Generating Functions for 1(n) and 2(n)

Generating Functions for 1(n) and 2(n)

Das, Sabuj and Mohajan, Haradhan

Assistant Professor, Premier University, Chittagong, Bangladesh.

17 March 2014

Online at https://mpra.ub.uni-muenchen.de/83046/

MPRA Paper No. 83046, posted 02 Jan 2018 23:06 UTC

1

Generating Functions for β

1

( n ) and β

2

( n )

Sabuj Das

Senior Lecturer, Department Of Mathematics Raozan University College, Bangladesh.

Email: sabujdas.ctg@gmail.com

Haradhan Kumar Mohajan²

Assistant Professor, Premier University, Chittagong, Bangladesh Email: haradhan_km@yahoo.com

ABSTRACT

This paper shows how to prove the two Theorems, which are related to the terms β1(n) and β2(n) respectively Theorem: N(0,5,5n+1)= β¹(n)+N (5,5,5n+1) and Theorem: N(1,5,5n+1)= β²(n)+

N(2,5,5n+2).

Keywords: Generating functions, Jecobi’s triple product.

1. INTRODUCTION

We give the definitions of



, Rank of partition, N m,n , N



m,t,n



, z,  x_ (zx)_∞,

 

m

xn , ₁

 

n , ₂

 

n ,

 

m k x

x ; ⁵ collected from Partitions Yesterday and Today (Garvan 1979), Generalizations of Dyson’s rank (Garvan 1986), Ramanujan’s Lost Notebook (Andrews 1979). We generate the generating functions for 1

 

ⁿ , 2

 

ⁿ (Andrews 1979) and prove the Theorems N



0,5,5n1



1

 

ⁿ +



2,5,5n1



N and N



1,5,5n1



 ₂ n + N



2,5,5n2



. Finally we give two examples, which are related to the Theorem 1 and Theorem 2 respectively when n =2.

2. DEFINITIONS

: A partition.

Rank of partition: The largest part of a partition  minus the number of parts of .

2

 

mn

N , : The number of partitions of n with rank m.



mt n



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

 

n

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

  ⁿ



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.

 x_: The product of infinite factors is defined as follows:

  

x  ^ 1^x

 

1^x²



1^x³



... ^.

 zx : The product of infinite factors is defined as follows:

  

zx  ^ 1^zx

 

1^zx²



1^zx³



... ^.

 

m

xn : 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 .

 

m k x

x ; ⁵ : The product of m factors is defined as follows:



^{; 5}

 

m^{1 k}



^{1 k5}



^{1 k10}



^...



^{1 km1 5}



k x x x x x

x .

 

n

1 : The number of partitions of n into 1’s and parts congruent to 0 or –1 modulo 5 with the largest part 0mod55 times the number of 1’s  the smallest part 1



mod5



.

 n

2 : The number of partitions of n into 2’s and parts congruent to 0 or – 2 modulo 5 with the largest part 0



mod5



5 times the number of 2’s  the smallest part 2



mod5



.

3. GENERATING FUNCTIONS (FROM RAMANUJAN’S LOST NOTE BOOK)

From Ramanujan’s Lost Note Book (Andrews 1979), Mock Theta Functions (2) (Watson 1937), G.

E. Andrews and F. G. Garvan (Andrews and Garvan 1989), we quote the relations as follows:

















 

...

5 ) cos2 2 1 5 )(

cos2 2 1 (

...

) 1 )(

1 )(

1 ) (

(

4 2

2

3 2

n x x n x

x

x x x x

F  

3











2 /

5 cos2 2 1 1 ) (

n x x x x

f  _{   } ^{ ... }

5 ) cos2 2 1 5 )(

cos2 2 1

( ^{2 2 4}

4

n x x n x

x

x





,n1or 2.







 ( )

5 cos4 2 ) 5 ( cos2 4 ) ( )

( ⁵

2 5

1 5

1

x n C x x n B x x A x

F  

) 5 ( cos2 2 ⁵

3

x n D

x  . (1)

) 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  _{ } 







  



 



 

 x

x x n

n D

x ( )

5 . sin 2 4 ) 5 ( cos2

2 ^{5 2}

3    . (2)

  



 



 















 

...

1 1

1

...

1

6 2 4 2

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12

 

^{21 3}

 

^{21...7}



^2...

