Generating Functions for Generalized Mock Theta Functions

(1)

Published online April 2, 2013 (http://www.sciencepublishinggroup.com/j/pamj) doi: 10.11648/j.pamj.20130202.13

Generating functions for generalized mock theta

functions

Sameena Saba

Lucknow University, Lucknow, India

Email address:

[email protected] (S. Saba)

To cite this article:

Sameena Saba. Generating Functions for Generalized Mock Theta Functions, Pure and Applied Mathematics Journal. Vol. 2, No. 2, 2013, pp. 62-70. doi: 10.11648/j.pamj.20130202.13

Abstract:

We consider generalized mock theta functions and give generating functions for the partial generalized mock

theta functions.

Keywords:

Generating Function, Mock Theta Function And Hypergeometric Series

1. Introduction

Mock theta functions have been generalized in [6] [7] [9]. We give below the definitions for the generalized third or-der, fifth oror-der, sixth oror-der, eighth order and tenth order mock theta functions.

This paper aims at presenting generating functions for these generalized functions. As a tool we have used simple expansion formula of Srivastava [8]. The generating func-tions can be suitably applied to yield numerous further re-sults involving these generalized functions.

A great surge of activities in the theory of q-series has been witnessed in recent years with this objective in mind we give generating functions for a very important class of special functions- mock theta functions.

2. Definitions and Notations

A generalized basic hypergeometric function with base q

is defined as:

, ,

, , ; ,

; ;

; ; ; 1 ,

For ! 1, the above series is convergent for | | # 1, for $ , the above series is convergent for all and for

% ! 1, the series diverges for all except 0. Also,

, ,

, , ; , _'

; ;

; ; ;

'

1 ,

denotes the partial sum of the generalized basic hyper-geometric series.

Here we have used the standard q-shifted factorials de-fined as

; ( 1 )

)

; ( 1 ) _{, | | # 1.} )

1. Main result

We shall be using the following identity in getting the generating functions for generalized mock theta functions.

+' _, '

'

- , + . + , |+| # 1 1

This identity is easily deduced from the identity of Sri-vastava [8]:

, / 0

/0 ,

0

! /' /' ,

' 0

(2)

Take / + , |+| # 1.

, + 0

+0 _,

0

! 1 + +' _,

' 0

' Let 4 5 ∞,

+' _, '

'

1

1 + , + , + +

3. Definition of Generalized Functions

To distinguish third order mock theta functions from tenth order mock theta functions we put suffix R in the symbol for third order mock theta functions.

The generalized third order mock theta functions [9]:

78 +, ,, /, ; ₉_:∑ 9<=

< >?<@<A_B<_C <

C;=< BC =;=⁄ < ,

（2）

8 +, ,, /, ; ₉_:∑ 9<=

< >E<@<A_C<

BC =⁄ ;= < ,

（3）

F8 +, ,, /, ; ₉_:∑ 9<=< ><@<AC <@G

BC =⁄ ;= <@G ,

（4）

H8 +, ,, /, ; ₉_:∑ 9<=< > <@<AC <

B C =_{⁄ ;=}E _<@G, （5）

I8 +, ,, /, ; ₉_:∑ 9<= < >J<>?@<AB<C? <@G

C =⁄ ;= <@G B C =⁄ ;=E <@G,

（6）

K8 +, /, ; ₉_:∑ 9<=< >E<@<AC <

LC;=< L C;=<

（7）

and

M8 +, /, ; C?

=? ₉_:∑ 9<=

< >E<@<A_C?<

L C =⁄ ;= <@GL> C =⁄ ;= <@G,

（8）

where _{H N}OPE.

For / 1 and we have generalized five third or-der mock theta functions namely 7, , F,H, I of Andrews [2]. For + 0, / 1, , and the generalized functions 7, ,F and Kreduce to the third order mock theta functions of Ramanujan and I,H and M to the third order mock theta functions of Watson [10].

The generalized fifth order mock theta functions [6]:

7 +, ,, ; ₊1 + _;Q B , 9

+, ,, ; 1

+ +

B Q_{⁄ ;}

, 10

F +, ,, ; 1

+ +

⁄ B _{⁄ ;}

, 11

S +, ,, ; ₊1 + _{⁄ ;}T B U , 12

7 +, ,, ; ₊1 + _; B , 13

+, ,, ; T

U ₊1 +

Q B Q_{⁄ ;}

, 14

F +, ,, ; 1

+ +

⁄ B _{⁄ ;}

, 15

S +, ,, ; ₊1 + _{⁄ ;}Q B U , 16

K +, ,, ; ₊1 + B_; ; 17

And

K +, ,, ; ₊1 + B_; ; . 18

For + 0,, 1 and these generalized functions reduce to Ramanujan’s mock theta functions of order five.

