1. Introduction

Throughout the paper, we assume . We use the standard notation

and

Ramanujan’s general theta function is defined by

(1)

The Jacobi triple product identity [^[1], Entry 19] in Ramanujan’s notation is

(2)

Ramanujan defined the following three special cases of (1) [^[1], Entry 22]:

(3)

(4)

and

(5)

For convenience, we define

for a positive integer .

and

(6)

These functions satisfy the famous Rogers-Ramanujan identities

and

(7)

In ^[27], Ramanujan remarks, “I have now found an algebraic relation between and , viz.:

Another noteworthy formula is

Each of these formulae is the simplest of a large class." In his lost notebook ^[28], Ramanujan recorded forty beautiful modular relations involving the Rogers-Ramanujan functions without proofs. The forty identities were first brought before the mathematical world by B. J. Birch ^[18]. Many of these identities have been established by L. J. Rogers ^[30], G. N. Watson ^[34], D. Bressoud ^{[20, 21]}, A. J. F. Biagioli ^[17] and B. C. Berndt et al. ^[16] offered proofs of of the identities. Recently, in Chapter 8 of their book ^[11], G. E. Andrews and Berndt collected proofs for all forty identities. Most likely these proofs might have given by Ramanujan himself. A number of mathematicians tried to find new identities for the Rogers-Ramanujan functions similar to those which have been found by Ramanujan ^[28], including Berndt and H. Yesilyurt ^[15], Yesilyurt ^[36], S. Robins ^[29] and C. Gugg ^[23].

Two beautiful analogues of the Rogers-Ramanujan functions are the Göllnitz-Gordon functions, which are defined as

(8)

and

(9)

Identities (8) and (9) can be found in L. J. Slater’s list ^[31]. S.-S. Huang ^[26] has established a number of modular relations for and . S.-L. Chen and Huang ^[22] have derived some new modular relations involving and . N. D. Baruah, J. Bora and N. Saikia ^[14] offered new proofs of many of these modular relations, as well as establishing some new relations. E. X. W. Xia and X. M. Yao ^[35] offered new proofs of some modular relations established by Huang ^[26] and Chen and Huang ^[22]. They also established some new relations that involve only Göllnitz-Gordon functions.

In view of the Ramanujan’s forty identities, many of the Rogers-Ramanujan type functions were studied by many mathematicians. For example, septic analogues of the Rogers-Ramanujan functions were studied by H. Hahn [24, 25]^{[24, 25]}, nonic analogues of the Rogers-Ramanujan functions were studied by Baruah and Bora ^[13], cubic functions were studied by C. Adiga et al. ^{[4, 5, 9]}, another cubic functions were studied by K. R. Vasuki, G. Sharath and K. R. Rajanna ^[33], dodecic analogues of the Rogers-Ramanujan functions were studied by Baruah and Bora ^[12], Robins ^[29] and C. Gugg ^[23], another dodecic analogues of the Rogers-Ramanujan functions were studied by Vasuki and P. S. Guruprasad ^[32], Adiga, Vasuki and B. R. Srivatsa Kumar ^[10] established modular relations involving two functions of Rogers-Ramanujan type, the authors have studied two functions of order ten ^{[2, 3]} and more recently, Adiga et al. [6, 7, 8]^{[6, 7, 8]} have studied four functions of order fifteen. Almost all of these functions which have been studied so far can be found in Slater’s list ^[31].

In Chapter of his Ph.D. thesis, C. Gugg ^[23] considered the following four sextodecic analogues of the Rogers-Ramanujan functions:

(10)

(11)

(12)

and

(13)

Gugg ^[23] established twelve modular relations involving the functions , , and including the following two beautiful identities:

(14)

(15)

To prove his results, Gugg ^[23] applied a theorem of R. Blecksmith, J. Brillhart and I. Gerst ^[19] and also employed the method given by Bressoud in his thesis ^[20].

