MockThetaConjectures Das,SabujandMohajan,Haradhan MunichPersonalRePEcArchive

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Munich Personal RePEc Archive

Mock Theta Conjectures

Das, Sabuj and Mohajan, Haradhan

Raozan University College, Bangladesh., Premier University, Chittagong, Bangladesh

10 January 2014

Online at https://mpra.ub.uni-muenchen.de/55070/

MPRA Paper No. 55070, posted 05 Apr 2014 19:36 UTC

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22 Mock Theta Conjectures

Sabuj Das¹, Haradhan Kumar Mohajan²

1- Senior Lecturer, Department of Mathematics, Raozan University College, Bangladesh.

2- Premier University, Chittagong, Bangladesh.

Received: 02/24/2014 Accepted: 04/04/2014 Published: 04/05/2014

Abstract

This paper shows how to prove the two Theorems first and second mock theta conjectures respectively.

Keywords: Mock theta, rank of partition.

1. Introduction

¹

We give the definitions of

π

, rank of partition, ^N

( )

^m^,ⁿ ^,

(

^m^t ⁿ

)

N , , ,

ρ

0

( ) ⁿ

^,

ρ

1

( ) ⁿ

^{, z,}

( )

x∞,

( )

zx∞,

( )

m

x

n ,

( )

m k x

x ; ⁵ which are collected from Partitions Yesterday and Today [4], Generalizations of Dyson’s Rank [3], Ramanujan’s Lost Notebook [2]. We generate the generating functions for

( )

ⁿ

ρ

0 ^{, and}

ρ

1

( ) ⁿ

[2] and prove the two Theorems first and second mock theta conjectures respectively. Finally we give two numerical examples which are related to first and second mock theta conjectures respectively when n = 1.

2. Definitions π

: A partition.

Rank of partition: The largest part of a partition

π

^minus

the number of parts of

π

^.

( ) ^m ⁿ

N ,

: The number of partitions of n with rank m.

( ^m ^t ⁿ )

N , ,

: The number of partition of n with rank congruent to m modulo t.

( ) ⁿ

ρ

0 : The number of partitions of n with unique smallest part and all other parts ≤ the double of the smallest part.

( ) ⁿ

ρ

1 : The number of partitions of n with unique smallest part and all other parts ≤ one plus the double of the smallest part.

z : The set of complex numbers.

Corresponding author: Haradhan Kumar Mohajan, Premier University, Chittagong, Bangladesh, Email:

haradhan_km@yahoo.com.

( ) x

∞: The product of infinite factors is defined as follows:

( ) ( x

∞

⁼ 1 ⁻ x ) ( 1 ⁻ x

²

)( 1 ⁻ x

³

) ... ^∞

.

( ) zx

∞: The product of infinite factors is defined as follows:

( ) ( ^zx

^∞

⁼ ¹ ⁻ ^zx ) ( ¹ ⁻ ^zx

²

)( ¹ ⁻ ^zx

³

) ^... ^∞

^.

( ) ^x

ⁿ ^m: The product of m factors is defined as follows:

( ) (

m

⁼ ¹ ⁻

ⁿ

)( ¹ ⁻

ⁿ⁺¹

)( ¹ ⁻

ⁿ⁺²

) ( ^... ¹ ⁻

ⁿ⁺^m⁻¹

)

n

x x x x

x

.

( ^x

^k

^; ^x

⁵

)

^m: The product of m factors is defined as follows:

( ^x

^k

^; ^x

⁵

) (

^m

⁼ ¹ ⁻ ^x

^k

)( ¹ ⁻ ^x

^k⁺⁵

)( ¹ ⁻ ^x

^k⁺¹⁰

) ^... ( ¹ ⁻ ^x

^k⁺⁽^m⁻¹⁾⁵

)

^.

3. Mock Theta Functions (2)

We quote the relations below [1, 2]:

∞ +

− +

−

∞

−

= −

...

5 ) cos2 2 1 5 )(

cos2 2 1 (

)...

1 )(

1 ) (

(

4 2

2

3 2

n x x n x

x

x x x x

F π π ^.