The generalized sixth order mock theta functions [9]:

Φ +, ,, ; 1

+ 1 +

Q B _⁄ _;

;

⁄ , 19

Ψ +, ,, ; 1

+ 1 +

B _⁄ _;

;

⁄ , 20

ρ +, ,, ; 1

+ +

Q ⁄ B _;

;

⁄ , 21 σ +, ,, ;

1

2 + +

⁄ B _⁄ _;

;

⁄ , 22

λ +, ,, ; 1

+ + 1

B Q_⁄ _; ;

(3)

µ +, ,, ; 1

+ + 1

B Q_⁄ _;

;

⁄ 24

γ +, ,, ; ₊1 +_{H ;} Q_H_U B_; . 25

For + 0, , 1, we have the generalized functions of Choi.

The generalized eighth order mock theta functions:

d +, ,, ; 1

+ +

B _{⁄ ;}

; , 26

d +, ,, ; 1

+ +

B _{⁄ ;}

; , 27 e +, ,, ;

1

+ +

B _;

⁄ ; , 28

e +, ,, ; 1

+ +

B _;

⁄ ; , 29 f +, ,, ;

1

+ +

Q B _{⁄ ;}

; U , 30

f +, ,, ; 1

+ +

B _{⁄ ;}

; U , 31

g +, ,, ;

1 ! ₊2 + _{⁄ ;}B ; , 32

g +, ,, ; 1

+ +

B _;

⁄ ; . 33

For + 0, , 1, and , these generalized functions reduce to the mock theta functions of Gordon and McIntosh.

The generalized tenth order mock theta functions [6]:

h +, ,, ; ₊1 +

T _B

;

⁄ , 34

F +, ,, ; ₊1 +

Q _B

;

⁄ , 35

Χ +, ,, ; 1

+ 1 +

Q B

;

⁄ , 36 χ +, ,, ;

1

+ 1 +

B ;

⁄ . 37

For + 0, , 1 and these generalized functions reduce to tenth order mock theta functions of Ramanujan.

4. Generating Functions of Generalized

Partial Third Order Mock Theta

Functions

(i) Taking , =k >?k@kABkCk

C;=k BC =⁄ ;=k in (1)

+'₇

8' 0, ,, /, ; '

_; U l_{, ⁄ ;}, + + , 38

or

+'₇

8' 0, ,, /, ; '

, , ⁄ ; , +, l Q + . 39

Special case:

For / 1, , , , in (38) we get the generating function for partial mock theta function of Ramanujan

+'₇ 8' '

+

; ; +

, ; , + + . 40

We list the generating functions for other generalized and partial mock theta functions omitting calculations only giv-ing the value of , in paranthesis.

ii +' _8' _{0, ,, /, ;} '

(4)

r, _{, ⁄ ;}Q l in 1 t

Here _8' is the partial mock theta functions of Ra-manujan.

Special cases:

+' 8'

' o , o ; , +

+ .

/ 1, , , in 41

ii +'_F8' _{0, ,, /, ;}

' 1 , ⁄

u _n, ⁄ _{, ,} ⁄ ; , + lq + . 42

r, _{, ;} l in 1 t

Here F8' is the partial mock theta functions of Ra-manujan.

Special cases:

+'_F8' '

1 n Q⁄ _, Q⁄ ; , + Qq + .

/ 1, , , in 42

( )

(

)

2

2 1

2 2 2

3 1

2 2

2 2 ₂

3 1 m 0

n 0

iv

0, , , ;

1 ; ,

.

,

1 in(1)

;

∞

∞ −

− + =

+

=









φ

_

₋

_





+























₌









₋









_

_







∑

m Rm

β n

r r rβ r r

r

t ν

α β z q

q

q tz q

t

iαz

α z

_q

q

z

α

α z

_q

q

(43)

Here H8' is the partial mock theta functions of Ra-manujan.

