From Dedekind’s Level 12 Identities to Combinatorial Structures of Colored Partitions

Abstract

The Dedekind $\eta$-function plays an important role in number theory, particularly in the study of modular forms, q-series, and partition identities. In this paper, we investigate several level-12 $\eta$-function identities and examine their combinatorial implications. These identities are obtained from algebraic transformations of known expansions involving mock theta functions, which were originally introduced by Srinivasa Ramanujan. By employing classical q-series techniques and modular transformations, we derive identities that reveal interesting relationships among $\eta$-functions. We further interpret these identities combinatorially to establish correspondences between specific classes of colored partitions with prescribed color restrictions. These results provide new insights into the structure of colored partition functions and highlight the interplay between mock theta functions, Dedekind $\eta$-function identities, and combinatorial partition theory. Our findings contribute to a deeper understanding of the connections between modular forms and colored partitions and suggest further directions for research in number theory and combinatorics.

1. Introduction

The Dedekind eta function and its products are important in the theory of modular forms, q-series, and integer partitions. Specifically, eta-product identities involving certain levels have been demonstrated to be capable of representing arithmetic and combinatorial information, such as congruences, dissections, and colored partition interpretations. Such identities have been determined at levels 6, 8, 10, and 14 in classical and recent works, showing strong links between modular objects and partition theory [1,2,3,4,5,6,7,8].

A major source of eta-product identities arises from Ramanujan’s mock theta functions (MTFs). Since the foundational work of Zwegers, MTFs have been understood as components of harmonic Maass forms, allowing their coefficients to be studied using modular and analytic techniques. Among these, second-order MTFs have attracted attention due to their rich arithmetic structure and their ability to generate nontrivial q-product identities through coefficient extraction and algebraic transformation.

This article is concerned with level-12 eta-function identities of the second-order mock theta function (MTF) $B\left(q\right)$. Isolating certain arithmetic progressions of the coefficients of the $B\left(q\right)$ and using known transformation formulas, we have three explicit eta-product identities at level 12.

In addition to their form of analysis, the derived identities also have combinatorial direct interpretations in colored partitions. The function-generating techniques have been used to translate every eta-product identity into a precise relation between colored partition counting functions with the specified congruence classes modulo 12 and constant color multiplicities. This gives a tangible connection between MTFs, the eta product, and colored partition theory. This study has the following contributions:

• We extract three identities of the eta product of level 12 using the second-order MTF $B\left(q\right)$.
• We make such identities clear in the form of Dedekind eta functions and infinite q-products.
• We derive three aligned identities of colored partition that are each premised on generating function arguments and examples.
• We illustrate a structure that can be extended to give colored partition interpretations of eta-product identities induced by other MTFs and higher levels.

Besides their combinatorical importance, these identities have possible links to applications in the real world, including coding theory, statistical mechanics, and cryptographic constructions, in which partition functions and modular forms tend to be involved. Although the paper is dedicated to the theoretical construction and combinatorial explanations, these links can serve as possibilities to develop the applied research and do the practical work.

The paper is organized as follows. Section 2 introduces the necessary notation and preliminary results. In Section 3, we derive the level-12 eta-function identities from the MTF $B\left(q\right)$. Section 4 is devoted to the combinatorial interpretation of these identities in terms of colored partitions, along with explicit examples. Finally, Section 5 concludes with remarks and possible directions for future research.

2. Preliminaries

In this section, we briefly introduce the notation and preliminary results required for the derivation of the level-12 eta-function identities and their combinatorial interpretations.

Throughout this study, we take $\left(\right|q|<1)$. The infinite q-Pochhammer symbol is defined by

$${(a;q)}_{\infty }=\prod _{s=0}^{\infty }(1-a{q}^s).$$

In particular,

$${(q;q)}_{\infty }=\prod _{s=1}^{\infty }(1-q^s).$$

The Dedekind eta function is defined, for $(\tau \in \mathbb{H}),$ by

$$\eta \left(\tau \right)=q^{1/24}{(q;q)}_{\infty },\mathrm{where}q=e^{2\pi i\tau }.$$

Many q-series identities can be expressed in terms of eta products, which play an important role in the study of modular forms and partition-generating functions.