The main purpose of this paper is to establish several modular relations that are analogues of Ramanujan’s forty identities involving , , and . Many of these identities that we find have partition theoretic interpretations. In the last section, we extract partition theoretic interpretations for some of these modular relations. To prove our results, we use the idea of Watson ^[34] which he has used to prove some of Ramanujan’s forty identities.

2. Main Results

In this section, we present a list of new modular relations for the functions , , and , which we establish in Section 4. For simplicity, we use the notations , , and , for a positive integer .

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

(26)

(27)

(28)

(29)

(30)

(31)

Identity (18) is the corrected version of identity (7.2.12) found in ^[23].

The following two identities are relations involving some combinations of the functions defined in (10)–(13), and the Göllnitz-Gordon functions and :

(32)

(33)

3. Some Preliminary Results

The function satisfies the following basic properties ^[1]:

(34)

(35)

(36)

and, if is an integer,

(37)

Lemma 3.1 We have

This lemma is a consequence of the Jacobi triple product identity (2) and Entry 24 of ^[1].

The following identity follows easily from Entry 31 found in ^[1]

(38)

Using (38) one can easily establish the following lemma:

Lemma 3.2 We have

(39)

The following lemma can be found in [1, Entry 30(ii) and (iii)]:

Lemma 3.3 We have

(40)

(41)

Lemma 3.4 Let and , . Here denote the largest integer less than or equal to . Then,

For a proof of Lemma, 3.4 see ^[3].

4. Proofs of the Main Results

We prove our main results using ideas similar to those of Watson ^[34]. In all proofs, one expresses the left sides of the identities in terms of theta functions by using (10), (11), (12) and (13). After clearing fractions, we see that the right side can be expressed as a product of two theta functions, say with summations indices and One then tries to find a change of indices of the form

or

so that the product on the right side decomposes into the requisite sum of two products of theta functions on the left side.

Proof of (14). Using (10), (11), (12), (13) and Lemma 3.1, we see that (14) is equivalent to

(42)

We have

(43)

In this representation, we make the change of indices by setting

where and have values selected from the set . Then

It follows easily that , and so and , where Thus, there is one-to-one correspondence between the set of all pairs of integers and triples of integers , From (43), we find that

which is same as (42).

Proof of(15). Using (10), (11), (12), (13) and Lemma 3.1, we see that (15) is equivalent to

(44)

We have

(45)

In these representations, we make the change of indices by setting

where and have values selected from the set . Then

It follows easily that , and so and , where Thus, there is one-to-one correspondence between the set of all pairs of integers and triples of integers , From (45), we find that

which is same as (44).

The proofs of the identities (16)–(23) are very similar to those above, so we omit the details.

Proof of (24). Using (10), (11), (12), (13) and Lemma 3.1, we see that (24) is equivalent to

(46)

Now changing to , and then applying Lemma 3.2 in the resulting identity, we may rewrite (46) in the form

(47)

Thus we need only to establish (47). We have

(48)

In these representations, we make the change of indices by setting

where and have values selected from the set . Then

It follows easily that , and so and , where Thus, there is one-to-one correspondence between the set of all pairs of integers and triples of integers , From (48), we find that

(49)

Using the same change of indices for the product , we find that

(50)

Subtracting (50) from (49), we deduce the desired result.

In a similar way, one can prove the identities (25)–(31).

Proof of(32) and (33). Using (10), (11), (12), (13), Lemma 3.1 and , , we see that (32) and (33) are equivalent, respectively, to

(51)

and

(52)

Applying Lemma 3.3, we obtain

and

We may rewrite (51) and (52) in the form

(53)

and

(54)

Thus to establish (32) and (33), it is suffices to prove (53) and (54).

We have

(55)

In this representation, we make the change of indices by setting

where and have values selected from the set . Then

It follows easily that , and so and , where Thus, there is one-to-one correspondence between the set of all pairs of integers and triples of integers , From (55), we find that

We make the same argument for the product , to find that

Using the above two identities, we deduce the desired results.

5. Applications to the Theory of Partitions

Some of our modular relations yield theorems in the theory of partitions. In this section, we present partition theoretic interpretations of some of our modular relations.