+ +

− +

=

2 /

5 cos2 2 1 1 ) (

n x x x x

f

π

, ...

5 ) cos2 2 1 5 )(

cos2 2 1

( ² ² ⁴

4

∞ + +

− +

− n x

x n x

x

π π

2 or

= 1

n

.

Journal web link: http://www.jett.dormaj.com

Enviro. Treat. Tech.

ISSN: 2309-1185

(3)

23

− +

−

= ( )

5 cos4 2 ) 5 ( cos2 4 ) ( )

( ⁵

2 5

1 5

1

x n C x x n B x x A x

F π π

) 5 ( cos 2 2

⁵

3

x n D

x π

. (1)

− +

+





 − Φ

′ = ( )

5 cos2 2 ) ( ) 5 ( sin 2 4 ) ( )

( ⁵

2 5

1 5 2

1

x n C x x B x n x

x A x

f π π







 +

x x x n

n D

x ( )

5 . sin 2 4 ) 5 ( cos2

2 ⁵ ²

3 π π ψ . (2)

( ) ⁼ ( ¹ ⁻ ¹ ⁻ )

²

( ¹ ⁻ ⁻

⁴

) ( ⁺

²

¹ ⁻ ⁺ ^...

⁶

) ^∞

²

^... ^∞

9 3 2

x x x

x x x x

A

( ) ( )( )( )

( ) ⁻ ⁻ ( ⁻ ⁻ )( ⁻ ⁻ ) ^∞ ^∞

= 1 1 1 ...

...

1 1 1

6 4

15 10 5

x x x

x x x x

B

( ) ( )( )( )

( ⁻ ⁻ )( ⁻ ⁻ )( ⁻ ⁻ ) ^∞ ^∞

= 1 1 1 ...

...

1 1 1

7 3 2

15 10 5

x x x

x x x x

C

( )

⁼

(

¹⁻¹²⁻

) (

²¹⁻⁻ ³⁺

) (

²¹⁻⁺^...⁷^∞

)

²^...^∞

7 4

x x x

x x x x

D

( ) ⁼ ⁻ ⁺

^^^

⁻ ⁺ ( ⁻ ) ( ⁻

⁴

)( ⁻

⁶

) ⁺

5

1 1 1 1 1 1

x x x

φ

( ¹ ⁻ ) ( ¹ ⁻

⁴

)( ¹ ⁻

⁶

)( ¹ ⁻

⁹

)( ¹ ⁻

¹¹

) ⁺ ^... ^∞

^^^

20

x x x x x

x

,

But we get;

− +

− ( )

5 cos 4 2 ) 5 ( cos 2 4 )

( x

⁵

x B x

⁵

x

²

C x

⁵

A π π

) 5 ( cos 2

2 x

³

π D x

⁵

− +

−

=

² ² ³

2

⁵

5 cos 2 5 2

cos 4 5 2

cos 2 4

1 x π x π x π x

∞ +

−

+ ...

5 cos 2 5 2

cos 2

4 x

⁶ ²

π x

⁸

π x

¹⁰

( ) ⁼ ⁻ ⁺

^^^

⁻ ⁺ ( ⁻ )( ⁻ )( ⁻ ) ⁺

Ψ

₂ ₂ ⁵₃ ₇

1 1 1 1

1 1

x x x

( ¹ ⁻

²

)( ¹ ⁻

³

)( ¹ ⁻

⁷

)( ¹ ⁻

⁸

)( ¹ ⁻

¹²

) ⁺ ^... ^∞

^^^

20

x x x x x

x

.

Now,

(

⁻

)(

⁻

) (

⁺ ⁻

)(

⁻

)(

⁻

)

⁺ ^∞

− + ...

1 1 1 1

1

1 ³ ⁴ ⁵

5 3

2 3

x x x

x x

x x x

x

( ) ^x + − ^A ( ) ^x

= 3 φ 1

^.

And,

( ⁻ )( ⁻ ) ( ⁺ ⁻ )( ⁻ )( ⁻ ) ⁺ ^∞

− + ...