For a positive integer $\left(t\right)$, we use the shorthand notation

$$\begin{array}{cc}\hfill U_t& ={(q^t;q^t)}_{\infty }=q^{{\displaystyle \frac{-t}{24}}}\eta \left(t\tau \right).\hfill \end{array}$$

More generally, for positive integers t and s, we write

$$\begin{array}{cc}\hfill U_t^s& ={(q^t;q^t)}_{\infty }^s,U_{(t_1,t_2,\dots ,t_n)}^{(s_1,s_2,\dots ,s_n)}=U_{t_1}^{s_1}{U}_{t_2}^{s_2}\dots U_{t_n}^{s_n}.\hfill \end{array}$$

The general theta function of Ramanujan is defined by [9]:

$$f(x,y):=\sum _{s=-\infty }^{\infty }{x}^{s(s+1)/2}{y}^{s(s-1)/2},$$

which satisfies Jacobi’s triple product identity

$$f(x,y)={(-x;xy)}_{\infty }{(-y;xy)}_{\infty }{(xy;xy)}_{\infty }$$

Several classical theta functions arise as special cases, including

$$\varphi \left(q\right)=f(q,q),\psi \left(q\right)=f(q,q^3),f(-q)=f(-q,-q^2).$$

We make essential use of the second-order MTF $B\left(q\right)$, as given in [10], defined by

$$B\left(q\right):=\sum _{s=0}^{\infty }{\displaystyle \frac{q^s{(-q;q^2)}_s}{{(q;q^2)}_{s+1}}},$$

with the expansion:

$$B\left(q\right)=\sum _{s=0}^{\infty }b\left(s\right)q^s,$$

where $b\left(s\right)$ represents the coefficient of $q^s$ in $B\left(q\right)$. In this study, we focus on the subsequences $b(6s+2)$ and $b(6s+4)$, which give rise to explicit eta-product identities at level 12.

Integer Partitions: Let s be a nonnegative integer. A partition of s is a way of expressing s as a sum of positive integers, where each summand is called a part. As an example, 4 may be partitioned as $4=3+1$, in which the parts of the partition are 3 and 1. The complete list of partitions of 4 is:

$$4,3+1,2+2,2+1+1,1+1+1+1.$$

An integer may have several distinct partitions and the total number depends on the value of the integer. The function $p\left(s\right)$, known as the partition function, counts the number of such partitions for a given integer s. As $s\to \infty$, the asymptotic behavior of $p\left(s\right)$ is described by Hardy and Ramanujan’s formula [11]:

$$p\left(s\right)\sim {\displaystyle \frac{1}{4\sqrt{3}s}}{e}^{\pi \sqrt{2s/3}}.$$

(1)

The generating function of the integer partitions is given as follows:

$$\sum _{s=0}^{\infty }p\left(s\right)q^s={\displaystyle \frac{1}{{(q;q)}_{\infty }}}.$$

(2)

Partitions have attracted the interest of researchers in many fields because they have wide uses in mathematics, cryptography, computer science, statistics, and physics (see, for example, [12,13,14,15,16]).

For the combinatorial interpretation, we employ colored partitions. A colored partition of a non-negative integer is a partition in which each part is assigned one of a prescribed number of colors. The generating function for the number of colored partitions $p_{\kappa }\left(s\right)$ of an integer s using $\kappa$ colors is given by:

$$\sum _{s=0}^{\infty }{p}_{\kappa }\left(s\right)q^s=\prod _{n=1}^{\infty }{\left({\displaystyle \frac{1}{1-q^n}}\right)}^{\kappa },\mathrm{for}\left|q\right|<1,$$

representing the fact that the n part size contributes $\kappa$ different choices, one for each color.

This concept was formally introduced by Huang [17], who developed the foundation for analyzing such partitions using generating functions and modular forms. Variations in the definition of colored partitions may appear across different studies, depending on specific constraints or applications.

Merca [18] extended this idea by permitting the integer s to appear in s different color instances in his analysis of partition sums. Bandyopadhyay and Baruah [19], along with other researchers [1,20,21,22,23], introduced s as a color variable alongside $s+t$ colors to derive results on the s-colored partition function.

A particularly significant category of colored partitions allows each partition component to be assigned any of the k available colors. For instance, when partitioning the integer 3 using two colors, purple and white, the following 10 distinct partitions arise:

$$\begin{array}{cc}\hfill 1_p+& 1_p+1_p,1_p+1_p+1_w,1_p+1_w+1_w,1_w+1_w+1_w,\hfill \\ \hfill & 1_p+2_p,1_p+2_w,1_w+2_p,1_w+2_w,3_p,3_w.\hfill \end{array}$$

Here, the subscripts p and w denote the colors purple and white, respectively. That is, parts such as $1_{\mathrm{p}}$, $1_{\mathrm{w}}$ represent the integer 1 with the colors pink and white, respectively. Likewise, $2_{\mathrm{p}}$, $2_{\mathrm{w}}$, and $3_{\mathrm{p}}$, $3_{\mathrm{w}}$ are the integers 2 and 3, which are differentiated by the colors. In this system, where each part can be assigned any of exactly k colors without restriction, it leads to the classification of such partitions as uniformly k-colored partitions.