7 4

x x x

x x x x

D ,

    ^ 



^



^



^

 



 ⁵_{4 6}

1 1 1 1 1 1

x x x

x x x

  

    





 







 ...

1 1 1 1

1 ^{4 6 9 11}

20

x x x x x

x .

But we get;





 ( )

5 cos4 2 ) 5 ( cos2 4 )

(x⁵ x B x⁵ x² C x⁵

A  

) 5 ( cos2

2x³  D x⁵











 ^{2 2 3} 2 ⁵

5 cos2 5 2

cos4 5 2

cos 2 4

1 x  x  x  x







 ...

5 cos2 5 2

cos 2

4x^{6 2}  x⁸  x¹⁰

   



^



^



^



^

 





 _{2 2} ⁵_{3 7}

1 1 1 1 1 1

x x x

x x x

     

 









 ...

1 1 1 1

1 ^{2 3 7 8 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

4

 x  A x

3 1 . And,

   

 











^{ }



 

 ...

1 1 1 1

1

1 ^{3 4 5}

3 3

2 2

x x x

x x

x x x

x

 

^{x xD}

 

^x



3 .

We assume without loss of generality that n = 1. Let _exp²⁵

i

  , then we may write the definitions of F(x) and ′(x) as;

   

 

 

 



 

x x x x

F ₁





and,

  ^       



      



 

1

1

1 ...1 1

1 1 1

2

n

n n

n

x x

x x

x x

f    

   



^

 



1

1

2

n n n

n

x x

x



 ,

where we have used the relations;

 

a ₀ 1,

  

a n^{ 1}a



^1ax



^...



^1ax^n1



, for n1 and,

    

¹



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 ′(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;



ⁿ



ⁿ



ⁿ



n

x x

z

zx  



^{  }

 1 1 ^{1 1} 1

1

5

 

1 ^21

 



 





n n n n n

x

z (3)















... z^2x z^1 1 zx z²x³ ... . where z  0 and x< 1.

Replacing x by x⁵ and z by x^3 we get from (3);



ⁿ



ⁿ



ⁿ



n

x x

x^{5 3 5 2 5}

1

1 1

1  

^{  }

















 ... x¹¹ x³ 1 x² x⁹ ...













1 x² x³ x⁹ x¹¹ ... .

Again replacing x by x⁵ and z by x^3 equation (3) becomes;



ⁿ



ⁿ



ⁿ



n

x x

x^{5 4 5 1 5}

1

1 1

1  

^{  }

















... x¹³ x⁴ 1 x x⁷ ...













1 x x⁴ x⁷ x¹³ ... . In fact we have;

     



^{5 4}

 

^{2 5 1}



²

5 2 5 3 5

1 1 1

1 1

1







 

  





 

 n n

n n

n

n x x

x x

x x

A ,

   



^{5 4}



^{5 1}



5

1 1 1

1







  

 

 _n ⁿ _n

n x x

x x

B ,

   



^{5 3}



^{5 2}



5

11 1

1







  

 

 _n ⁿ _n

n x x

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

x x

D .

6 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 (Garvan 1986);

 

^{ }

   



^



 





    



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 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^2m5 x^2m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

n x

x x x x

 

^{ }



^{ }





^





 









1

1 23

1

n n

m t n n mn

n

nx 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



7

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

x d tn t b N d tn t a

N   



 ^

 , , , ,

0

. The generating function 

 

^x is of the form;

 

 



^

 

^



^



^

 



 ⁵_{4 6}

1 1 1 1 1 1

x x x

x x x



 

^1

   

^{1 4 1 6 1 9 1 11}



^..._^

20

x x x x x

x ,



^{   }

 

^{    }



^





 1 1 x x² ... x⁵1 x x² ...



^{1x4...}



^{1x6...}



^...



















x x² x³ x⁴ 2x⁵ 2x⁶ 2x⁷ 2x⁸ ...

   

 

ⁿ

n

x n N n

N1,5,5 2,5,5

0



^







 

0

2 ,

r1

 _.

The generating function A(x) is defined as;

 

^



^11



²



^{1 4}

 

^{21...6}



^2...

9 3 2

x x

x

x x x x

A



^{    }



^{   }



^

 1 x² x³ x⁹ ... 1 2x 3x² ...