1 1 1 1

1

³ ⁴ ⁵

3 3

2 2

x x x

x x

x x x

x

( ) ^x + ^xD ( ) ^x

Ψ

= 3

.

We assume without loss of generality that n = 1. Let

5 2

exp

πi

ζ

= , then we may write the definitions of F(x) and ƒ′(x) as;

( ) ( )

− ∞

∞

= ∞

x x x x

F ₁

ζ ζ

and

( ) ^∑ ( ) ( ) ( )( )

∞

= − − − − − −

+

′ =

1

1 ...1 1

1 1 1

2

n

n n

n

x x

f

ζ ζ ζ ζ

( ) ( )

∑

^∞

= −

=

1

1 2

n n n

n

x x

x

ζ

^,

where we have used the relations;

( ) a

0

= ¹

,

( ) ( )( ^a

ⁿ

⁼ ¹ ⁻ â ¹ ⁻ âx ) ^... ( ¹ ⁻ âx

ⁿ⁻¹

)

^{, for}

ⁿ ^≥ ¹

and

( ) ( ) (

¹

)

1

⁻

∞

∞ =

∞

=

→

= ∏ −

ⁿ

n n

n

a ax

Lim

a

^.

After replacing x by x⁵ we see that (1) and (2) are identities for F(x) and ƒ′(x). We note that the numerators in the definitions of A(x) and D(x) are theta series in x and hence may be written as infinite products using Jecobi’s triple product identity;

(

^zxⁿ

) (

^z ^xⁿ

) ⁽

^xⁿ

⁾

n∞ − − − − −

∏= 1 1 1 1 1

1

( ) ¹

⁽²⁺¹⁾

∞

−∞

=

−

∏

=

ⁿ ⁿ ⁿ ⁿ

n

x

z

(3)

(4)

24 ∞

− +

= ... z

⁻²

x z

⁻¹

1 zx z

²

x

³

...

. where z ≠ 0 and x< 1.

Replacing x by

x

⁵ and z by

x

⁻³ we get from (3);

(

^x ⁿ

)(

^x ⁿ

)(

^x ⁿ

)

n

1 5 2 1 5

3 1 5

1 − − − − −

∞

∏=

∞

− +

− + +

= ... x

¹¹

1 x

²

x

⁹

...

∞

− + +

−

= 1 x

²

x

³

x

⁹

x

¹¹

...

^.

Again replacing x by x⁵ and z by

x

⁻³ (3) becomes;

(

ⁿ

)(

ⁿ

)(

ⁿ

)

n

x x

x

⁵ ⁴ ⁵ ¹ ⁵

1

1 1

1 − − −

∏

^∞ ⁻ ⁻

=

∞

− +

= ... x

¹³

x

⁴

1 x x

⁷

...

∞

− + +

−

= 1 x x

⁴

x

⁷

x

¹³

...

In fact we have;

( ) ( )( )( )

(

⁵ ⁴

) (

² ⁵ ¹

)

²

5 2 5 3 5

1

1 1

1

−

∞ −

=

− −

−

∏ −

=

n n

n

x x

A

^,

( ) ( )

(

⁵ ⁴

)(

⁵ ¹

)

5

1

1 1

1

−

∞

=

− −

∏ −

=

_n ⁿ _n

n

x x

B

,

( ) ( )

(

⁵ ³

)(

⁵ ²

)

5

1

1 1

1

−

∞

=

− −

∏ −

=

_n ⁿ _n

n

x x

C

^,

( ) ( )( )( )

(

⁵ ³

)(

⁵ ²

)

5 1 5 4 5

1 1 1

1 1

1

−

∞ −

= − −

−

∏ −

= ⁿ _n ⁿ _n ⁿ

n x x

x x

D ^.

3.1 Rank of a Partition

The rank of a partition is defined as the largest part minus the number of parts. Thus the partition 6 + 5 + 2 + 1 + 1 + 1 + 1 of 17 has rank, 6–7 = –1 and the conjugated partition, 7 + 3 + 2 + 2 + 2 + 1 has rank, 7–6 = 1. i.e., the rank of a partition and that of the conjugate partition differ only in sign. The rank of a partition of 5 belongs to any one of the residues (mod 5) and we have exactly 5 residues. There is similar result for all partitions of 7 leading to (mod 7).