Several authors, including [2,7,24,25,26,27,28,29,30,31], as well as [32], have conducted research based on this concept, alongside numerous other published works. The modified version of this partition method allows certain parts to be selected from k colors, while others use alternative designations. If the integer 1 is allowed only in two colors (say, orange (o) and brown (b)) and other integers (2) can only be presented in one color (uncolored), then the partitions of 2 are:

$$2,1_o+1_o,1_b+1_b,\mathrm{and}{1}_o+1_b,$$

where the $1_{\mathrm{o}}$ and $1_{\mathrm{b}}$ are the integer 1, given the colors orange and brown, respectively. This methodology has been widely utilized to establish connections between q-product identities and colored partitions, as demonstrated in the works of researchers [3,4,33]. Modern partitioning techniques provide deeper insights into number-based combinatorial structures.

Using the formula from [25], one can calculate the colored partitions of a non-negative integer s into k distinct colors:

$$p_k\left(s\right)={\displaystyle \frac{1}{s!}}\left|\begin{array}{ccccc}k\sigma \left(1\right)& k\sigma \left(2\right)& k\sigma \left(3\right)& \cdots & k\sigma \left(s\right)\\ -1& k\sigma \left(1\right)& k\sigma \left(2\right)& \cdots & k\sigma (s-1)\\ 0& -2& k\sigma \left(1\right)& \cdots & k\sigma (s-2)\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0\cdots & -(s-1)& k\sigma \left(1\right)\end{array}\right|,$$

(3)

where the divisor function, $\sigma \left(s\right)$, is defined as the sum of all positive divisors of s:

$$\sigma \left(s\right)=\sum _{d\mid s}d.$$

(4)

To describe congruence conditions modulo (12), we use the following notation:

$$\begin{array}{c}\hfill {(q^{s\pm };q^p)}_{\infty }={(q^s,q^{s-p};q^p)}_{\infty }\mathrm{for}s<p,s,p\in \mathbb{N}.\end{array}$$

(5)

We introduce a convenient notation to classify parts according to their congruence classes modulo 12. The following superscripts are used to distinguish the corresponding residue classes:

• Parts congruent to $1, 5, 7,$ or $11 (mod 12)$ (equivalently, of the form $12n\pm 1$ or $12n\pm 5$) are labeled with the superscript a.
• Parts congruent to 2 or $10 (mod 12)$ (that is, of the form $12n\pm 2$) are indicated by the superscript b.
• Parts congruent to 3 or $9 (mod 12)$ (corresponding to $12n\pm 3$) are denoted by the superscript c.
• Parts congruent to 4 or $8 (mod 12)$ (i.e., of the form $12n\pm 4$) are represented using the superscript d.
• Parts congruent to $6 (mod 12)$ (that is, of the form $12n\pm 6$) are assigned the superscript g.

Thus, we have:

$$\begin{array}{cc}\hfill & q^a=q^{1\pm },q^{5\pm },q^b=q^{2\pm },q^c=q^{3\pm },q^d=q^{4\pm },\mathrm{and}{q}^g=q^6,\hfill \end{array}$$

(6)

where:

$$\begin{array}{cc}\hfill & {(q^{1\pm };q^{12})}_{\infty }={(q^1,q^{11};q^{12})}_{\infty },{(q^{2\pm };q^{12})}_{\infty }={(q^2,q^{10};q^{12})}_{\infty },\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill & {(q^{3\pm };q^{12})}_{\infty }={(q^3,q^9;q^{12})}_{\infty },{(q^{4\pm };q^{12})}_{\infty }={(q^4,q^8;q^{12})}_{\infty },\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill & {(q^{5\pm };q^{12})}_{\infty }={(q^5,q^7;q^{12})}_{\infty }.\hfill \end{array}$$

(9)

Finally, we make use of the following identities, which play a recurring role in the derivation of our subsequent results.

Lemma 1.

We have

$$\begin{array}{cc}\hfill & {\displaystyle \frac{U_3^3}{U_1}}={\displaystyle \frac{U_4^3{U}_6^2}{U_2^2{U}_{12}}}+q{\displaystyle \frac{U_{12}^3}{U_4}},\hfill \end{array}$$

(10)

$$\begin{array}{c}\hfill {\displaystyle \frac{U_3}{U_1^3}}={\displaystyle \frac{U_4^6{U}_6^3}{U_2^9{U}_{12}^2}}+3q{\displaystyle \frac{U_4^2{U}_6{U}_{12}^2}{U_2^7}}.\end{array}$$

(11)

These identities are due to Baruah and Ojah [34] and Xia and Yao [35] and will be used to extract eta-product identities from the MTF $B\left(q\right)$.

