Keywords: at most, auxiliary function, convenient, expansion, minimal difference, operator, Ramanujan’s device
Turkish Journal of Analysis and Number Theory, 2015 3 (2),
pp 3742.
DOI: 10.12691/tjant321
Received February 12, 2015; Revised March 28, 2015; Accepted April 01, 2015
Copyright © 2015 Science and Education Publishing. All Rights Reserved.
1. Introduction
In this article, we give some related definitions of , , , , , and . We describe the generating functions for , , , , and , and establish the Remarks 1 and 2 with numerical examples and also prove the Corollaries 1 and 2 with the help of Jacobi’s triple product identity ^{[3]}. We transfer the auxiliary function into another auxiliary function with the help of Ramanujan’s device of the introduction of a second parameter a ^{[5]},
to
where k =1, and a = x, it is used in proving The RogersRamanujan Identity 1. We prove The RogersRamanujan Identities with the help of auxiliary functions.
2. Some Related Definitions
^{[7]}: The number of partitions of n like: 4, 3+1, 2+2, 2+1+1, 1+1+1+1 P (4)=5.
^{[6]}: The number of partitions of n into parts each of which is of one of the forms 5m + 1 and .
: The number of partitions of into m parts at most.
: The number of partitions of n into parts of the forms 5m + 2 and 5m + 3.
: The number of partitions of n into parts without repetitions or parts whose minimal difference is 2.
: The number of partitions of into m parts at most.
: The number of partitions of n into parts not less than 2 and with minimal difference 2.
3. Generating Functions for C'(n) and C''(n)
In this section we describe the generating functions for and respectively. The generating function for is of the form ^{[5]};
 (1) 
where the coefficient of is the number of partitions of n into parts each of which is of one of these forms 5m + 1 and 5m + 4.
Now we consider a special function, which is given below:
It is convenient to define . The coefficient of in the above expansion is the number of partitions of into m parts at most. Another special function, which is defined as;
 (2) 
where the coefficient is the number of partitions of n into parts without repetitions or parts, whose minimal difference is 2.
From (1) and (2) we can establish the following Remark:
Remark 1:
 (3) 
i.e., the number of partitions of n with minimal difference 2 is equal to the number of partitions of n into parts of the forms 5m + 1 and 5m + 4.
Example 1: For n = 11, there are 7 partitions of 11 that are enumerated by of above statement, which are given bellow ^{[6]}:
There are 7 partitions of 11 are enumerated by of above statement, which are given bellow:
Hence, .
We can conclude that, .
which will be proved later as identity 1, it is known as The RogersRamanujan identity 1.
The generating function for is of the form ^{[1]};
 (4) 
where the coefficient is the number of partitions of n into parts of the forms 5m + 2 and .
Now we consider a special function, which is of the form ^{[1]};
where the coefficient of in the above expansion is the number of partitions of into m parts at most.
Another special function, which is defined as;
 (5) 
where the coefficient is the number of partitions of n into parts not less than 2 and with minimal difference 2.
From (4) and (5) we can establish the following Remark:
Remarks 2:
 (6) 
i.e., the number of partitions of n into parts not less than 2 and with minimal difference 2 is equal to the number of partitions of n into parts of the forms 5m + 2 and 5m + 3.
Example 2: If n = 11, the four partitions of 11 into parts not less than 2 and with minimal difference 2 are given below:
Hence, .
Again the four partitions of 11 into parts of the form 5m + 2 and 5m + 3 are given as;
Hence, .
∴ .
We can conclude that, .
i.e.,
which will be proved later as identity 2, it is known as The RogersRamanujan identity 2.
Now we give two Corollaries, which are related to the Jacobi’s triple product identity ^{[3]}.
Corollary 1:
Proof: From Jacobi’s Theorem ^{[2]} we have;
for all z except z = 0, if .
If we write for x, for z and replace n by n + 1 on the left hand side we obtain;
Hence, the Corollary.
Corollary 2:
Proof: From Jacobi’s Theorem we have;
for all z except z = 0, when .
If we write for x, for z and replace n by n + 1 on the left hand side we obtain;
Hence the Corollary.
4. The RogersRamanujan Identities
First we transfer the following auxiliary function into another auxiliary function. Let us consider the auxiliary function ^{[1, 2]} with and .
 (7) 
it is known as Ramanujan’s device of the introduction of a second parameter a, where k is 0, 1 or 2 and ,
Hence,
where ,
 (8) 
which is another auxiliary function, and it is used in proving The RogersRamanujan Identities ^{[1]}.
But from (7) we can easily verify that with k = 1, 2 and a = x.
 (9) 
 (10) 
From (8) we can also find that, if k =1 and a = x, then;
 (11) 
Again for k = 2 and a = x, we get;
 (12) 
Now we can consider the following Remark ^{[2]}.
Remark 3: , where the operator η is defined by η f(a) = f(ax), and k = 1 or 2.
Proof: From (8) we have;
where , and ,
It is convenient to define , . We have;
Now we have, and , hence,
In the second sum on the right hand side of the Identity we change n into n + 1. Thus,
We have and so,
Hence, the Remark.
The RogersRamanujan Identities
Identity 1 ^{[4]}:
Identity 2 ^{[4]}:
Proof: From (8) we have;
 (13) 
where .
From above Remark we have;
where the operator η is defined by η f(a) = f(ax), and k = 1 or 2. In particular
 (14) 
So we have,
 (15) 
Suppose now that;
 (16) 
where the coefficients depend on x only. Substituting this into (15), we obtain;
Hence, equating the coefficients of various powers of a from both sides we get;
From (13) and (16), we have for k = 2;
If a = x, then;
Therefore,
Hence the Identity 1.
Again from (13), (14) and (16) we have with k = 1,
If a = x, then we have;
Therefore,
Hence the Identity 2.
5. Conclusion
In this study, we have shown with the help of a numerical example when n=11, and also have shown with the help of a numerical example when n =11. We have transferred the auxiliary function into another auxiliary function with the help of Ramanujan’s device of the introduction of a second parameter a,
i.e.,
to
where k =2, and a = x, it is used in proving The RogersRamanujan Identity 2. Finally we have proved The RogerRamanujan Identities with the help of auxiliary function,
where .
References
[1]  Andrews, G.E, “An Introduction to Ramanujan’s Lost Notebook”, American Mathmatical Monthly, 86: 89108. 1979. 
 In article  CrossRef 

[2]  Hardy, G.H. and Wright, E.M. “Introduction to the Theory of Numbers”, 4^{th} Edition, Oxford, Clarendon Press, 1965. 
 In article  

[3]  Jacobi, C.G.J. (1829), “Fundamenta Nova Theoriae Functionum Ellipticarum (in Latin), Konigsberg Borntraeger”, Cambridge University Press, 2012. 
 In article  

[4]  Baxter, R.J., “Exactly Solved Model in Statistical Models”, London, Academic Press, 1982. 
 In article  

[5]  Ramanujan, S., “Congruence Properties of Partitions”, Math, Z. 9: 147153. 1921. 
 In article  CrossRef 

[6]  Ramanujan, S., “Some Properties of P(n), Number of Partitions of n”, Proc. of the Cam. Philo. Society XIX, 207210. 1919. 
 In article  

[7]  Das, S. and Mohajan, H.K., “Generating Function for P(n,p,*) and P(n, *,p)”, Amer. Rev. of Math. and Sta. 2(1): 3335. 2014. 
 In article  
