Shuffle Product Formulas and Combinatorial Identities

Abstract

We study shuffle product structures on words in three letters, extending the classical framework of multiple zeta values. Using an evaluation map that relates admissible words to iterated integrals, we translate shuffle identities into combinatorial identities. This yields a general multinomial-type formula and a symmetric decomposition of binomial coefficients. We also establish two new shuffle product formulas, leading to further combinatorial identities and revealing a structural connection between shuffle algebra and combinatorics.

1. Introduction

For a given positive integer r, we define the multiple zeta values (MZVs) by [1,2,3]

$$\zeta ({\alpha }_1,\dots ,{\alpha }_r)=\sum _{1\le n_1<n_2<\cdots <n_r}{\displaystyle \frac{1}{n_1^{{\alpha }_1}{n}_2^{{\alpha }_2}\cdots n_r^{{\alpha }_r}}},$$

where ${\alpha }_1,{\alpha }_2,\dots ,{\alpha }_r$ are positive integers and ${\alpha }_r\ge 2$ to ensure convergence. We refer to such convergent multi-indices $\mathit{\alpha }=({\alpha }_1,\dots ,{\alpha }_r)$ as admissible indices. The quantity $\left|\mathit{\alpha }\right|={\alpha }_1+{\alpha }_2+\cdots +{\alpha }_r$ is called the weight, and $\ell \left(\mathit{\alpha }\right)=r$ is called the depth. Throughout the paper, we denote by $\left|\mathit{\alpha }\right|$ and $\ell \left(\mathit{\alpha }\right)$ the weight and the depth of a multi-index $\mathit{\alpha }$, respectively. We use the shorthand notation ${\left\{a\right\}}^k$ for k repetitions of a; for instance, $\zeta ({\left\{2\right\}}^3,5)=\zeta (2,2,2,5)$.

These objects admit both series and iterated integral representations, which are connected through shuffle product structures [1,4,5,6,7]. These perspectives have played a fundamental role in understanding the structural relations of multiple zeta values. The study of such algebraic structures, including shuffle and stuffle products, has led to numerous identities in the theory of multiple zeta values (see, e.g., [8,9]). Moreover, these structures also appear in broader contexts such as mathematical physics and knot theory (see, e.g., [10,11]). Moreover, shuffle identities provide a natural bridge between algebraic expressions and combinatorial formulas [12,13,14].

We introduce a new family of “zeta-like” series, referred to as the multiple $\rho$-values (see [15,16,17]) and denoted by $\rho ({\alpha }_1+1,\dots ,{\alpha }_r+1)$, defined by

$$\rho ({\alpha }_1+1,\dots ,{\alpha }_r+1)=\sum _{1\le n_1<\cdots <n_r}{\displaystyle \frac{1}{{\left(n_1\right)}_{{\alpha }_1+1}{(n_2+{\alpha }_1)}_{{\alpha }_2+1}\cdots {(n_r+{\alpha }_1+\cdots +{\alpha }_{r-1})}_{{\alpha }_r+1}}},$$

where the rising factorial is defined by ${\left(x\right)}_0=1$ and ${\left(x\right)}_n=x(x+1)\cdots (x+n-1)$ for $n\in \mathbb{N}$. Here, ${\alpha }_r\ge 1$ is required to guarantee convergence.

These families naturally arise from iterated integrals involving the differential forms considered in this paper.

In this paper, we extend the classical shuffle algebra by introducing an additional letter z, corresponding to the regular differential form $dt$. This extension enlarges the class of admissible words and reveals new structures beyond the classical multiple zeta value setting. Within this framework, we define an evaluation map Z that assigns to each admissible word an iterated integral, thereby providing a bridge between algebraic expressions and explicit numerical or combinatorial quantities.

The introduction of the map Z is not merely a formal extension of the classical setting. Similar evaluation maps have appeared in the literature in the context of shuffle algebras on two letters, mainly in connection with multiple zeta values [1,12,18,19]. These constructions are closely related to iterated integral representations and their algebraic structures [4,7].

The present work extends this framework by incorporating an additional letter z, which enables a direct connection between shuffle product structures and explicit combinatorial identities.

Although some of the resulting combinatorial identities may be proved by direct or elementary methods, the approach via shuffle products and the map Z provides a systematic and conceptual framework that explains their structure and allows for generalizations.

Moreover, the inclusion of the letter z leads to new classes of zeta-like values, such as the $\mathcal{E}$-values, which do not arise in the classical theory. These values naturally extend previously studied $\rho$-type values and fit into the same unified framework.

Our first main contribution is to establish a systematic procedure for converting shuffle product identities into combinatorial identities via the evaluation map Z. In particular, by applying Z to the identities in Equations (4) and (7), we derive explicit combinatorial formulas involving binomial and multinomial coefficients. This provides a unified combinatorial interpretation of classical shuffle identities (see Theorems 1 and 2).

Our second contribution is the derivation of two new shuffle product formulas. The first describes insertion processes within words and leads to explicit combinatorial expansions, while the second provides a global decomposition of shuffle products in terms of permutations and multinomial distributions. These results extend the shuffle framework and reveal new structural features of the algebra.

Our third contribution is the application of these new shuffle formulas to derive nontrivial combinatorial identities. In particular, we obtain symmetric decompositions of binomial coefficients and general summation formulas involving multiple parameters. These identities further illustrate that shuffle algebra provides a powerful framework for generating and interpreting combinatorial structures.

Our approach leads to explicit combinatorial identities involving nonnegative integer parameters. For example, we obtain the general formula

$$\sum _{p_1+\cdots +p_s=p}{\displaystyle \frac{{\prod }_{i=1}^s\left(\genfrac{}{}{0pt}{}{a_i+p_i}{p_i}\right)}{{\prod }_{k=2}^s((a_k+p_k)+\cdots +(a_s+p_s))}}={\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{a_1+\cdots +a_s+p}{p}\right)}{{\prod }_{k=2}^s(a_k+\cdots +a_s)}},$$

as well as the following symmetric decomposition of binomial coefficients:

$$\left({\displaystyle \genfrac{}{}{0pt}{}{p_1+p_2+q_1+q_2}{p_1+q_1}}\right)=\sum _{k=0}^{p_2}\left({\displaystyle \genfrac{}{}{0pt}{}{p_1+k}{p_1}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{q_1+q_2+p_2-k-1}{q_1-1}}\right)+\sum _{k=0}^{p_1}\left({\displaystyle \genfrac{}{}{0pt}{}{p_2+k}{p_2}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{q_1+q_2+p_1-k-1}{q_2-1}}\right).$$

The first identity yields a general multinomial-type formula, while the second yields a symmetric decomposition of binomial coefficients (see Theorem 5 and Proposition 1).

These results reveal a structural mechanism by which shuffle algebra encodes combinatorial identities.

The paper is organized as follows. In Section 2, we introduce the algebraic setting and the evaluation map Z. In Section 3, we derive combinatorial identities from known shuffle product formulas. Section 4 is devoted to the proof of two new shuffle product formulas. In Section 5, we apply these formulas to obtain new combinatorial identities. Finally, Section 6 concludes the paper with a discussion of further directions.