3. Derivation of Identities

This section is devoted to the derivation of level-12 eta-function identities arising from the second-order MTF $B\left(q\right)$.

Our colored partition results are based on the algebraic basis of the identities (12), (13), and (14). These identities emerge from an analytical investigation of MTFs, which occupy a central position in contemporary studies of modular forms, partition theory, and q-series. Originally introduced by Ramanujan, MTFs display remarkable modular behavior and encode substantial arithmetic content within their Fourier coefficients. The primary incentive for examining these identities lies in the observation that certain coefficients of MTFs reveal highly structured patterns and modular transformations when extracted through algebraic methods. We obtain closed formulas, in terms of products of q-shifted factorials, by applying a particular series of coefficients, as in a second-order MTF, to obtain $b(6s+2)$ and $b(6s+4)$. These expressions, when simplified by Lemma 1, lead to the algebraic identities (12), (13), and (14).

Instead of being developed independently, the identities are obtained by modifying well-known mock theta expansions, leading to the discovery of new relations of modular type. In this sense, they serve as an effective bridge connecting analytic techniques in q-series with combinatorial frameworks. In Section 4, these analytic identities are reinterpreted in terms of colored partitions, yielding a clear and natural combinatorial description of the underlying modular phenomena.

$$\begin{array}{cc}\hfill & q{U}_1{U}_2^2{U}_{12}^4+U_1{U}_4^4{U}_6^2-U_2^2{U}_3^3{U}_4{U}_{12}=0.\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill & 3q{U}_1^3{U}_2^2{U}_4^2{U}_6{U}_{12}^4+U_1^3{U}_4^6{U}_6^3-U_2^9{U}_3{U}_{12}^2=0.\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill & 27q^3{U}_1^9{U}_2^6{U}_4^6{U}_6^3{U}_{12}^{12}+27q^2{U}_1^9{U}_2^4{U}_4^{10}{U}_6^5{U}_{12}^8+9q{U}_1^9{U}_2^2{U}_4^{14}{U}_6^7{U}_{12}^4+U_1^9{U}_4^{18}{U}_6^9-\hfill \\ & U_2^{27}{U}_3^3{U}_{12}^6=0.\hfill \end{array}$$

(14)

This is because identities (12), (13), and (14) arise from algebraic rearrangements of the second-order MTF $B\left(q\right)$ and its associated coefficient sequences. Similar techniques have been employed in the literature to extract modular-type identities from MTFs (see [10,34,35]). The novelty here lies in the systematic use of these identities to obtain level-12 eta-product representations that admit direct combinatorial interpretations in terms of colored partitions.

Application of Identity (12) to Function $b(6s+4)$

From (12), we have

$$\begin{array}{c}\hfill q{U}_1{U}_2^2{U}_{12}^4+U_1{U}_4^4{U}_6^2=U_2^2{U}_3^3{U}_4{U}_{12},\end{array}$$

(15)

where

$$\begin{array}{cc}\hfill U_t^s& ={(q^t;q^t)}_{\infty }^s,\hfill \end{array}$$

for positive integers t and s.

By multiplying on both sides of Equation (15) by $\left(9{\displaystyle \frac{U_2^2{U}_3{U}_6}{U_1^8{U}_4{U}_{12}}}\right)$, we obtain

$$\begin{array}{c}\hfill 9q{\displaystyle \frac{U_2^4{U}_3{U}_6{U}_{12}^3}{U_1^7{U}_4}}+9{\displaystyle \frac{U_2^2{U}_3{U}_4^3{U}_6^3}{U_1^7{U}_{12}}}=9{\displaystyle \frac{U_2^4{U}_3^4{U}_6}{U_1^8}}\end{array}$$

(16)

$$\begin{array}{c}\hfill 9{\displaystyle \frac{U_2^4{U}_3{U}_6}{U_1^7}}\left(q{\displaystyle \frac{U_{12}^3}{U_4}}+{\displaystyle \frac{U_4^3{U}_6^2}{U_2^2{U}_{12}}}\right)=9{\displaystyle \frac{U_2^4{U}_3^4{U}_6}{U_1^8}}\end{array}$$

(17)

By using Equation (10) and [10,36], we obtain

$$\begin{array}{cc}\hfill \sum _{s=0}^{\infty }b(6s+4)q^s& =9\left({\displaystyle \frac{U_2^4{U}_3^4{U}_6}{U_1^8}}\right)\hfill \end{array}$$