The generating function for the rank is of the form [3];

( )

⁽ ⁾

( ) ( )

∑

^∞

=

∞ −

= +

− −

− ∏ −

−

1

1 1

1 23

1

1 1

1

n

j j n n m n n

n

x x x

( )

⁽ ⁾ ⁽ ⁾

( )

∑

^∞

∑

=

∞

= + +

− +









−

=

1 0

1 2 23 1 2 23

1

n k

m k n n m n n

n

x x P k x

( ⁺ ⁺ ⁺ ^∞ ) ( ⁻ ⁺ ⁺ ^∞ )

= x

^m⁺¹

0 . x

^m⁺²

x

^m⁺³

... x

²^m⁺⁵

x

²^m⁺⁶

...

( )

ⁿ

n

x n m N ,

0

∑

^∞

=

^.

The generating function for

^N ( ^m ^, ^t ^, ⁿ )

is of the form;

( )

⁽ ⁾

(

⁽ ⁾

) ( )

∑

^∞

≠−∞

=

∞ −

= + −

−

− ∏

− +

1

1 1

1

23

1 1 1

n n

j tn j

m t n n mn

n

x

x x x x

( )

⁽ ⁾

(

⁽ ⁾

)

∑

^∞

≠−∞

=

+ −

× +

−

=

1

1 23

1

n n

m t n n mn

n

x x x

( ) ∑

^∞

( )

=

∞ + + +

0

2

...

1

k

k tn

tn

x P k x

x

( )

ⁿ

n

x n t m N , ,

0

∑

^∞

=

^;

which shows that all the coefficients of

x

⁻ⁿ (where n is any positive integer) are zero.

Now we define the generating function;

( ) ^d

r

_a for

^N ( ^a ^, ^t ^, ^tn ⁺ ^d )

where

( ) ( ) ( )

ⁿ

a n

a

d r d t N a t tn d x

r , , ,

0

+

∏

=

^∞

= , and

( ) ^d ^r ( ) ^d ^t ^r ( ) ^d ^r ( ) ^d

r

_a_,_b

=

_a_,_b

, =

_a

−

_b .

( ) ( )

{ }

ⁿ

n

N a , t , tn d N b , t , tn d x

0

+ − +

∏

=

^∞

= .

The generating function

φ ( ) ^x

is of the form;

( ) ⁼ ⁻ ⁺

^^^

⁻ ⁺ ( ⁻ ) ( ⁻

⁴

)( ⁻

⁶

) ⁺

5

1 1 1 1 1 1

x x x

φ

( ) ( )( )( )( )



∞

− +

−

− ...

1 1 1 1

1 ⁴ ⁶ ⁹ ¹¹

20

x x x x x

x _,

( ¹ ^... ) ( ¹ ^... )

1 + + +

²

+ ∞ +

⁵

+ +

²

+ ∞

−

= x x x x x

(5)

25 ( ¹ ⁺ ^x

⁴

⁺ ^... ^∞ )( ¹ ⁺ ^x

⁶

⁺ ^... ^∞ ) ⁺ ^... ^∞

∞ + + + + + + + +

= x x

²

x

³

x

⁴

2 x

⁵

2 x

⁶

2 x

⁷

2 x

⁸

...

( ) ( )

{ }

ⁿ

n

x n N n

N 1,5,5 2,5,5

0

∑

^∞

=

−

=

( ) ⁰

2 ,

r

1

=

^.

The generating function A(x) is defined as;

( ) ⁼ ( ¹ ⁻ ¹ ⁻ )

²

( ¹ ⁻ ⁻

⁴

) ( ⁺

²

¹ ⁻ ⁺ ^...

⁶

) ^∞

²

^... ^∞

9 3 2

x x

x

x x x x

A

( ¹ ⁻

²

⁻

³

⁺

⁹

⁺ ^... ^∞ )( ¹ ⁺ ² ⁺ ³

²

⁺ ^... ^∞ )

= x x x x x

( ¹ ⁺ ² ^x

⁴

⁺ ³ ^x

⁸

⁺ ^... ^∞ ) ^... ^∞

∞ + + + + +

= 1 2 x 2 x

²

x

³

2 x

⁴

...