Special cases:

∑ +'_H8' '

= no Q⁄ , o Q⁄ ; , + q ∑ + . (43) / 1, , , in

v +'_I8' _{0, ,, /, ;}

' _U

U ₁ _{⁄ 1 ,} _⁄ Q

u n _{, ,} _{⁄ ; , +,} U l Q_q _{+ . 44}

w, _{C =}=_{⁄ ;=}k >Jk@kA>?_k@G_{B C =}B_{⁄ ;=}kCE?k@?_k@G in 1 x

Here I8' is the partial mock theta functions of Ra-manujan.

Special cases:

+'_I8' '

1

1 n Q_, Q; , + Uq + .

/ 1, , , in (44)

vi +'_K8' _{0, /, ;} '

H , H ; , + l + . 45

w, _LC;==k >Ek@kA_k _{L C;=}C k_k in 1 x

Here K8' is the partial mock theta functions of Ra-manujan.

Special cases:

+'_K8'

' nH , H ; ,+ q + .

/ 1, , , in 45

vii +'_M8' _{0, /, ;}

' _U

U _{1 H} _{⁄ 1 H} _⁄

u _{nH ,H} ; , + U l _q _{+ . 46}

r, _H _{⁄ ;} Q _Hl U_{⁄ ;} in 1 t

Here M8' is the partial mock theta functions of Ra-manujan.

Special cases:

+'_M8' '

1

(5)

u _nH _Q_{, H} _Q; , + U_q ₊

/ 1, , , in 46

5. Generating Functions of Generalized

Partial Fifth Order Mock Theta

Functions

i +'₇_' _{0, ,, ;} '

, 0; , + B + . 47

w, =k >Ek@ky_C;=_kCk in 1 x

Here 7_' is the partial mock theta functions of Ra-manujan.

Special cases:

+'₇ '

' , 0; , +

+ .

/ 1, , , in 47

ii +'

' 0, ,, ; '

n , _0,0Q⁄ ; , + B _q _{+ . 48}

r, =k @k@kyz =E⁄ ;= {C k

Ck in 1 t

Here _' is the partial mock theta functions of Ra-manujan.

Special cases:

+' '

' n ,0,0 ; , + q + .

/ 1, , , in 48

iii +'_F _' _{0, ,, ;} '

B

n , _{0,0 ; , +}⁄ B _q _{+ . 49}

r, =k @Ek⁄ @kyz = C⁄ ;={k

Ck in 1 t

Here F_' is the partial mock theta functions of Ra-manujan.

Special cases:

+'_F ' '

,

0,0 ; , + + .

/ 1, , , in 49

iv +'_S_' _{0, ,, ;} '

n ⁄ ,0; ,+ U B Q_q _{+ . 50}

w, = k >Jk@ky_{C =}_{⁄ ;=} C_k?k in 1 x

Here S_' is the partial mock theta functions of Ra-manujan.

Special cases:

+'_S '

' n ,0; ,+ q + .

/ 1, , , in 50

v +'₇

' 0, ,, ; '

, 0; , + B + . 51

w, =k > k@ky_C;=_kCk in 1 x

Here 7_' is the partial mock theta functions of Ra-manujan.

Special cases:

+'₇_'

' , 0; , +

+ .

/ 1, , , in 51

vi +'

' 0, ,, ; '

T

U n ,

Q_⁄

0,0 ; , + B Uq + . 52

r, =k @Ek@kyz =E⁄ ;= {C k

Ck in 1 t

Here _' is the partial mock theta functions of Ra-manujan.

Special cases:

+' ' '

(6)

/ 1, , , in 52

vii +'_F

' 0, ,, ; '

n , _{0,0 ; , +}⁄ B _q _{+ . 53}

r, =k @k ⁄ @kyz = C⁄ ;={k

Ck in 1 t

Here F_' is the partial mock theta functions of Ra-manujan.

Special cases:

+'_F_' '

,

0,0 ; , + + .

/ 1, , , in 53

vii +'_S

' 0, ,, ; '

1

1 ⁄ n , 0; , + U B q + . 54

w, =_{C =}k >Ek@ky_{⁄ ;=} _k@GC?k in 1 x

Here S_' is the partial mock theta functions of Ra-manujan.

Special cases:

+'_S ' '

1

1 n Q_{, 0; , +} Uq + .