Application of Identity (13) to Function $b(6s+2)$

From (13), we have

$$\begin{array}{c}\hfill 3q{U}_1^3{U}_2^2{U}_4^2{U}_6{U}_{12}^4+U_1^3{U}_4^6{U}_6^3=U_2^9{U}_3{U}_{12}^2.\end{array}$$

(18)

By multiplying on both sides of Equation (18) by $\left(4{\displaystyle \frac{U_2{U}_3}{U_1^{10}{U}_6{U}_{12}^2}}\right)$, we get

$$\begin{array}{c}\hfill 12q{\displaystyle \frac{U_2^3{U}_3{U}_4^2{U}_{12}^2}{U_1^7}}+4{\displaystyle \frac{U_2{U}_3{U}_4^6{U}_6^2}{U_1^7{U}_{12}^2}}=4{\displaystyle \frac{U_2^{10}{U}_3^2}{U_1^{10}{U}_6}}\end{array}$$

(19)

$$\begin{array}{c}\hfill 4{\displaystyle \frac{U_2^{10}{U}_3}{U_1^7{U}_6}}\left(3q{\displaystyle \frac{U_4^2{U}_6{U}_{12}^2}{U_2^7}}+{\displaystyle \frac{U_4^6{U}_6^3}{U_2^9{U}_{12}^2}}\right)=4{\displaystyle \frac{U_2^{10}{U}_3^2}{U_1^{10}{U}_6}}\end{array}$$

(20)

By using (11) and [10,36], we obtain

$$\begin{array}{cc}\hfill \sum _{s=0}^{\infty }b(6s+2)q^s& =4\left({\displaystyle \frac{U_2^{10}{U}_3^2}{U_1^{10}{U}_6}}\right)\hfill \end{array}$$

Application of Identity (14) to Function $b(6s+4)$

From (14), we have

$$\begin{array}{cc}\hfill & 27q^3{U}_1^9{U}_2^6{U}_4^6{U}_6^3{U}_{12}^{12}+27q^2{U}_1^9{U}_2^4{U}_4^{10}{U}_6^5{U}_{12}^8+9q{U}_1^9{U}_2^2{U}_4^{14}{U}_6^7{U}_{12}^4+\hfill \\ & U_1^9{U}_4^{18}{U}_6^9=U_2^{27}{U}_3^3{U}_{12}^6.\hfill \end{array}$$

(21)

By multiplying on both sides of Equation (21) by $\left(9{\displaystyle \frac{U_3{U}_6}{U_1^8{U}_2^{23}{U}_{12}^6}}\right)$ and taking common $\left(9U_1{U}_2^4{U}_3{U}_6\right)$ from L.H.S, we obtain

$$\begin{array}{c}\hfill 9U_1{U}_2^4{U}_3{U}_6{\left({\displaystyle \frac{U_4^6{U}_6^3}{U_2^9{U}_{12}^2}}+3q{\displaystyle \frac{U_4^2{U}_6{U}_{12}^2}{U_2^7}}\right)}^3=9\left({\displaystyle \frac{U_2^4{U}_3^4{U}_6}{U_1^8}}\right)\end{array}$$

By using Equation (11) and [10,36], we obtain

$$\begin{array}{cc}\hfill \sum _{s=0}^{\infty }b(6s+4)q^s& =9\left({\displaystyle \frac{U_2^4{U}_3^4{U}_6}{U_1^8}}\right)\hfill \end{array}$$

We discuss the notation to be used in the colored partition functions before starting the principal identities. In this section, the terms ${\gamma }_1\left(m\right)$, ${\gamma }_2\left(m\right)$, ${\gamma }_3\left(m\right)$, … represent enumeration functions that count the number of colored partitions of the integer m with some congruence and coloring constraints. The parameters a, b, and c are the residue classes modulo 12 as in (6). These parameters are assigned numerical coefficients that determine the number of different colors of parts that are in the respective residue classes. This description is provided to help the reader interpret the combinatorial implications of the identities that were obtained in this work.

As noted earlier, $12n+a$, $12n+b$, and $12n+c$ are expressions that are known to be integers in particular residue classes modulo 12 as stated in (6).

Remark 1

(q-notation with color indices). Subscripts on variable q are used in the generating functions of this paper to specify the colors allocated to a component (part). Specifically, the notation $q_j$ represents that there are j different colors of the specific part. As an example, the character $q_6$ is the generating variable of parts that can exist in 6 distinct colors. This notation offers an easy method of encoding the multiplicities of colors directly in the q-products expressions, giving the generating functions of colored partitions.

In addition, when rewriting certain $\eta$-product expressions, we adopt the notation $q_0^a$ to indicate that parts of type a do not appear in the corresponding infinite product expansion.