2. Algebraic Framework and the Evaluation Map

Let $\mathbb{Q}⟨x,y,z⟩$ be the $\mathbb{Q}$-algebra of polynomials in three non-commutative variables, graded by degree, where x, y, and z are each assigned a degree of 1. The algebra $\mathbb{Q}⟨x,y,z⟩$ is identified with the graded $\mathbb{Q}$-vector space $\mathfrak{H}$, which is spanned by the monomials in the variables x, y, and z (see [1,12,20,21]).

Let ${\mathfrak{H}}^0$ be the subalgebra of $\mathfrak{H}$ generated by words not beginning with y and not ending with x. The words in ${\mathfrak{H}}^0$ are called admissible words.

Let $Z:{\mathfrak{H}}^0\to \mathbb{R}$ be the $\mathbb{Q}$-linear map defined by

$$Z(u_1{u}_2\cdots u_k)={\int }_{0<t_1<\cdots <t_k<1}{\omega }_{u_1}\left(t_1\right)\cdots {\omega }_{u_k}\left(t_k\right),$$

(1)

where $u_i\in \{x,y,z\}$ and

$${\omega }_x\left(t\right)={\displaystyle \frac{dt}{1-t}},{\omega }_y\left(t\right)={\displaystyle \frac{dt}{t}},{\omega }_z\left(t\right)=dt.$$

Since the word $u_1{u}_2\cdots u_k$ is admissible, the integral is convergent. For notational convenience, powers such as $y^a$ indicate repeated concatenation of the letter y, and the corresponding iterated integral follows the order of the letters in the word. The classical multiple zeta values are recovered as follows:

$$Z(x{y}^{a_1}x{y}^{a_2}\cdots x{y}^{a_r})=\zeta (a_1+1,a_2+1,\dots ,a_r+1).$$

We also obtain evaluations corresponding to known extensions such as $\rho$-values:

$$Z(x{z}^{a_1}x{z}^{a_2}\cdots x{z}^{a_r})=\rho (a_1+1,a_2+1,\dots ,a_r+1).$$

The inclusion of the letter z further leads to new types of evaluations that do not arise in the classical multiple zeta value setting:

$$Z(z{y}^{a_1}z{y}^{a_2}\cdots z{y}^{a_r})={\displaystyle \frac{1}{1^{a_1+1}{2}^{a_2+1}\cdots r^{a_r+1}}}.$$

Motivated by the above observations, we introduce a new class of zeta-like values, which naturally arise from the extended shuffle algebra and generalize the $\rho$-type values.

$$\begin{array}{c}\mathcal{E}(a_0,a_1,\dots ,a_r):=Z(z^{a_0}x{z}^{a_1}\cdots x{z}^{a_r})\hfill \\ :={\displaystyle \frac{1}{a_0!}}\sum _{1\le n_1<n_2<\cdots <n_r}{\displaystyle \frac{1}{{(n_1+a_0)}_{a_1+1}{(n_2+a_0+a_1)}_{a_2+1}\cdots {(n_r+a_0+\cdots +a_{r-1})}_{a_r+1}}}.\hfill \end{array}$$

(2)

In particular, when $a_0=0$, we recover the $\rho$-values:

$$\mathcal{E}(0,a_1,\dots ,a_r)=\rho (a_1+1,\dots ,a_r+1).$$

The value of $\mathcal{E}(a_0,\dots ,a_r)$ can be explicitly evaluated via the change of variables $t=1-u$ in the iterated integral representation, which transforms

$$Z(z^{a_0}x{z}^{a_1}\cdots x{z}^{a_r})\mathrm{into}Z(z^{a_r}y{z}^{a_{r-1}}\cdots y{z}^{a_0}).$$

Hence,

$$\mathcal{E}(a_0,a_1,\dots ,a_r)={\displaystyle \frac{1}{\left|\mathit{a}\right|!{\prod }_{k=1}^r(a_k+\cdots +a_r)}}.$$

(3)

This explicit evaluation will be used in Section 3 to derive combinatorial identities.

Let us define the bilinear product ш (the shuffle product) on $\mathfrak{H}$ by the rules

$$1шw=wш1=w,$$

for any word w, and

$$u_1{w}_1шu_2{w}_2=u_1(w_1шu_2{w}_2)+u_2(u_1{w}_1шw_2),$$

for any words $w_1$, $w_2$, any letters $u_i=x$, y, or z ($i=1,2$), and then extend the above rules to the whole algebra $\mathfrak{H}$ by linearity. It is well known that the shuffle product is commutative and associative [19,22]. We denote the algebras $({\mathfrak{H}}^0,+,ш)$ by ${\mathfrak{H}}_{ш}^0$. By the standard shuffle product identity of iterated integrals, the evaluation map Z is again an algebra homomorphism for the multiplication ш (see [19]):

$$Z(w_1шw_2)=Z\left(w_1\right)Z\left(w_2\right).$$

We recall two useful identities in the shuffle algebra $(\mathfrak{H},+,ш)$: the first one is that for any nonnegative integers $a,b,c$, and d, we have ([20] Euqation (4.4)):

$$\begin{array}{cc}\hfill u_1^{a+1}{u}_2^c& ш u_1^{b+1}{u}_2^d=\sum _{p+q=a+b}\left({\displaystyle \genfrac{}{}{0pt}{}{p}{a}}\right)u_1^{p+1}\sum _{\left|\mathit{\alpha }\right|=c+d}{u}_2^{{\alpha }_0}{u}_1{u}_2^{{\alpha }_1}\cdots u_1{u}_2^{{\alpha }_{q+1}}\left({\displaystyle \genfrac{}{}{0pt}{}{{\alpha }_{q+1}}{d}}\right)\hfill \\ \hfill & +\sum _{p+q=a+b}\left({\displaystyle \genfrac{}{}{0pt}{}{p}{b}}\right)u_1^{p+1}\sum _{\left|\mathit{\alpha }\right|=c+d}{u}_2^{{\alpha }_0}{u}_1{u}_2^{{\alpha }_1}\cdots u_1{u}_2^{{\alpha }_{q+1}}\left({\displaystyle \genfrac{}{}{0pt}{}{{\alpha }_{q+1}}{c}}\right),\hfill \end{array}$$

(4)

where $u_1,u_2\in \{x,y,z\}$ are distinct letters. Throughout this paper, all summation indices are assumed to be nonnegative integers. For example, we use the notation ${\sum }_{a_1+a_2=a}$ to indicate ${\sum }_{{\displaystyle \genfrac{}{}{0pt}{}{a_1+a_2=a}{a_i\ge 0}}}$.

Let $u_1=x,u_2=y$ and $a=b=0$, $c=p,d=q$ in the above formula, we have

$$x{y}^pшx{y}^q=\sum _{{\displaystyle \genfrac{}{}{0pt}{}{{\alpha }_1+{\alpha }_2=p+q}{{\alpha }_i\ge 0}}}\left[\left({\displaystyle \genfrac{}{}{0pt}{}{{\alpha }_2}{p}}\right)+\left({\displaystyle \genfrac{}{}{0pt}{}{{\alpha }_2}{q}}\right)\right]x{y}^{{\alpha }_1}x{y}^{{\alpha }_2}.$$

(5)

This gives Euler’s decomposition formula (ref. [21])

$$\zeta (p+1)\zeta (q+1)=\sum _{{\displaystyle \genfrac{}{}{0pt}{}{{\alpha }_1+{\alpha }_2=p+q+1}{{\alpha }_i\ge 1}}}\left[\left({\displaystyle \genfrac{}{}{0pt}{}{{\alpha }_2}{p}}\right)+\left({\displaystyle \genfrac{}{}{0pt}{}{{\alpha }_2}{q}}\right)\right]\zeta ({\alpha }_1,{\alpha }_2+1).$$

(6)

Here, when applying the map Z, the admissibility conditions are taken into account in the standard way, leading to a slight adjustment of the parameters in the resulting expression.