( ) ( )

{

∑

^∞

=

+

− +

=

0

5 , 5 , 2 5

, 5 , 0 1

n

n N n N

( ¹ ^, ⁵ ^, ⁵ ⁿ ) ² ^N ( ² ^, ⁵ ^, ⁵ ⁿ )} ^x

²

N −

( ) ( )

{ ⁻ } ⁺

+

= ∑

^∞

=

n n

x n N

n N

0

5 , 5 , 2 5

, 5 , 0 1

( ) ( )

{ }

ⁿ

n

x n N n

N 1 , 5 , 5 2 , 5 , 5 2

0

∑

^∞

=

−

( ) ⁰ ² ( ) ⁰

1 + r

₀_,₂

+ r

₁_,₂

=

^.

The generating function is of the form;

(

⁵ ⁴

)(

⁵ ¹

)

5

1

1 1

1

−

∞

=

− −

∏

_n

−

ⁿ _n

n

x x

x

( ⁻ )( ⁺ ⁺ ^∞ )( ⁺ ⁺ ^∞ )

∏

=

^∞ ⁻ ⁻

=

1

⁵

1

⁵ ⁴

... 1

⁵ ¹

...

1

n n

x x

x

( ) ( ⁻ ⁺ ⁻ ) ( ⁺ ⁻ ) ⁺ ⁺ ⁺ ^∞

= 1 0 3 2 x 12 11 x

²

x

³

2 x

⁴

...

( ) ( )

{ }

ⁿ

n

x n N n

∑

^∞

N

=

+

− +

=

0

1 5 , 5 , 2 1 5 , 5 , 0

( ) ¹

2 ,

r

0

=

^.

The generating function is of the form;

(

⁵ ³

)(

⁵ ²

)

5

1

1 1

1

−

∞

=

− −

∏

_n

−

ⁿ _n

n

x x

x

( ¹

⁵

)( ¹

⁵ ³ ¹⁰ ⁶

^... )

1

− + + + ∞

∏

=

^∞ ⁻ ⁻

=

n n

x x x

( ) ( − + − ) ( + − ) + ∞

= 1 0 3 3 x 16 15 x

²

...

( ) ( )

{ + − + } ×

= ∑

^∞

=

n n

x n N n

N 0 , 5 , 5 2 2 , 5 , 5 2

0

( ¹ ⁺ ^x

⁵ⁿ⁻²

⁺ ^x

¹⁰ⁿ⁻⁴

⁺ ^... ^∞ )

( ) ²

2 ,

r

1

=

^.

The generating function Ψ(x) is of the form;

( ) ⁼ ⁻ ⁺

^^^

⁻ ⁺ ( ⁻ )( ⁻ )( ⁻ ) ⁺

Ψ

₂ ₂ ⁵₃ ₇

1 1 1 1

1 1

x x x

( ¹ ⁻

²

)( ¹ ⁻

³

)( ¹ ⁻

⁷

)( ¹ ⁻

⁸

)( ¹ ⁻

¹²

) ⁺ ^... ^∞

^^^

20

x x x x x

x

( ⁺ ⁺ ⁺ ^∞ ) ( ⁺ ⁺ ⁺ ^∞ ) ^×

+

−

= 1 1 x

²

x

⁴

... x

⁵

1 x

²

...

( ¹ ⁺ ^x

³

⁺ ^x

⁶

⁺ ^... ^∞ )( ¹ ⁺ ^x

⁷

⁺ ^... ^∞ ) ⁺ ^... ^∞

∞ + + + + + + +

= x

²

x

⁴

x

⁶

x

⁷

2 x

⁸

x

⁹

2 x

¹⁰

...