4. Colored Partition Interpretations

This section presents identities associated with level-12 $\eta$-functions and demonstrates how these analytic identities admit direct combinatorial interpretations in terms of colored partitions.

In particular, identity (12) is first translated into a generating function identity for colored partitions, leading directly to the initial combinatorial result stated below.

Theorem 1.

Let $s,t\in \mathbb{N}$. Define ${\gamma }_1\left(s\right)$ as the number of partitions of s into parts of the form $12t+c$, $12t+d$, and $12t+g$, each appearing in two colors. Similarly, let ${\gamma }_2\left(s\right)$ count the partitions of s into parts of the form $12t+b$, $12t+c$, and $12t+g$, also appearing in two colors. Finally, let ${\gamma }_3\left(s\right)$ denote the number of partitions of s into parts of the form $12t+a$ and $12t+b$ in one color, while $12t+d$ appears in two colors.

Therefore, for all $s\ge 1$, the identity takes the form:

$${\gamma }_1(s-1)+{\gamma }_2\left(s\right)-{\gamma }_3\left(s\right)=0.$$

Proof. 

Dividing Equation (12) by $U_1^7$, we obtain:

$$\begin{array}{c}\hfill q{\displaystyle \frac{U_1{U}_2^2{U}_{12}^4}{U_1^7}}+{\displaystyle \frac{U_1{U}_4^4{U}_6^2}{U_1^7}}-{\displaystyle \frac{U_2^2{U}_3^3{U}_4{U}_{12}}{U_1^7}}=0.\end{array}$$

(22)

By considering the expression ${\displaystyle \frac{U_2^2{U}_{12}^4}{U_1^6}}$ from Equations (5) and (6), we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{U_2^2{U}_{12}^4}{U_1^6}}& ={\displaystyle \frac{1}{{(q_6^{1\pm ,5\pm },q_4^{2\pm },q_6^{3\pm },q_4^{4\pm },q_4^6;q^{12})}_{\infty }}}={\displaystyle \frac{1}{{(q_6^a,q_4^b,q_6^c,q_4^d,q_4^g;q^{12})}_{\infty }}}.\hfill \end{array}$$

Similarly,

$$\begin{array}{cc}\hfill {\displaystyle \frac{U_4^4{U}_6^2}{U_1^6}}& ={\displaystyle \frac{1}{{(q_6^a,q_6^b,q_6^c,q_2^d,q_4^g;q^{12})}_{\infty }}},\hfill \\ \hfill {\displaystyle \frac{U_2^2{U}_3^3{U}_4{U}_{12}}{U_1^7}}& ={\displaystyle \frac{1}{{(q_7^a,q_5^b,q_4^c,q_4^d,q_2^g;q^{12})}_{\infty }}}.\hfill \end{array}$$

Substituting these expressions into (22), we obtain:

$$\begin{array}{c}\hfill {\displaystyle \frac{q}{{(q_6^a,q_4^b,q_6^c,q_4^d,q_4^g;q^{12})}_{\infty }}}+{\displaystyle \frac{1}{{(q_6^a,q_6^b,q_6^c,q_2^d,q_4^g;q^{12})}_{\infty }}}-{\displaystyle \frac{1}{{(q_7^a,q_5^b,q_4^c,q_4^d,q_2^g;q^{12})}_{\infty }}}=0.\end{array}$$

Multiplying by ${(q_6^a,q_4^b,q_4^c,q_2^d,q_2^g;q^{12})}_{\infty }$ yields:

$$\begin{array}{c}\hfill {\displaystyle \frac{q}{{(q_0^a,q_0^b,q_2^c,q_2^d,q_2^g;q^{12})}_{\infty }}}+{\displaystyle \frac{1}{{(q_0^a,q_2^b,q_2^c,q_0^d,q_2^g;q^{12})}_{\infty }}}-{\displaystyle \frac{1}{{(q_1^a,q_1^b,q_0^c,q_2^d,q_0^g;q^{12})}_{\infty }}}=0.\end{array}$$

(23)

We recognize Equation (23) as generating functions for ${\gamma }_i\left(s\right)$, where $i=1,2,3$, leading to the identity:

$$\begin{array}{c}\hfill \sum _{s=0}^{\infty }\left({\gamma }_1\left(s\right)q^{s+1}+{\gamma }_2\left(s\right)q^s-{\gamma }_3\left(s\right)q^s\right)=0.\end{array}$$

Setting ${\gamma }_i\left(0\right)=1$, where $i=1,2,3,$ we extract coefficients of $q^s$ to obtain the desired result. □

Example 1.