Another useful identity is stated as follows.

Let $\mathit{a}=(a_1,\dots ,a_n)$ be an n-tuple of nonnegative integers with $\left|\mathit{a}\right|=w$, and let $u_1,u_2\in \{x,y,z\}$ be distinct letters (see ([20] Equation (4.6))). Then

$$\begin{array}{cc}\hfill & u_1{u}_2^{a_1}шu_1{u}_2^{a_2}ш\cdots шu_1{u}_2^{a_n}\hfill \\ \hfill & =\sum _{\left|\mathit{d}\right|=w}(u_1{u}_2^{d_1})(u_1{u}_2^{d_2})\cdots (u_1{u}_2^{d_n})\sum _{\sigma \in S_n}{\sigma }_{\mathit{a}}\left\{\prod _{j=2}^n\left({\displaystyle \genfrac{}{}{0pt}{}{{\sum }_{k=j}^n{d}_k-{\sum }_{k=j+1}^n{a}_k}{a_j}}\right)\right\}.\hfill \end{array}$$

(7)

Here $S_n$ denotes the symmetric group on n elements. We write permutations in one-line notation, i.e.,

$$\sigma =\left(\sigma \right(1),\dots ,\sigma (n\left)\right).$$

For $\mathit{a}=(a_1,\dots ,a_n)$, we define the action of $\sigma$ by

$${\sigma }_{\mathit{a}}=(a_{\sigma \left(1\right)},\dots ,a_{\sigma \left(n\right)}).$$

The corresponding formula for alternating multiple zeta values was obtained in [20] (Theorem 4.2). These identities will serve as key tools for deriving the general shuffle formulas in Section 3.

3. Combinatorial Identities from Shuffle Product Formulas

Within our algebraic framework, shuffle product identities can be systematically translated into combinatorial identities via the evaluation map Z.

We now derive explicit formulas arising from the shuffle product relations in Section 2.

Theorem 1 

(Two-variable case). Let $a,b$ be nonnegative integers and $c,d$ be positive integers. Then

$$\begin{array}{cc}\hfill \rho ({\left\{1\right\}}^a,& c+1)\rho ({\left\{1\right\}}^b,d+1)\hfill \\ \hfill & =\sum _{p+q=a+b}\left({\displaystyle \genfrac{}{}{0pt}{}{p}{a}}\right)\sum _{\left|\mathit{\alpha }\right|=c+d-1}\rho ({\left\{1\right\}}^p,{\alpha }_0+1,\dots ,{\alpha }_q+1,{\alpha }_{q+1}+2)\left({\displaystyle \genfrac{}{}{0pt}{}{{\alpha }_{q+1}+1}{d}}\right)\hfill \\ \hfill & +\sum _{p+q=a+b}\left({\displaystyle \genfrac{}{}{0pt}{}{p}{b}}\right)\sum _{\left|\mathit{\alpha }\right|=c+d-1}\rho ({\left\{1\right\}}^p,{\alpha }_0+1,\dots ,{\alpha }_q+1,{\alpha }_{q+1}+2)\left({\displaystyle \genfrac{}{}{0pt}{}{{\alpha }_{q+1}+1}{c}}\right).\hfill \end{array}$$

(8)

Proof. 

By substituting $u_1=x$ and $u_2=z$ into Equation (4) and applying the map Z, we obtain the corresponding identity in terms of iterated integrals. The admissibility conditions are taken into account in the standard way, leading to a slight adjustment of the parameters in the resulting expression. □

Substituting the explicit values of the $\rho$-functions yields

$$\begin{array}{cc}\hfill \left({\displaystyle \genfrac{}{}{0pt}{}{c+d}{c}}\right){\left({\displaystyle \frac{1}{c}}+{\displaystyle \frac{1}{d}}\right)}^{a+b+2}{c}^{b+1}{d}^{a+1}& =\sum _{\begin{array}{c}p+q=a+b\\ {\alpha }_0+\cdots +{\alpha }_{q+1}=c+d-1\end{array}}{\displaystyle \frac{{(c+d)}^{q+1}\left(\genfrac{}{}{0pt}{}{p}{a}\right)\left(\genfrac{}{}{0pt}{}{{\alpha }_{q+1}+1}{d}\right)}{{\prod }_{k=1}^{q+1}\left({\alpha }_k+{\alpha }_{k+1}+\cdots +{\alpha }_{q+1}+1\right)}}\hfill \\ \hfill & +\sum _{\begin{array}{c}p+q=a+b\\ {\alpha }_0+\cdots +{\alpha }_{q+1}=c+d-1\end{array}}{\displaystyle \frac{{(c+d)}^{q+1}\left(\genfrac{}{}{0pt}{}{p}{b}\right)\left(\genfrac{}{}{0pt}{}{{\alpha }_{q+1}+1}{c}\right)}{{\prod }_{k=1}^{q+1}\left({\alpha }_k+{\alpha }_{k+1}+\cdots +{\alpha }_{q+1}+1\right)}}.\hfill \end{array}$$

(9)

As a special case, when $a=b=0$, we obtain

$$\left({\displaystyle \genfrac{}{}{0pt}{}{c+d}{c}}\right)\left({\displaystyle \frac{1}{c}}+{\displaystyle \frac{1}{d}}\right)=\sum _{{\alpha }_0+{\alpha }_1=c+d-1}{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{{\alpha }_1+1}{d}\right)+\left(\genfrac{}{}{0pt}{}{{\alpha }_1+1}{c}\right)}{{\alpha }_1+1}}.$$

(10)

As another special case, taking $a=1$ and $b=0$, we obtain

$$\begin{array}{cc}\hfill \left({\displaystyle \genfrac{}{}{0pt}{}{c+d}{c}}\right){\left({\displaystyle \frac{1}{c}}+{\displaystyle \frac{1}{d}}\right)}^{3}c{d}^2& =(c+d)\sum _{{\alpha }_0+{\alpha }_1=c+d-1}{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{{\alpha }_1+1}{d}\right)+\left(\genfrac{}{}{0pt}{}{{\alpha }_1+1}{c}\right)}{{\alpha }_1+1}}\hfill \\ \hfill & +{(c+d)}^2\sum _{{\alpha }_0+{\alpha }_1+{\alpha }_2=c+d-1}{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{{\alpha }_2+1}{c}\right)}{({\alpha }_1+{\alpha }_2+1)({\alpha }_2+1)}}.\hfill \end{array}$$

Applying Equation (10), we obtain

$$\sum _{{\alpha }_0+{\alpha }_1+{\alpha }_2=c+d-1}{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{{\alpha }_2+1}{c}\right)}{({\alpha }_1+{\alpha }_2+1)({\alpha }_2+1)}}={\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{c+d}{c}\right)}{c^2}}.$$

(11)

Substitute $u_1=z$ and $u_2=y$ in Equation (4), and apply Z map we obtain the following identity.