Definition 5.1 A positive integer has colors if there are copies of available and all of them are viewed as distinct objects. Partitions of positive integer into parts with colors are called “colored partitions".

For example, if is allowed to have two colors, say (red) and (green), then all the colored partitions of are , , , , , and . It is easy to see that

is the generating function for the number of partitions of where all the parts are congruent to and have colors. For simplicity, we define

where and are positive integers with and

Theorem 5.2 Let denote the number of partitions of into parts congruent to , , , where the parts congruent to , have two colors. Let denote the number of partitions of into parts congruent to , , , , where the parts congruent to , have two colors. Let denote the number of partitions of into parts congruent to , , , , where the parts congruent to , have two colors. Then, for any positive integer ,

Proof. Using (2), one can easily verify that (32) is equivalent to

(56)

The three quotients of (56) represent the generating functions for , and , respectively. Hence, (56) is equivalent to

where we set . Equating coefficients of () on both sides yields the desired result.

Example 5.3 The following table illustrates the case in Theorem 5.2

Table 1. Example for Theorem 5.2

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Theorem 5.4 Let denote the number of partitions of into parts congruent to , , , where the parts congruent to , have two colors. Let denote the number of partitions of into parts congruent to , , , , where the parts congruent to , have two colors. Let denote the number of partitions of into parts congruent to , , , , where the parts congruent to , have two colors. Then, for any positive integer ,

Proof. Using (2), one can easily verify that (33) is equivalent to

(57)

The three quotients of (57) represent the generating functions for , and , respectively. Hence, (57) is equivalent to

where we set . Equating coefficients of () on both sides yields the desired result.

Example 5.5 The following table illustrates the case in Theorem 5.4

Table 2. Example for Theorem 5.4

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In a similar way one can prove the following four theorems using the modular relations (14), (16), (18) and (19), respectively:

Theorem 5.6 Let denote the number of partitions of into parts not congruent to , , , , where the parts congruent to , have two colors and , have three colors. Let denote the number of partitions of into parts not congruent to , , , , where the parts congruent to , have two colors and , have three colors. Let denote the number of partitions of into parts not congruent to , , , , where the parts congruent to , have two colors and , have three colors. Let denote the number of partitions of into parts not congruent to , , , , where the parts congruent to , have two colors and , have three colors. Let denote the number of partitions of into odd parts with two colors. Then, for any positive integer ,

Theorem 5.7 Let denote the number of partitions of into parts congruent to , , , , , where the parts congruent to , , , have two colors and have three colors. Let denote the number of partitions of into parts congruent to , , , , , where the parts congruent to , , , have two colors and have three colors. Let denote the number of partitions of into parts congruent to , , , , , where the parts congruent to , , , have two colors and have three colors. Let denote the number of partitions of into parts congruent to , , , , , where the parts congruent to , , , have two colors and have three colors. Let denote the number of partitions of into parts congruent to , , , , , , , , , , , . Then, for any positive integer ,

Theorem 5.8 Let denote the number of partitions of into parts congruent to , , , , , , , . Let denote the number of partitions of into parts congruent to , , , , , , , . Let denote the number of partitions of into parts congruent to , , , , , , , . Let denote the number of partitions of into parts congruent to , , , , , , , . Then, for any positive integer ,

Theorem 5.9 Let denote the number of partitions of into parts not congruent to , , , , , , , , , where the parts congruent to , , , have two colors, , have three colors and have four colors. Let denote the number of partitions of into parts not congruent to , , , , , , , , , where the parts congruent to , , , have two colors, , have three colors and have four colors. Let denote the number of partitions of into parts not congruent to , , , , , , , , , where the parts congruent to , , , have two colors, , have three colors and have four colors. Let denote the number of partitions of into parts not congruent to , , , , , ,, , , where the parts congruent to , , , have two colors, , have three colors and have four colors. Let denote the number of partitions of into parts not congruent to , , , , , where the parts congruent to , , , , , have two colors. Then, for any positive integer ,

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