For $s=1$, the following table verifies the theorem:

${\gamma }_1\left(0\right)=1$	${\gamma }_2\left(1\right)=0$
${\gamma }_3\left(1\right)=1_r$

Using a similar approach, identity (13) is employed to derive another colored partition relation involving different color multiplicities, leading to the following theorem.

Theorem 2.

Let $s,t\in \mathbb{N}$. Define ${\gamma }_1\left(s\right)$ as the total number of partitions of s such that every part has the form $12t+b$ and $12t+g$ with four colors, and $12t+d$ with two colors. Similarly, let ${\gamma }_2\left(s\right)$ denote the partitions of s where each part is of the form $12t+b$ with six colors and $12t+g$ with four colors. Finally, let ${\gamma }_3\left(s\right)$ represent the partitions of s into parts of the form $12t+a$ with three colors and $12t+c$ with two colors.

Then, for $s\ge 1$, we have

$$3{\gamma }_1(s-1)+{\gamma }_2\left(s\right)-{\gamma }_3\left(s\right)=0.$$

Proof. 

Dividing Equation (13) by $U_1^{12}$ and applying a similar approach as in the previous theorem, we obtain the desired result:

$$\begin{array}{cc}\hfill & 3{\displaystyle \frac{q}{{(q_9^a,q_7^b,q_9^c,q_5^d,q_6^g;q^{12})}_{\infty }}}+{\displaystyle \frac{1}{{(q_9^a,q_9^b,q_9^c,q_3^d,q_6^g;q^{12})}_{\infty }}}-{\displaystyle \frac{1}{{(q_{12}^a,q_3^b,q_{11}^c,q_3^d,q_2^g;q^{12})}_{\infty }}}=0.\hfill \end{array}$$

Multiplying the above equation by the common factor ${(q_9^a,q_3^b,q_9^c,q_3^d,q_2^g;q^{12})}_{\infty }$, we obtain:

$$\begin{array}{cc}\hfill & 3{\displaystyle \frac{q}{{(q_0^a,q_4^b,q_0^c,q_2^d,q_4^g;q^{12})}_{\infty }}}+{\displaystyle \frac{1}{{(q_0^a,q_6^b,q_0^c,q_0^d,q_4^g;q^{12})}_{\infty }}}-{\displaystyle \frac{1}{{(q_3^a,q_0^b,q_2^c,q_0^d,q_0^g;q^{12})}_{\infty }}}=0.\hfill \end{array}$$

(24)

From Equation (24), we identify the generating functions for ${\gamma }_i\left(s\right)$, where $i=1,2,3$. Hence, the identity is equivalent to:

$$\sum _{s=0}^{\infty }\left(3{\gamma }_1\left(s\right)q^{s+1}+{\gamma }_2\left(s\right)q^s-{\gamma }_3\left(s\right)q^s\right)=0.$$

Using initial results ${\gamma }_i\left(0\right)=1$, where $i=1,2,3$, the result follows by determining coefficients of $q^s$ in the above equation. □

Example 2.

For $s=1$, the following table verifies the theorem:

${\gamma }_1\left(0\right)=1$:

${\gamma }_2\left(1\right)=0$:

${\gamma }_3\left(1\right)=3$:    $1_r,1_b,1_g.$

Finally, identity (14), which involves higher powers of q and eta products, yields a more intricate colored partition identity with multiple shifted terms, as stated below.

Theorem 3.

Let $s,t\in \mathbb{N}$. Define ${\gamma }_1\left(s\right)$ to count partitions of s where parts are of the form $12t+b$ and $12t+g$ with 12 colors, and $12t+d$ with 6 colors. Define ${\gamma }_2\left(s\right)$ to count partitions of s where parts are of the form $12t+b$ with 14 colors, $12t+d$ with 4 colors, and $12t+g$ with 12 colors. Let ${\gamma }_3\left(s\right)$ count partitions where parts are of the form $12t+b$ with 16 colors, $12t+d$ with 2 colors, and $12t+g$ with 12 colors. Furthermore, let ${\gamma }_4\left(s\right)$ denote partitions where parts are of the form $12t+b$ with 18 colors and $12t+g$ with 12 colors, and ${\gamma }_5\left(s\right)$ count partitions of s where parts are of the form $12t+a$ with 9 colors and $12t+c$ with 6 colors.

Then, the following identity holds:

$$27{\gamma }_1(s-3)+27{\gamma }_2(s-2)+9{\gamma }_3(s-1)+{\gamma }_4\left(s\right)-{\gamma }_5\left(s\right)=0,fors\ge 3.$$

Proof. 