$$\left({\displaystyle \genfrac{}{}{0pt}{}{a+b+2}{a+1}}\right){\displaystyle \frac{1}{{(a+1)}^c{(b+1)}^d}}=\sum _{\begin{array}{c}p+q=a+b\\ {\alpha }_0+\cdots +{\alpha }_{q+1}=c+d\end{array}}{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{p}{a}\right)\left(\genfrac{}{}{0pt}{}{{\alpha }_{q+1}}{d}\right)+\left(\genfrac{}{}{0pt}{}{p}{b}\right)\left(\genfrac{}{}{0pt}{}{{\alpha }_{q+1}}{c}\right)}{{\prod }_{k=0}^{q+1}{(p+k+1)}^{{\alpha }_k}}}.$$

Theorem 2 

(n-variable case). For an n-tuple $\mathit{a}=(a_1,a_2,\dots ,a_n)$ of positive integers with $\left|\mathit{a}\right|=w$, we get

$$\begin{array}{cc}\hfill & \rho (a_1+1)\rho (a_2+1)\cdots \rho (a_n+1)\hfill \\ \hfill & =\sum _{\left|\mathit{d}\right|=w-1}\rho (d_1+1,\dots ,d_{n-1}+1,d_n+2)\sum _{\sigma \in S_n}{\sigma }_{\mathit{a}}\left\{\prod _{j=2}^n\left({\displaystyle \genfrac{}{}{0pt}{}{1+{\sum }_{k=j}^n{d}_k-{\sum }_{k=j+1}^n{a}_k}{a_j}}\right)\right\},\hfill \end{array}$$

(12)

where $S_n$ is the symmetric group on n elements, and ${\sigma }_{\mathit{a}}$ denotes the permutation of $(a_1,\dots ,a_n)$ induced by $\sigma \in S_n$.

Proof. 

Let $u_1=x$ and $u_2=z$ in Equation (7). Applying the map Z and using the same admissibility considerations as above, we obtain the desired formula. □

Using the explicit values of the multiple $\rho$ values, we obtain

$$\left({\displaystyle \genfrac{}{}{0pt}{}{w}{a_1,\dots ,a_n}}\right){\displaystyle \frac{w}{a_1{a}_2\cdots a_n}}=\sum _{\left|\mathit{d}\right|=w-1}{\displaystyle \frac{{\sum }_{\sigma \in S_n}{\sigma }_{\mathit{a}}\left\{{\prod }_{j=2}^n(\genfrac{}{}{0pt}{}{1+{\sum }_{k=j}^n{d}_k-{\sum }_{k=j+1}^n{a}_k}{a_j})\right\}}{{\prod }_{i=2}^n(d_i+d_{i+1}+\cdots +d_n+1)}}.$$

The case $n=2$ gives Equation (10). When $n=3$ the formula becomes

$$\left({\displaystyle \genfrac{}{}{0pt}{}{a_1+a_2+a_3}{a_1,a_2,a_3}}\right){\displaystyle \frac{a_1+a_2+a_3}{a_1{a}_2{a}_3}}=\sum _{d_1+d_2+d_3=w-1}{\displaystyle \frac{{\sum }_{\sigma \in S_3}{\sigma }_{\mathit{a}}\left\{(\genfrac{}{}{0pt}{}{1+d_2+d_3-a_3}{a_2})(\genfrac{}{}{0pt}{}{1+d_3}{a_3})\right\}}{(d_2+d_3+1)(d_3+1)}}.$$

Applying $u_1=z$ and $u_2=y$ in Equation (7) we obtain the following identity:

$$1=\sum _{\left|\mathit{d}\right|=w}\prod _{\ell =1}^n{\ell }^{-d_{\ell }-1}\sum _{\sigma \in S_n}{\sigma }_{\mathit{a}}\left\{\prod _{j=2}^n\left({\displaystyle \genfrac{}{}{0pt}{}{{\sum }_{k=j}^n{d}_k-{\sum }_{k=j+1}^n{a}_k}{a_j}}\right)\right\}.$$

(13)

These identities follow from combining shuffle product expansions with explicit evaluations of the corresponding iterated integrals. These identities reveal a structured correspondence between shuffle algebra expansions and rational combinatorial sums.

4. New Shuffle Product Formulas

In this section, we give two new shuffle product formulas. The following shuffle identity appears to be new.

Theorem 3. 

Let $(a_1,a_2,\dots ,a_s)$ be a tuple of nonnegative integers with $a_s\ge 1$, and let p be a positive integer. Then

$$\begin{array}{cc}\hfill x{z}^pшz^{a_1}x{z}^{a_2}x\cdots x{z}^{a_s}=& \sum _{\begin{array}{c}1\le k\le s\\ a_{k1}+a_{k2}=a_k\\ p_k+p_{k+1}+\cdots +p_s=p\end{array}}\left({\displaystyle \genfrac{}{}{0pt}{}{a_{k2}+p_k}{p_k}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{a_{k+1}+p_{k+1}}{p_{k+1}}}\right)\cdots \left({\displaystyle \genfrac{}{}{0pt}{}{a_s+p_s}{p_s}}\right)\hfill \\ \hfill & z^{a_1}x{z}^{a_2}\cdots x{z}^{a_{k1}}x{z}^{a_{k2}+p_k}x{z}^{a_{k+1}+p_{k+1}}\cdots x{z}^{a_s+p_s},\hfill \end{array}$$

where all $p_k,\dots ,p_s$ are nonnegative integers.

Proof. 

We give a direct combinatorial proof based on the definition of the shuffle product.

Let

$$w=z^{a_1}x{z}^{a_2}x\cdots x{z}^{a_s}.$$

When forming the shuffle product $x{z}^pшw$, the relative order of the letters inside each word must be preserved. Since the word $x{z}^p$ contains exactly one letter x, this letter must appear in the shuffled word before all p copies of z coming from $x{z}^p$.

Hence the letter x from $x{z}^p$ must be inserted into one of the s blocks

$$z^{a_1},z^{a_2},\dots ,z^{a_s}.$$

Suppose that it is inserted into the k-th block $z^{a_k}$, where $1\le k\le s$. Then this block is split into two parts:

$$z^{a_k}=z^{a_{k1}}{z}^{a_{k2}},a_{k1}+a_{k2}=a_k,$$

so that in the resulting shuffled word the inserted x appears between these two parts. Therefore the shuffled word must have the form

$$z^{a_1}x{z}^{a_2}\cdots x{z}^{a_{k1}}x{z}^{a_{k2}+p_k}x{z}^{a_{k+1}+p_{k+1}}\cdots x{z}^{a_s+p_s},$$

where $p_k,p_{k+1},\dots ,p_s$ are nonnegative integers satisfying

$$p_k+p_{k+1}+\cdots +p_s=p.$$

Here $p_j$ denotes the number of z’s from $x{z}^p$ that are placed into the j-th z-block after the insertion of x, for $j=k,k+1,\dots ,s$.

Now we count how many shuffles produce this particular word. For the k-th block, after splitting $z^{a_k}$ into $z^{a_{k1}}$ and $z^{a_{k2}}$, the $p_k$ copies of z from $x{z}^p$ are shuffled with the $a_{k2}$ copies of z lying to the right of the inserted x. Since all letters involved are z’s, the number of such interlacings is

$$\left({\displaystyle \genfrac{}{}{0pt}{}{a_{k2}+p_k}{p_k}}\right).$$

Similarly, for each $j=k+1,\dots ,s$, the $p_j$ copies of z from $x{z}^p$ are shuffled with the block $z^{a_j}$, giving

$$\left({\displaystyle \genfrac{}{}{0pt}{}{a_j+p_j}{p_j}}\right)$$

possible interlacings.