. Hence

( ) ₌ ₊ ₊ ₊ ₊ ₊ ₊ ₊ ₊ _∞

Ψ x x

³

x

⁴

x

⁵

x

⁶

2 x

⁷

x

⁸

2 x

⁹

...

x x

( ) ( )

{ }

ⁿ

n

x n N n

N 2 , 5 , 5 3 0 , 5 , 5 3

0

∑

^∞

=

+

− +

=

( ) ³

0 ,

r

2

=

and

(6)

26 ( ) ( )

x r

₀_,₂

3 = − Ψ x

^.

The generating function D(x) is of the form;

( ) ⁼ ( ¹ ⁻ ¹

²

⁻ ) (

²

¹ ⁻ ⁻

³

⁺ ) (

²

¹ ⁻ ⁺ ^...

⁷

^∞ )

²

^... ^∞

7 4

x x

x

x x x x

D

( ⁻ ⁻ ⁺ ⁺ ^∞ )( ⁺ ⁺ ⁺ ^∞ )

= 1 x x

⁴

x

⁷

... 1 2 x

²

3 x

⁴

...

( ¹ ⁺ ² ^x

³

⁺ ^... ^∞ )( ¹ ⁺ ² ^x

⁷

⁺ ^... ^∞ ) ^... ^∞

∞ + + +

−

= 1 x 2 x

²

0 . x

³

...

( ) ( )

{

∑

^∞

=

+ +

− +

=

0

3 5 , 5 , 1 3 5 , 5 , 0

n

n N n

N

( ⁿ ) ( ^N ⁿ )} ^x

ⁿ

N 0 , 5 , 5 + 3 − 2 , 5 , 5 + 3

( ) ( )

{ ⁺ ⁻ ⁺ } ⁺

= ∑

^∞

=0

3 5 , 5 , 1 3

5 , 5 , 0

n

x

n

n N n

N

( ) ( )

{ }

∑

^∞

=

+

− +

0

3 5 , 5 , 2 3 5 , 5 , 0

n

x

n

n N n

N

( ) ³

0,2

( ) ³

1 ,

0

r

r +

=

^.

3.2 Mock Theta Conjectures

ρ

0

( ) ⁿ

is of the form [5];

∑

^∞

( )

= + +

+

0 1

1 1 2

n n

x x

(

⁻

)(

⁻

) (

⁺ ⁻

)(

⁻

)(

⁻

)

⁺ ^∞

− +

= ...

1 1 1 1

1

1 ³ ⁴ ⁵

5 3

2 3

x x x

x x

x x x

x

( ⁺ ⁺ ⁺ ^∞ ) ( ⁺ ⁺ ⁺ ⁺ ^∞ ) ^×

= x 1 x x

²

... x

³

1 x

²

x

⁴

...

( ¹ ⁺ ^x

³

⁺ ^x

⁶

⁺ ^... ^∞ ) ⁺ ^... ^∞

∞ + + + + + +

= x x

²

2 x

³

x

⁴

3 x

⁵

2 x

⁷

...

( )

ⁿ

n

x

∑

^∞

n

=

0

ρ

0 ^, ⁽⁴⁾

which is convenient to define

ρ

0

( ) ⁰ = ⁰

Now we prove the Theorem, which is known as First Mock Theta Conjecture.

Theorem 1:

^N ( 1 , 5 , 5 ⁿ ) ( = ^N 0 , 5 , 5 ⁿ ) + ρ

0

( ) ⁿ

^{, where}

( ) ⁿ

ρ

0 is the number of partitions of n with unique smallest part and all other parts ≤ the double of the smallest part.

Proof: From (4) we have;

( ⁻ )( ) ( )( ⁻ ⁺ ⁻ ⁻ )( ⁻ ) ⁺ ^∞

− + ...

1 1 1 1

1

³ ⁴ ⁵

5 3

2 3

x x x

x x

x x x

x

( )

ⁿ

n

x n

0

∑

^∞ 0

=

= ρ

∞ + + + + + +

⇒ x x

²

2 x

³

x

⁴

3 x

⁵

2 x

⁶

...