^{12x43x8...}



^...













1 2x 2x² x³ 2x⁴ ...

   



^













0

5 , 5 , 2 5

, 5 , 0 1

n

n N

n

N N



1,5,5n



2N



2,5,5n



x²

   



^



^







^



n n

x n N

n

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

0

   

 

ⁿ

n

x n N n

N1,5,5 2,5,5 2

0



^





 

0 2

 

0 1r_0,2  r_1,2

 .

8 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

n n

x x

x





 

 

 

 



   

 1 0 3 2x 12 11x² x³ 2x⁴ ...

   

 

ⁿ

n

x n N

n



^ N











0

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

 

¹

2 ,

r0

 .

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

n n

x x

x





 

 

 

 



 

 1 0 3 3 x 16 15 x² ...

   



  









^



n n

x n N

n

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

0



^{1x5n2x10n4...}



 

²

2 ,

r1

 .

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

 

 



^



^



^



^

 





 _{2 2} ⁵_{3 7}

1 1 1 1

1 1

x x x

x

x x



^{1 2}



^{1 3}



^{1 7}



^{1 8}



^{1 12}



^..._^

20

x x x x x

x



^{1 ...}

 

^{1 ...}



1  ²  ⁴    ⁵  ²  



 x x x x



^{1x3x6...}



^{1x7...}



^...

















x² x⁴ x⁶ x⁷ 2x⁸ x⁹ 2x¹⁰ ... .

9 Hence,

 _{         }

 x x³ x⁴ x⁵ x⁶ 2x⁷ x⁸ 2x⁹ ...

x

x

   

 

ⁿ

n

x n N n

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

0



^











 3

0 ,

r2



and,

   

x r_0,2 3  x .

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

 

^



^112

 

^{21 3}

 

^{21...7}



^2...

7 4

x x

x

x x x x

D



^{    }



^{   }



^

 1 x x⁴ x⁷ ... 1 2x² 3x⁴ ...



^{12x3...}



^{12x7 ...}



^...











1 x 2x² 0.x³ ...

   



^















0

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

n

n N n

N N



0,5,5n3

 

N 2,5,5n3



xⁿ

   

     





^

0

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

n

xn

n N n

N



^

     









0

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

n

xn

n N n

N

 

3 _0,2

 

3

1 ,

0 r

r 



4. THE GENERATING FUNCTIONS FOR ₁ n AND ₂

 

n

First we shall establish the following identity, which is used in proving the Theorems. If a and t are both real numbers with a < 1 and t < 1, we have;

        

 

^{ }

 

^



^

 





...

1 1 1

...

1 1

1

2 2

tx tx t

atx atx

at t

at

10

  



^{  }

 

^{   }



^

 1 at 1 atx ... 1 t t² ...



^{1txt2x2 ...}



^{1t4x4...}



^...

   



^{        }



^



1 t 1 x x² ... a1 x x² ... ^t2

 

^{1x2x22x3...}

 

^{a12x3x2...}



^



^{ 22 32 4...}



^...

2 x x x x

a



^

 

^{   }



^



1 1 a t1 x x² ...

 

^{1a 1ax}



^t2



^{1x2x22x3...}



^...

    



^

 

^



^

 

 

 ₂ ²

1 1

1 1 1

1 1

x x

t ax a

x t

a

    

 

^{1 1}

  

^{1 11 ...}

1

3 2

3 2

x x x

t ax ax a

   



^





0

n n

n n

x t a

i.e.,

 

   



^

 

 

 

0

n n

n n

x t a t

at . (4)

The generating function for ₁

 

n is defined as;

   



^

  

0

5 4 5 5

5; ;

n

n n

n

x x x x

x



^



^



^



^

 1 1 1 ...

1

14 9

4 x x

x



^1x5



^1x91



^1x14



^...



^{1 5}



^{1 10}



^{1 14}



^{......}

2

x x

x

x



















1 x x² x³ 2x⁴ x⁵ 2x⁶ 3x⁸ ...

 

ⁿ

n

x n

0



^ 1



  , (5)

were we have assumed ₁

 

0 1.

The generating function for ₂

 

n is defined as;

   



^

  

0

5 3 5 5 5

2

;

n

;

n n

n

x x x x

x