/ 1, , , in 54

6. Generating Functions of Generalized

Partial Sixth Order Mock Theta

Functions

i +'_Φ' _{0, ,, ;} '

n , ⁄_{⁄ ,} ; , + B _q _{+ . 55}

r, k=k >Ek@kyC kzC =⁄ ;= {k

C =⁄ ;= k in 1 t

Here Φ' is the partial mock theta functions of Ra-manujan.

Special cases:

+'_Φ' '

n _,, ; , + q + .

/ 1, , , in 55

ii +'_Ψ' _{0, ,, ;} '

1 ! ⁄ n , ⁄, ; , + Bq + . 56

r, k=k >k@kyC k@GzC =⁄ ;= {k

C =⁄ ;= k@G in 1 t

Here Ψ' is the partial mock theta functions of Rama-nujan.

Special cases:

+'_Ψ '

' 1 !

n _,, _Q; , + Q_q _{+ .}

/ 1, , , in 56

iii +'_ρ' _{0, ,, ;} '

1

1 ⁄

u n _{p , p ; , +}, B _q _{+ . 57}

|, =k k>E ⁄ @kyCk C;=k

C =⁄ ;= k@G in 1 }

Here ρ' is the partial mock theta functions of Ra-manujan.

Special cases:

+'_ρ' '

1

1 n

,

Q⁄ _, Q⁄ ; , + q + .

/ 1, , , in 57

iv +'_γ' _{0, ,, ;} '

,

H , HU ; , + B + . 58

w, _{L C;=}=k >Ek@ky_k_L?_C;=Ck_k in 1 x

Here γ' is the partial mock theta functions of Ra-manujan.

Special cases:

+'_γ' '

(7)

/ 1, , , in 58

v +'_σ' _{0, ,, ;} '

1 ! ⁄

2 1 ⁄ n

,

p , p ; , + Bq + . 59

w, =zk >k{⁄ @kyCk@G C =⁄ ;=k@G

C =⁄ ;= k@G in 1 x

Here σ' is the partial mock theta functions of Ra-manujan.

Special cases:

+'_σ' '

1 n

,

Q⁄ _, Q⁄ ; , + q + .

/ 1, , , in 59

7. Generating Functions of Generalized

Partial Eighth Order Mock Theta

Functions

i +'_S

' 0, ,, ; '

n , _{, 0 ; , +}⁄ B _q _{+ . 60}

r, =k > k@kyCkz C =⁄ ;= {k

C ;= k in 1 t

Here S _' is the partial mock theta functions of Gor-don and McIntosh.

Special cases:

+'_S ' '

_{n , ,0; , + q + .}

/ 1, , , in 60

ii +'_S _' _{0, ,, ;} '

n , _{, 0 ; , +}⁄ B _q _{+ . 61}

r, =k @kyCkz C =⁄ ;= {k

C ;= k in 1 t

Here S _' is the partial mock theta functions of Gor-don and McIntosh.

Special cases:

+'_S '

' n , ,0; , +

Q_q _{+ .}

/ 1, , , in 61

iii +'_T_' _{0, ,, ;} '

1 ! ⁄ n , ,0; , + B q + . 62

r, =k @k@kyCk@z C ;= {k

C =⁄ ;= k@G in 1 t

Here T_' is the partial mock theta functions of Gor-don and McIntosh.

Special cases:

+'_T_'

' 1 !

n ,Q_{, 0 ; , +} Uq + .

/ 1, , , in 62

iv +'_T

' 0, ,, ; '

1

1 ! ⁄ n , ,0; , + Bq + . 63

r, =k >k@kyCkz C ;= {k

C =⁄ ;= k@G in 1 t

Here T_' is the partial mock theta functions of Gor-don and McIntosh.

Special cases:

+'_T ' '

1

1 ! n ,Q_{, 0 ; , + q} + .

/ 1, , , in 63

v +'_U _' _{0, ,, ;} '

n ,_{o , o ; , +}⁄ B _q _{+ . 64}

r, =k >Ek@kyCkz C =⁄ ;= {k

C = ;=? k in 1 t

Here U _' is the partial mock theta functions of Gor-don and McIntosh.

Special cases:

+'_U ' '

(8)

/ 1, , , in 64

vi +'_U

' 0, ,, ; '

1 ! n ,o , o⁄ ; , + Bq + . 65

r, =k >k@kyCk@Gz C =⁄ ;= {k

C ;=? k@G in 1 t

Here U_' is the partial mock theta functions of Gor-don and McIntosh.