( )

ⁿ

n

x n

0

∑

^∞ 0

=

= ρ

( ) ^x + − ^A ( ) ^x

⇒ 3 φ 1 ( )

ⁿ

n

x n

0

∑

^∞ 0

=

= ρ

(by above)

( )

⁰ ¹

(

¹

( )

⁰

( )

⁰

)

3r₁_,₂ + − +r₀_,₂ +r₁_,₂

( )

ⁿ

n

x n

0

∑

^∞ 0

=

ρ

(by above)

( ) ⁰

0,2

( ) ⁰

2 ,

1

r

r −

⇒ ( )

ⁿ

n

x

∑

^∞

n

=

0

ρ

0 (by above)

( ) ( )

{

∑

^∞

=

+

−

⇒

0

5 , 5 , 2 5

, 5 , 1

n

n N n N

( ⁿ ) ( ^N ⁿ )} ^x

ⁿ

N 0 , 5 , 5 − 2 , 5 , 5 ( )

ⁿ

n

x

∑

^∞

n

=

0

ρ

0

( ) ( )

{ }

ⁿ

n

x n N n

N 1 , 5 , 5 0 , 5 , 5

0

∑

^∞

=

−

⇒ ( )

ⁿ

n

x n

0

∑

^∞ 0

=

= ρ

^.

Equating the coefficient of

x

ⁿ on both sides, we get

( ⁿ ) ^N ( ⁿ ) ( ) ⁿ

N 1 , 5 , 5 = 0 , 5 , 5 + ρ

₀ . Hence the Theorem.

ρ

1

( ) ⁿ

is defined as;

(7)

27 ∑

^∞

( )

=

−

1 1

n n

x x

(

⁻

)(

⁻

) (

⁺ ⁻

)(

⁻

)(

⁻

)

⁺ ^∞

− +

= ...

1 1 1 1

1 1

1

5 4 3

2 3

2 x x x

x x

x x x

∞ + + + + + + + + +

=1 2x 2x² 3x³ 3x⁴ 4x⁵ 4x⁶ 6x⁷ 4x⁸ ...

( )

{ }

ⁿ

n

x

∑

^∞

n

=

+

=

0

1

1 ρ

, if we assume

ρ

1

( ) ⁰

. (5) Now we prove the Theorem, which is known as Second Mock Theta Conjecture.

Theorem 2:

²N

(

²^,⁵^,⁵n+³

)

=N

(

¹^,⁵^,⁵n+³

) (

+N⁰^,⁵^,⁵n+³

)

+ρ1

( )

n +¹^, where ρ1

( )

ⁿ is the number of partitions of n with unique smallest part and all other parts ≤ one plus the double of the smallest part.

Proof: We have;

( ⁻ )( ) ( )( ⁻ ⁺ ⁻ ⁻ )( ) ⁻ ⁺ ^∞

− + ...

1 1 1 1

1 1

1

5 4 3

2 3

2

x x x

x x

x x x

( ) ( ) ^D ^x

x x +

= 3 Ψ

∞ + + + + + + +

⇒ 1 2 x 2 x

²

2 x

³

2 x

⁴

3 x

⁵

3 x

⁶

...

( )

{ }

ⁿ

n

x n 1

0

∑

^∞ 1

=

+

= ρ

( )

{ }

ⁿ

n

x

∑

^∞

n

=

+

⇒

0

1

1 ρ ( ) ( ) ^D ^x

x x +

= 3 Ψ

, (by above)

( ) ³ ⁺ ( ) ³ ⁺ ( ) ³ ⁼

3 r

₂_,₀

r

₀_,₁

r

₀_,₂

{ ( ) }

ⁿ

n

x n 1

0

∑

^∞ 1

=

ρ +

( ) ( )

{

+ − +

(

+

)

−

⇒

∑

^∞

=

3 5 , 5 , 0 5 , 5 , 0 3 3 5 , 5 , 2 3

0

n N n N n

N

n

( ⁿ ) ( ^N ⁿ ) ( ^N ⁿ )} ^x

ⁿ

N 1 , 5 , 5 + 3 + 0 , 5 , 5 + 3 − 2 , 5 , 5 + 3

( )

{ }

ⁿ

n

x

∑

^∞

n

=

+

=

0

1

1 ρ

Equating the coefficient of

x

ⁿ on both sides, we get;

( ² ^, ⁵ ^, ⁵ ³ ) ( ⁰ ^, ⁵ ^, ⁵ ) ( ¹ ^, ⁵ ^, ⁵ ³ )

2 N n + − N n − N n +

( ) ¹

1

+

= ρ n

(

²^,⁵^,⁵ ³

) (

⁰^,⁵^,⁵

) (

¹^,⁵^,⁵ ³

)

2N n+ =N n +N n+ +ρ1

( )

n +¹^. Hence the Theorem.