Because these choices are independent, the total number of shuffles leading to the above word is

$$\left({\displaystyle \genfrac{}{}{0pt}{}{a_{k2}+p_k}{p_k}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{a_{k+1}+p_{k+1}}{p_{k+1}}}\right)\cdots \left({\displaystyle \genfrac{}{}{0pt}{}{a_s+p_s}{p_s}}\right).$$

Summing over all possible positions k, all decompositions $a_k=a_{k1}+a_{k2}$, and all nonnegative integers $p_k,\dots ,p_s$ with

$$p_k+\cdots +p_s=p,$$

we obtain

$$\begin{array}{cc}\hfill x{z}^pшz^{a_1}x{z}^{a_2}x\cdots x{z}^{a_s}=& \sum _{\begin{array}{c}1\le k\le s\\ a_{k1}+a_{k2}=a_k\\ p_k+p_{k+1}+\cdots +p_s=p\end{array}}\left({\displaystyle \genfrac{}{}{0pt}{}{a_{k2}+p_k}{p_k}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{a_{k+1}+p_{k+1}}{p_{k+1}}}\right)\cdots \left({\displaystyle \genfrac{}{}{0pt}{}{a_s+p_s}{p_s}}\right)\hfill \\ \hfill & z^{a_1}x{z}^{a_2}\cdots x{z}^{a_{k1}}x{z}^{a_{k2}+p_k}x{z}^{a_{k+1}+p_{k+1}}\cdots x{z}^{a_s+p_s},\hfill \end{array}$$

which is the desired identity. □

We illustrate the identity with the first few cases.

Example 1 

(The case $s=1$).

$$x{z}^pшz^a=\sum _{a_1+a_2=a}\left({\displaystyle \genfrac{}{}{0pt}{}{a_2+p}{p}}\right)z^{a_1}x{z}^{a_2+p}.$$

(14)

Example 2 

(The case $s=2$).

$$\begin{array}{cc}\hfill x{z}^pшz^{a_1}x{z}^{a_2}& =\sum _{\begin{array}{c}{a}_{11}+a_{12}=a_1\\ p_1+p_2=p\end{array}}\left({\displaystyle \genfrac{}{}{0pt}{}{a_{12}+p_1}{p_1}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{a_2+p_2}{p_2}}\right)z^{a_{11}}x{z}^{a_{12}+p_1}x{z}^{a_2+p_2}\hfill \\ \hfill & +\sum _{a_{21}+a_{22}=a_2}\left({\displaystyle \genfrac{}{}{0pt}{}{a_{22}+p}{p}}\right)z^{a_1}x{z}^{a_{21}}x{z}^{a_{22}+p}.\hfill \end{array}$$

(15)

While the previous formula describes the insertion of a single x, the following result provides a global description of shuffle products involving multiple words.

This formula can be viewed as a refinement and extension of Equation (7), in which the shuffle structure is described in terms of permutations and multinomial distributions.

We now give an explicit combinatorial formula for the n-fold shuffle product of words of the form $z^{p}x{z}^q$. This result provides a complete combinatorial description of such shuffle products and will be fundamental for deriving new identities in Section 5.

Theorem 4. 

Let $p_i,q_i\ge 0$ for $1\le i\le n$, and set

$$w_i=z^{p_i}x{z}^{q_i}.$$

Then

$$w_1ш\cdots шw_n=\sum _{\sigma \in S_n}\sum _{m_{i,r}\ge 0}\left(\prod _{r=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{A_r}{m_{1,r},\dots ,m_{n,r}}}\right)\right)z^{A_0}x{z}^{A_1}x\cdots x{z}^{A_n},$$

where $S_n$ is the symmetric group of n objects, $A_r={\sum }_{i=1}^n{m}_{i,r}(0\le r\le n)$, and the inner sum ranges over all families of nonnegative integers

$$m_{i,r}(1\le i\le n,0\le r\le n)$$

satisfying, for each $k=1,\dots ,n$,

$$\sum _{r=0}^{k-1}{m}_{\sigma \left(k\right),r}=p_{\sigma \left(k\right)},\sum _{r=k}^n{m}_{\sigma \left(k\right),r}=q_{\sigma \left(k\right)}.$$

Proof. 

Fix a permutation $\sigma \in S_n$, which determines the relative order of the n occurrences of x in the shuffle.

For each $i=1,\dots ,n$, suppose that the letter x from the word $w_i=z^{p_i}x{z}^{q_i}$ appears in the k-th position among all x’s, that is, $i=\sigma \left(k\right)$. Then the $p_i$ copies of z preceding this x must be placed among the blocks before it, namely blocks $0,1,\dots ,k-1$, while the $q_i$ copies of z following this x must be placed among the blocks $k,k+1,\dots ,n$.

Accordingly, for each i and each block r, let $m_{i,r}$ denote the number of z’s from the word $w_i$ that are assigned to the r-th block. Then the above constraints are equivalent to

$$\sum _{r=0}^{k-1}{m}_{\sigma \left(k\right),r}=p_{\sigma \left(k\right)},\sum _{r=k}^n{m}_{\sigma \left(k\right),r}=q_{\sigma \left(k\right)},(1\le k\le n).$$

Conversely, any family of nonnegative integers $m_{i,r}$ satisfying these conditions determines a unique way to distribute all z’s into the $n+1$ blocks between consecutive x’s.

For each block r, the total number of z’s is

$$A_r=\sum _{i=1}^n{m}_{i,r}.$$

The letters coming from different words are interleaved arbitrarily within each block, and the number of such interleavings is given by the multinomial coefficient

$$\left({\displaystyle \genfrac{}{}{0pt}{}{A_r}{m_{1,r},\dots ,m_{n,r}}}\right).$$

Since the choices for different blocks are independent, the total number of shuffle realizations corresponding to a fixed $\sigma$ and a fixed family $\left\{m_{i,r}\right\}$ is the product

$$\prod _{r=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{A_r}{m_{1,r},\dots ,m_{n,r}}}\right).$$

Summing over all admissible families $\left\{m_{i,r}\right\}$ and all permutations $\sigma \in S_n$ yields the stated formula. □

Example 3 

(The case $n=2$). Let

$$w_1=z^{p_1}x{z}^{q_1},w_2=z^{p_2}x{z}^{q_2}.$$

Then

$$w_1шw_2=\sum _{\sigma \in S_2}\sum _{m_{i,r}\ge 0}\left(\prod _{r=0}^2\left({\displaystyle \genfrac{}{}{0pt}{}{A_r}{m_{1,r},m_{2,r}}}\right)\right)z^{A_0}x{z}^{A_1}x{z}^{A_2},$$

where

$$A_r=m_{1,r}+m_{2,r}(0\le r\le 2),$$

and the inner sum is subject to

$$\sum _{r=0}^{k-1}{m}_{\sigma \left(k\right),r}=p_{\sigma \left(k\right)},\sum _{r=k}^2{m}_{\sigma \left(k\right),r}=q_{\sigma \left(k\right)}(k=1,2).$$

Since $S_2=\{(1,2),(2,1)\}$, the shuffle product splits into two contributions.

