  2



q n 1 (q; q) n

n 0

(q; q 2 ) n  2

D5, n (q)   (1  q n 1 )

(5.3)

.

(5.4)

.

(5.5)

REFERENCES [1].

[2].

[3].

[4].

[5].

[6].

[7].

[8].

[9].

[10].

[11].

[12].

Agarwal, R. P., Resonance of Ramanujans Mathematics II. New Age International Pvt. Ltd. Publishers, New Delhi, 1996.

Andrews, G. E., Mordell integrals and Ramanujan's "lost" notebook, Analytic number theory (Philadelphia, Pa., 1980), 10-18, Lecture Notes in Math., 899, Springer, Berlin-New York, 1981.

Andrews, G. E. and Hickerson, D., Ramanujan‟s „Lost‟Notebook VII: “The sixth order Mock theta functions”, Adv. in Math., 89, (1991), 60-105.

Gordon, B. and McIntosh, R.J., Some eighth order Mock theta functions, J. London Math. Soc. 62, (2), (2000), 321335.

Gordon, B. and McIntosh, R.J., Modular transformations of Ramanujan‟s fifth and seventh order mock theta functions, Rankin memorial issues. Ramanujan J. 7, (2003), 193-222.

Choi, Y. S., Tenth order mock theta functions in Ramanujan's lost notebook, Invent. Math. 136, (1999), 497-569.

Eltikali, M., Paul, A. and et al, A new spectrum of Hikami's Mock theta functions, Int. J. of Education and Science Research Review, vol. II, Issue 6, (2015), 32-42.

Hikami, K., Mock (false) theta functions as quantum invariants, Regular and Chaotic Dynamics, 10, ( 2005), 509530.

Hikami, K., Transformation formula of the 2nd order Mock theta functions, Lett. Math. Phys. 75, (2006), 93-98.

Gasper, G. and Rahman, M., Basic Hypergeometric Series, Cambridge University Press, Cambridge, 1990.

Ramanujan, S., The Lost Notebook and other Unpublished papers, Narosa Publishing House, New Delhi, 1988.

Singh, S. P. and Mishra, B. P., On certain involving Mock theta functions, J. of Ramanujan Society of Math. and Math. Sc., vol. 1, no. 1, (2012), 07-16.

| IJMER | ISSN: 2249–6645

www.ijmer.com

| Vol. 7 | Iss. 5 | May. 2017 | 51 |