4. Illustrative Examples

Here we give two examples, which are related to first and second mock theta conjectures respectively.

Example 1

For n = 2, we have;

N

(

¹^,⁵^,¹⁰

)

=⁹ with the relevant partitions are: 8 + 2, 6 + 1 + 1 + 1 + 1, 5 + 3 + 1 + 1, 5 + 2 + 2 + 1, 4 + 4 + 2, 4 + 3 + 3, 3 + 2 + 1 + 1 + 1 + 1 + 1, 2 + 2 + 2 + 2 + 1 + 1, 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1.

But N

(

⁰^,⁵^,¹⁰

)

=⁸ with the relevant partitions are:

8 + 1 + 1, 7 + 3, 5 + 2 + 1 + 1 + 1, 4 + 4 + 1 + 1, 4 + 3 + 2 + 1, 4 + 2 + 2 + 2, 3 + 1 + 1 + 1 + 1 + 1 + 1 + 1, 2 + 2 + 2 + 1 + 1 + 1 + 1.

Again,

ρ

0

( ) ² = ¹

with the relevant partition being 2.

( ¹ ^, ⁵ ^, ¹⁰ ) ( = N ⁰ ^, ⁵ ^, ¹⁰ ) + ρ

0

( ) ²

N

.

Example 2

For n = 1, we have;

N ( 2 , 5 , 8 ) = 5

with the relevant partitions are: 8, 5 + 2 + 1, 4 + 4, 3 + 1 + 1 + 1 + 1 + 1, 2 + 2 + 2 + 1 + 1.

But N(1, 5, 8) = 4 with the relevant partitions are: 5 + 1 + 1 + 1, 4 + 3 + 1, 4 + 2 + 2, 2 + 2 + 1 + 1 + 1 + 1, and N(0, 5, 8) = 4 with the relevant partitions are: 7 + 1, 4 + 2 + 1 + 1, 3 + 3 + 2, 2 + 1 + 1 + 1 + 1 + 1 + 1.

Again ρ1

( )

¹ =¹ with the relevant partition being 1.

Therefore, 2N(2, 5, 8) = 2×5 = 10 = 4 + 4 + 1 + 1 =

( ) ( ¹ ^, ⁵ ^, ⁸ ^N ⁰ ^, ⁵ ^, ⁸ )

N +

⁺

ρ

1

( )

¹ =¹^.

5. Conclusion

We have verified for any positive integer of n in two Theorems first and second mock theta conjectures. But we have seen these for n = 2 or 1 respectively.

6. Acknowledgment

It is a great pleasure to express our sincerest gratitude to our respected Professor Md. Fazlee Hossain, Department of Mathematics, University of Chittagong, Bangladesh. We will remain ever grateful to our respected Late Professor Dr. Jamal Nazrul Islam, JNIRCMPS, University of Chittagong, Bangladesh.

(8)

28 References

1- Andrews, G.E. and Garvan, F.G. Ramanuj’s Lost Notebook VI; the Mock Theta Conjectures, Advances in Math., 73, 1989: 242–255.

2- Andrews, G.E., An Introduction to Ramanujan’s Lost Notebook, Amer. Math. Monthly, 86. 1979. 89-108.

3- Garvan, F.G. Generalizations of Dyson’s Rank, Ph. D.

thesis, Pennsylvania State University, 1986.

4- Agarwal, A.K. Partitions Yesterday and Today, New Zealand Math. Soc., Wellington, 1979.

5- Watson, G.N., The Mock Theta Functions (2) Proc.

London Math. Soc., 42, 1937: 274–304.

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