Case 1: 

$\sigma =(1,2)$. Then

$$m_{1,0}=p_1,m_{1,1}+m_{1,2}=q_1,$$

$$m_{2,0}+m_{2,1}=p_2,m_{2,2}=q_2.$$

Write

$$i=m_{2,0}(0\le i\le p_2),j=m_{1,1}(0\le j\le q_1).$$

Then

$$m_{2,1}=p_2-i,m_{1,2}=q_1-j,$$

and hence

$$A_0=p_1+i,A_1=p_2-i+j,A_2=q_1-j+q_2.$$

Therefore the contribution of $\sigma =(1,2)$ is

$$\sum _{i=0}^{p_2}\sum _{j=0}^{q_1}\left({\displaystyle \genfrac{}{}{0pt}{}{p_1+i}{p_1}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{p_2-i+j}{j}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{q_1-j+q_2}{q_1-j}}\right)z^{p_1+i}x{z}^{p_2-i+j}x{z}^{q_1-j+q_2}.$$

Case 2: $\sigma =(2,1)$. Then

$$m_{2,0}=p_2,m_{2,1}+m_{2,2}=q_2,$$

$$m_{1,0}+m_{1,1}=p_1,m_{1,2}=q_1.$$

Write

$$i=m_{1,0}(0\le i\le p_1),j=m_{2,1}(0\le j\le q_2).$$

Then

$$m_{1,1}=p_1-i,m_{2,2}=q_2-j,$$

and hence

$$A_0=p_2+i,A_1=p_1-i+j,A_2=q_1+q_2-j.$$

Therefore the contribution of $\sigma =(2,1)$ is

$$\sum _{i=0}^{p_1}\sum _{j=0}^{q_2}\left({\displaystyle \genfrac{}{}{0pt}{}{p_2+i}{p_2}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{p_1-i+j}{j}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{q_1+q_2-j}{q_1}}\right)z^{p_2+i}x{z}^{p_1-i+j}x{z}^{q_1+q_2-j}.$$

Combining the two cases, we obtain

$$\begin{array}{cc}\hfill z^{p_1}x{z}^{q_1}шz^{p_2}x{z}^{q_2}=& \sum _{i=0}^{p_2}\sum _{j=0}^{q_1}\left({\displaystyle \genfrac{}{}{0pt}{}{p_1+i}{p_1}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{p_2-i+j}{j}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{q_1+q_2-j}{q_2}}\right)z^{p_1+i}x{z}^{p_2-i+j}x{z}^{q_1+q_2-j}\hfill \\ \hfill & +\sum _{i=0}^{p_1}\sum _{j=0}^{q_2}\left({\displaystyle \genfrac{}{}{0pt}{}{p_2+i}{p_2}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{p_1-i+j}{j}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{q_1+q_2-j}{q_1}}\right)z^{p_2+i}x{z}^{p_1-i+j}x{z}^{q_1+q_2-j}.\hfill \end{array}$$

(16)

Example 4 

(The case $n=3$). Let

$$w_i=z^{p_i}x{z}^{q_i}(i=1,2,3).$$

Then

$$w_1шw_2шw_3=\sum _{\sigma \in S_3}\sum _{m_{i,r}\ge 0}\left(\prod _{r=0}^3\left({\displaystyle \genfrac{}{}{0pt}{}{A_r}{m_{1,r},m_{2,r},m_{3,r}}}\right)\right)z^{A_0}x{z}^{A_1}x{z}^{A_2}x{z}^{A_3},$$

where

$$A_r=m_{1,r}+m_{2,r}+m_{3,r}(0\le r\le 3),$$

and the inner sum is subject to

$$\sum _{r=0}^{k-1}{m}_{\sigma \left(k\right),r}=p_{\sigma \left(k\right)},\sum _{r=k}^3{m}_{\sigma \left(k\right),r}=q_{\sigma \left(k\right)}(k=1,2,3).$$

Since $|S_3|=6$, the shuffle product splits into six contributions corresponding to

$$\sigma =(1,2,3),(1,3,2),(2,1,3),(2,3,1),(3,1,2),(3,2,1).$$

We illustrate the computation by considering a representative case.

Case: 

$\sigma =(1,2,3)$. Then

$$m_{1,0}=p_1,m_{1,1}+m_{1,2}+m_{1,3}=q_1,$$

$$m_{2,0}+m_{2,1}=p_2,m_{2,2}+m_{2,3}=q_2,$$

$$m_{3,0}+m_{3,1}+m_{3,2}=p_3,m_{3,3}=q_3.$$

Write

$$i=m_{2,0},j=m_{1,1},u=m_{3,0},v=m_{3,1},\ell =m_{2,2},$$

so that

$$0\le i\le p_2,0\le j\le q_1,u,v\ge 0,u+v\le p_3,0\le \ell \le q_2.$$

Then

$$m_{2,1}=p_2-i,m_{1,2}+m_{1,3}=q_1-j,$$

$$m_{3,2}=p_3-u-v,m_{2,3}=q_2-\ell .$$

Now split the remaining quantity $q_1-j$ by writing

$$s=m_{1,2},0\le s\le q_1-j,$$

so that

$$m_{1,3}=q_1-j-s.$$

Hence

$$A_0=p_1+i+u,$$

$$A_1=j+(p_2-i)+v,$$

$$A_2=s+\ell +(p_3-u-v),$$

$$A_3=(q_1-j-s)+(q_2-\ell )+q_3.$$

Therefore the contribution of $\sigma =(1,2,3)$ is

$$\begin{gathered} \sum _{i=0}^{p_2}\sum _{j=0}^{q_1}\sum _{\begin{array}{c}u,v\ge 0\\ u+v\le p_3\end{array}}\sum _{\ell =0}^{q_2}\sum _{s=0}^{q_1-j}\left({\displaystyle \genfrac{}{}{0pt}{}{p_1+i+u}{p_1,i,u}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{j+p_2-i+v}{j,p_2-i,v}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{s+\ell +p_3-u-v}{s,\ell ,p_3-u-v}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{q_1-j-s+q_2-\ell +q_3}{q_1-j-s,q_2-\ell ,q_3}}\right) \\ ·z^{p_1+i+u}x{z}^{j+p_2-i+v}x{z}^{s+\ell +p_3-u-v}x{z}^{q_1-j-s+q_2-\ell +q_3}. \end{gathered}$$

The remaining five cases are obtained analogously by permuting the roles of

$$(p_1,q_1),(p_2,q_2),(p_3,q_3)$$

according to the corresponding permutation $\sigma \in S_3$.

Consequently,

$$\begin{array}{cc}\hfill z^{p_1}x{z}^{q_1}шz^{p_2}x{z}^{q_2}шz^{p_3}x{z}^{q_3}=& {\mathcal{C}}_{(1,2,3)}+{\mathcal{C}}_{(1,3,2)}+{\mathcal{C}}_{(2,1,3)}\hfill \\ & +{\mathcal{C}}_{(2,3,1)}+{\mathcal{C}}_{(3,1,2)}+{\mathcal{C}}_{(3,2,1)},\hfill \end{array}$$

where ${\mathcal{C}}_{\sigma }$ denotes the contribution corresponding to the permutation σ.