The result follows by dividing Equation (14) by $U_1^{36}$.

$$\begin{array}{cc}\hfill & 27q^3{\displaystyle \frac{U_1^9{U}_2^6{U}_4^6{U}_6^3{U}_{12}^{12}}{U_1^{36}}}+27q^2{\displaystyle \frac{U_1^9{U}_2^4{U}_4^{10}{U}_6^5{U}_{12}^8}{U_1^{36}}}+9q{\displaystyle \frac{U_1^9{U}_2^2{U}_4^{14}{U}_6^7{U}_{12}^4}{U_1^{36}}}+{\displaystyle \frac{U_1^9{U}_4^{18}{U}_6^9}{U_1^{36}}}-\hfill \\ & {\displaystyle \frac{U_2^{27}{U}_3^3{U}_{12}^6}{U_1^{36}}}=0.\hfill \end{array}$$

(25)

By applying Equations (5) and (6) to (25), we obtain:

$$\begin{array}{cc}\hfill & 27{\displaystyle \frac{q^3}{{(q_{27}^a,q_{21}^b,q_{27}^c,q_{15}^d,q_{18}^g;q^{12})}_{\infty }}}+27{\displaystyle \frac{q^2}{{(q_{27}^a,q_{23}^b,q_{27}^c,q_{13}^d,q_{18}^g;q^{12})}_{\infty }}}+9{\displaystyle \frac{q}{{(q_{27}^a,q_{25}^b,q_{27}^c,q_{11}^d,q_{18}^g;q^{12})}_{\infty }}}\hfill \\ \hfill & +{\displaystyle \frac{1}{{(q_{27}^a,q_{27}^b,q_{27}^c,q_9^d,q_{18}^g;q^{12})}_{\infty }}}-{\displaystyle \frac{1}{{(q_{36}^a,q_9^b,q_{33}^c,q_9^d,q_6^g;q^{12})}_{\infty }}}=0.\hfill \end{array}$$

Multiplying both sides by the common factor ${(q_{27}^a,q_9^b,q_{27}^c,q_9^d,q_6^g;q^{12})}_{\infty }$, we obtain:

$$\begin{array}{cc}\hfill & 27{\displaystyle \frac{q^3}{{(q_0^a,q_{12}^b,q_0^c,q_6^d,q_{12}^g;q^{12})}_{\infty }}}+27{\displaystyle \frac{q^2}{{(q_0^a,q_{14}^b,q_0^c,q_4^d,q_{12}^g;q^{12})}_{\infty }}}+9{\displaystyle \frac{q}{{(q_0^a,q_{16}^b,q_0^c,q_2^d,q_{12}^g;q^{12})}_{\infty }}}\hfill \\ & +{\displaystyle \frac{1}{{(q_0^a,q_{18}^b,q_0^c,q_0^d,q_{12}^g;q^{12})}_{\infty }}}-{\displaystyle \frac{1}{{(q_9^a,q_0^b,q_6^c,q_0^d,q_0^g;q^{12})}_{\infty }}}=0.\hfill \end{array}$$

(26)

Equation (26) provides the generating functions for ${\gamma }_i\left(s\right)$, where $i=1,2,3,4,5$. Therefore, the identity takes the form:

$$\begin{array}{c}\hfill \sum _{s=0}^{\infty }\left(27{\gamma }_1\left(s\right)q^{s+3}+27{\gamma }_2\left(s\right)q^{s+2}+9{\gamma }_3\left(s\right)q^{s+1}+{\gamma }_4\left(s\right)q^s-{\gamma }_5\left(s\right)q^s\right)=0.\end{array}$$

By choosing the initial results:

$${\gamma }_i\left(0\right)=1,\mathrm{where}i=1,2,3,4,5.$$

□

Remark 2.

The techniques developed in this paper are not restricted to level-12 identities. The same type of eta-product manipulation can be done on identities of other levels of MTFs or Ramanujan-type q-series. Specifically, the technique can be generalized to find colored partition interpretations of higher-level eta-function identities, which provides an early indication of a fruitful way forward in future research.

5. Conclusions

We have studied level-12 identities of the Dedekind $\eta$-function and have found their direct connections with colored partition theory. Using algebraic transformations of the second-order MTF $B\left(q\right)$, we were able to derive some structured eta-products identities and put them into the form of structured combinatorial interpretations in terms of colored partitions with given congruence properties. The principal accomplishment of the work is the filling in of analytic identities of q-series with partition structure with combinatorics, adding to both the theory of modular forms and colored partitions. The constructed identities illustrate the ability of modular-type relations to represent fined enumerative information.

Further research can be done on generalizing these methods to higher levels of eta-function identities, understanding similar outcomes with other MTFs, and coming up with techniques that can efficiently produce colored partition data of such identities. These indications can extend the relationship between modular forms, q-series, and combinatorial number theory.