Special cases:

+'_U_' '

1 ! no Q_{, o}, Q; , + Qq + .

/ 1, , , in 65

vii +'_V_' _{0, ,, ;} '

1 ! 2 _{n ,⁄ ,0; , +} B _q _{+ . 66}

r, =k > k@kyCkz C;= {k

C =⁄ ;= k in 1 t

Here V_' is the partial mock theta functions of Gor-don and McIntosh.

Special cases:

+'_V ' '

1 ! 2 _{n ,,0 ; , + q + .}

/ 1, , , in 66

viii +'_V

' 0, ,, ; '

1 ⁄ n , ,0; , + B q + . 67

r, =k @kyCk@Gz C;= {k

C =⁄ ;= k@G in 1 t

Here V_' is the partial mock theta functions of Gor-don and McIntosh.

Special cases:

+'_V_'

' 1

n ,Q_{, 0 ; , +} Qq + .

/ 1, , , in 67

8. Generating Functions of Generalized

Partial Tenth Order Mock Theta

Functions

i +'_φ' _{0, ,, ;} '

1

1 ⁄ n ⁄ _,, 0 ⁄ ; , + B q + . 68

w, =zk >Jk{⁄ @ky_{C =}_{⁄ ;=} _k@GCk in 1 x

Here φ' is the partial mock theta functions of Ra-manujan.

Special cases:

+'_φ' '

1

1 n Q⁄ _,, 0 Q⁄ ; , + q + .

/ 1, , , in 68

ii +'_F' _{0, ,, ;} '

1 ⁄ n ⁄ _,, 0 ⁄ ; , + B q + . 69

|, =k k>E ⁄ @ky_{C =}_{⁄ ;=} _k@GCk@G in 1 }

Here F' is the partial mock theta functions of Ra-manujan.

Special cases:

+'_F' '

1 n Q⁄ _,, 0 Q⁄ ; , + q + .

/ 1, , , in 69

iii +'_Χ' _{0, ,, ;} '

n _{⁄ ,}, 0 ; , + B _q _{+ . 70}

(9)

Here Χ' is the partial mock theta functions of Ra-manujan.

Special cases:

+'_Χ' '

n _,, 0 ; , + q + .

/ 1, , , in 70

iv +'_χ' _{0, ,, ;} '

1 ! ⁄ n ,, 0 ; , + Bq + . 71

w, k=_{C =}k >k@ky_{⁄ ;=} _k@GC k@G in 1 x

Here χ' is the partial mock theta functions of Ra-manujan.

Special cases:

+'_χ' '

1 ! n ,, 0 Q; , + Qq + . / 1, , , in 71

Acknowledgement

I am thankful to Dr. Bhaskar Srivastava for his help and guidance.

References

[1] R. P. Agarwal, Certain basic hypergeometric identities asso-ciated with mock theta functions, Quart. J. Math. (Oxford)

20 (1968), 121-128.

[2] G. E. Andrews, On basic hypergeometric series, mock theta functions and partitions (I), Quart. J. Math. (Oxford) (2) 17

(1966), 64-80.

[3] R. P. Agarwal, Certain basic hypergeometric identities asso-ciated with mock theta functions, Quart. J. Math. (Oxford) 20 (1968), 121-128.

[4] G. E. Andrews, On basic hypergeometric series, mock theta functions and partitions (I), Quart. J. Math. (Oxford) (2) 17 (1966), 64-80.

[5] S. Ramanujan, Collected Papers, Cambridge University Press, 1972, reprinted Chelsea, New York, 1962.

[6] T. M. Rassias, S.N. Singh and H.M. Srivastava, Some q-generating functions associated with basic multiple hyper-geometric series, Comp. and Math. with app. 27(1) (1994), 33-39.

[7] A. K. Srivastava, On partial sum of mock theta functions of order three, Proc. India Acad. Sci. 107 (1997), 1-12. [8] B. Srivastava, Ramanujan’s fifth order and tenth order mock

theta functions- A generalization (Communicated).

[9] B. Srivastava, On a generalization of Ramanujan’s seventh order mock theta functions (Accepted).

[10] B. Srivastava, Ramanujan’s mock theta functions, Math. J. Okayama Univ. 47 (2005), 163-174.

[11] S. Saba, A study of a generalization of Ramanujan’s sixth order and third order mock theta functions, Appl. Math. 2(5), (2012), 157-165.