5. Applications to Combinatorial Identities

Let $(a_1,a_2,\dots ,a_s)$ be a tuple of nonnegative integers with $a_s\ge 1$, and let p be a positive integer. For $1\le k\le s$, we let

$$A_k=a_k+a_{k+1}+\cdots +a_s.$$

Thus, $\left|\mathit{a}\right|=A_1$. Applying the map Z to the identity in Theorem 3, we obtain expressions in terms of the $\mathcal{E}$-values introduced in Section 2:

$$\begin{array}{cc}\hfill & \rho (p+1)\mathcal{E}(a_1,a_2,\dots ,a_s)\hfill \\ \hfill & =\sum _{k=1}^s\sum _{\begin{array}{c}{a}_{k1}+a_{k2}=a_k\\ p_k+\cdots +p_s=p\end{array}}\left({\displaystyle \genfrac{}{}{0pt}{}{a_{k2}+p_k}{p_k}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{a_{k+1}+p_{k+1}}{p_{k+1}}}\right)\cdots \left({\displaystyle \genfrac{}{}{0pt}{}{a_s+p_s}{p_s}}\right)\hfill \\ \hfill & \times \mathcal{E}(a_1,a_2,\dots ,a_{k-1},a_{k1},a_{k2}+p_k,a_{k+1}+p_{k+1},\dots ,a_s+ps)\hfill \end{array}$$

Using the explicit evaluation in Equation (3), this yields

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{p·p!·\left|\mathit{a}\right|!{\prod }_{j=2}^s{A}_j}}\hfill \\ \hfill & =\sum _{k=1}^s\sum _{\begin{array}{c}{a}_{k1}+a_{k2}=a_k\\ p_k+\cdots +p_s=p\end{array}}\left({\displaystyle \genfrac{}{}{0pt}{}{a_{k2}+p_k}{p_k}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{a_{k+1}+p_{k+1}}{p_{k+1}}}\right)\cdots \left({\displaystyle \genfrac{}{}{0pt}{}{a_s+p_s}{p_s}}\right)\hfill \\ \hfill & \times {\displaystyle \frac{1}{\left(\right|\mathit{a}|+p)!}}{\displaystyle \frac{1}{{\prod }_{j=2}^k(A_j+p)}}{\displaystyle \frac{1}{(a_{k2}+A_{k+1}+p)}}{\displaystyle \frac{1}{{\prod }_{j=k+1}^s(A_j+p_j+\cdots +p_s)}}.\hfill \end{array}$$

(17)

Multiplying both sides by

$$p\left(\right|\mathit{a}|+p)!\prod _{j=2}^s{A}_j,$$

we obtain

$$\begin{array}{cc}\hfill \left({\displaystyle \genfrac{}{}{0pt}{}{\left|\mathit{a}\right|+p}{p}}\right)& =p\prod _{j=2}^s{A}_j\sum _{k=1}^s\sum _{\begin{array}{c}{a}_{k1}+a_{k2}=a_k\\ p_k+\cdots +p_s=p\end{array}}\left({\displaystyle \genfrac{}{}{0pt}{}{a_{k2}+p_k}{p_k}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{a_{k+1}+p_{k+1}}{p_{k+1}}}\right)\cdots \left({\displaystyle \genfrac{}{}{0pt}{}{a_s+p_s}{p_s}}\right)\hfill \\ \hfill & \times {\displaystyle \frac{1}{{\prod }_{j=2}^k(A_j+p)}}{\displaystyle \frac{1}{(a_{k2}+A_{k+1}+p)}}{\displaystyle \frac{1}{{\prod }_{j=k+1}^s(A_j+p_j+\cdots +p_s)}}.\hfill \end{array}$$

(18)

When $s=1$, the above formula reduces to the hockey-stick identity [23]

$$\left({\displaystyle \genfrac{}{}{0pt}{}{a+p}{p}}\right)=\sum _{k=0}^a\left({\displaystyle \genfrac{}{}{0pt}{}{k+p-1}{p-1}}\right).$$

When $s=2$, Equation (18) becomes

$$\begin{array}{cc}\hfill \left({\displaystyle \genfrac{}{}{0pt}{}{a_1+a_2+p}{p}}\right)& =\sum _{a_{11}+a_{12}=a_1}{\displaystyle \frac{p}{a_{12}+a_2+p}}\sum _{p_1+p_2=p}\left({\displaystyle \genfrac{}{}{0pt}{}{a_{12}+p_1}{p_1}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{p_2+a_2-1}{p_2}}\right)\hfill \\ \hfill & +\sum _{a_{21}+a_{22}=a_2}\left({\displaystyle \genfrac{}{}{0pt}{}{a_{22}+p-1}{p-1}}\right){\displaystyle \frac{a_2}{a_2+p}}.\hfill \end{array}$$

Using the Vandermonde’s identity and the hockey-stick identity, we have

$$\sum _{a_{11}+a_{12}=a_1}\left({\displaystyle \genfrac{}{}{0pt}{}{a_{12}+a_2+p-1}{p-1}}\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{a_1+a_2+p}{p}}\right)-\left({\displaystyle \genfrac{}{}{0pt}{}{a_2+p-1}{p}}\right).$$

This may be viewed as a shifted version of the hockey-stick identity.

A careful inspection of Equation (18) suggests that the inner summation may admit a closed form. This observation leads to the following identity. Although Equation (18) is derived under the assumption $p\ge 1$, the resulting identity naturally extends to the case $p=0$.

Theorem 5. 

Let $(a_1,a_2,\dots ,a_s)$ be a tuple of nonnegative integers with $a_s\ge 1$, and let p be a nonnegative integer. Then

$$\sum _{p_1+\cdots +p_s=p}{\displaystyle \frac{(\genfrac{}{}{0pt}{}{a_1+p_1}{p_1})(\genfrac{}{}{0pt}{}{a_2+p_2}{p_2})\cdots (\genfrac{}{}{0pt}{}{a_s+p_s}{p_s})}{{\prod }_{k=2}^s\left((a_k+p_k)+\cdots +(a_s+p_s)\right)}}={\displaystyle \frac{(\genfrac{}{}{0pt}{}{a_1+\cdots +a_s+p}{p})}{{\prod }_{k=2}^s(a_k+\cdots +a_s)}}.$$

(19)

Proof. 

We argue by induction on s. The case $s=1$ is immediate.

Assume that the assertion holds for tuples of length $s-1$. Let

$$A_k=a_k+a_{k+1}+\cdots +a_s(1\le k\le s).$$

Then

$$\begin{array}{cc}\hfill & \sum _{p_1+\cdots +p_s=p}{\displaystyle \frac{(\genfrac{}{}{0pt}{}{a_1+p_1}{p_1})(\genfrac{}{}{0pt}{}{a_2+p_2}{p_2})\cdots (\genfrac{}{}{0pt}{}{a_s+p_s}{p_s})}{{\prod }_{k=2}^s\left((a_k+p_k)+\cdots +(a_s+p_s)\right)}}\hfill \\ \hfill & =\sum _{p_1=0}^p{\displaystyle \frac{(\genfrac{}{}{0pt}{}{a_1+p_1}{p_1})}{A_2+p-p_1}}\sum _{p_2+\cdots +p_s=p-p_1}{\displaystyle \frac{{\prod }_{i=2}^s(\genfrac{}{}{0pt}{}{a_i+p_i}{p_i})}{{\prod }_{k=3}^s(A_k+p_k+\cdots +p_s)}}.\hfill \end{array}$$

By the inductive hypothesis, the inner sum equals

$${\displaystyle \frac{(\genfrac{}{}{0pt}{}{A_2+p-p_1}{p-p_1})}{{\prod }_{k=3}^s{A}_k}}.$$

Hence the left-hand side becomes

$${\displaystyle \frac{1}{{\prod }_{k=3}^s{A}_k}}\sum _{p_1=0}^p{\displaystyle \frac{(\genfrac{}{}{0pt}{}{a_1+p_1}{p_1})(\genfrac{}{}{0pt}{}{A_2+p-p_1}{p-p_1})}{A_2+p-p_1}}.$$

Using

$${\displaystyle \frac{1}{A_2+p-p_1}}\left({\displaystyle \genfrac{}{}{0pt}{}{A_2+p-p_1}{p-p_1}}\right)={\displaystyle \frac{1}{A_2}}\left({\displaystyle \genfrac{}{}{0pt}{}{A_2+p-p_1-1}{p-p_1}}\right),$$

we obtain

$${\displaystyle \frac{1}{{\prod }_{k=2}^s{A}_k}}\sum _{p_1=0}^p\left({\displaystyle \genfrac{}{}{0pt}{}{a_1+p_1}{p_1}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{A_2+p-p_1-1}{p-p_1}}\right).$$

Applying Vandermonde’s identity yields

$${\displaystyle \frac{1}{{\prod }_{k=2}^s{A}_k}}\left({\displaystyle \genfrac{}{}{0pt}{}{a_1+A_2+p}{p}}\right)={\displaystyle \frac{1}{{\prod }_{k=2}^s{A}_k}}\left({\displaystyle \genfrac{}{}{0pt}{}{a_1+\cdots +a_s+p}{p}}\right),$$

which proves the theorem. □

Due to the complexity of Theorem 4, we restrict ourselves to the case $n=2$, which already yields a nontrivial combinatorial identity.

The following identity exhibits a symmetry between $(p_1,q_1)$ and $(p_2,q_2)$.

Proposition 1. 

For $p_1,p_2\ge 0$ and $q_1,q_2\ge 1$, we have

$$\left({\displaystyle \genfrac{}{}{0pt}{}{p_1+p_2+q_1+q_2}{p_1+q_1}}\right)=\sum _{k=0}^{p_2}\left({\displaystyle \genfrac{}{}{0pt}{}{p_1+k}{p_1}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{q_1+q_2+p_2-k-1}{q_1-1}}\right)+\sum _{k=0}^{p_1}\left({\displaystyle \genfrac{}{}{0pt}{}{p_2+k}{p_2}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{q_1+q_2+p_1-k-1}{q_2-1}}\right).$$

Proof. 

We apply the map Z to Equation (16) in the case $n=2$, where $p_1,p_2\ge 0$ and $q_1,q_2\ge 1$. This yields expressions in terms of the $\mathcal{E}$-values. Using the explicit evaluation in Equation (3), we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{(p_1+q_1)!(p_2+q_2)!q_1{q}_2}}& =\sum _{\begin{array}{c}{p}_{21}+p_{22}=p_2\\ q_{11}+q_{12}=q_1\end{array}}{\displaystyle \frac{(\genfrac{}{}{0pt}{}{p_1+p_{21}}{p_1})(\genfrac{}{}{0pt}{}{q_{11}+p_{22}}{q_{11}})(\genfrac{}{}{0pt}{}{q_2+q_{12}}{q_2})}{(p_1+p_2+q_1+q_2)!(q_2+q_{12})(q_1+q_2+p_{22})}}\hfill \\ \hfill & +\sum _{\begin{array}{c}{p}_{11}+p_{12}=p_1\\ q_{21}+q_{22}=q_2\end{array}}{\displaystyle \frac{(\genfrac{}{}{0pt}{}{p_2+p_{11}}{p_2})(\genfrac{}{}{0pt}{}{p_{12}+q_{21}}{p_{12}})(\genfrac{}{}{0pt}{}{q_{22}+q_1}{q_1})}{(p_1+p_2+q_1+q_2)!(q_{22}+q_1)(q_1+q_2+p_{12})}}.\hfill \end{array}$$

Multiplying both sides by $(p_1+p_2+q_1+q_2)!q_1{q}_2$, we obtain

$$\begin{array}{cc}\hfill \left({\displaystyle \genfrac{}{}{0pt}{}{p_1+p_2+q_1+q_2}{p_1+q_1}}\right)& =\sum _{\begin{array}{c}{p}_{21}+p_{22}=p_2\\ q_{11}+q_{12}=q_1\end{array}}{\displaystyle \frac{q_1(\genfrac{}{}{0pt}{}{p_1+p_{21}}{p_1})(\genfrac{}{}{0pt}{}{q_{11}+p_{22}}{p_{22}})(\genfrac{}{}{0pt}{}{q_2+q_{12}-1}{q_{12}})}{q_1+q_2+p_{22}}}\hfill \\ \hfill & +\sum _{\begin{array}{c}{p}_{11}+p_{12}=p_1\\ q_{21}+q_{22}=q_2\end{array}}{\displaystyle \frac{q_2(\genfrac{}{}{0pt}{}{p_2+p_{11}}{p_2})(\genfrac{}{}{0pt}{}{p_{12}+q_{21}}{p_{12}})(\genfrac{}{}{0pt}{}{q_1+q_{22}-1}{q_{22}})}{q_1+q_2+p_{12}}}.\hfill \end{array}$$

Applying Vandermonde’s identity to the inner sums, we obtain

$$\begin{array}{cc}\hfill \left({\displaystyle \genfrac{}{}{0pt}{}{p_1+p_2+q_1+q_2}{p_1+q_1}}\right)& =\sum _{k=0}^{p_2}\left({\displaystyle \genfrac{}{}{0pt}{}{p_1+k}{p_1}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{q_1+q_2+p_2-k-1}{q_1-1}}\right)\hfill \\ \hfill & +\sum _{k=0}^{p_1}\left({\displaystyle \genfrac{}{}{0pt}{}{p_2+k}{p_2}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{q_1+q_2+p_1-k-1}{q_2-1}}\right).\hfill \end{array}$$

(20)

This gives the stated symmetric decomposition of the binomial coefficient. □

These identities demonstrate how shuffle product formulas give rise to nontrivial decompositions of binomial coefficients.

6. Conclusions and Further Directions

In this paper, we studied shuffle product structures in a noncommutative algebra generated by the letters $x,y,z$, where the additional letter z corresponds to the regular differential form $dt$. This extension goes beyond the classical framework of multiple zeta values and provides a systematic way to derive new combinatorial identities.

We established an algebraic setting together with the evaluation map Z, which connects words in the shuffle algebra to iterated integrals. Within this framework, known shuffle product identities lead naturally to classical combinatorial formulas, while new phenomena arise from the inclusion of the letter z.

Using this framework, we first translated known shuffle product formulas into explicit combinatorial identities. In particular, by applying the evaluation map Z to classical shuffle identities, we obtained formulas involving binomial and multinomial coefficients, thereby providing a combinatorial interpretation of these algebraic relations.

We then introduced two new shuffle product formulas. The first provides a refined description of insertion processes within words, while the second gives a global combinatorial decomposition in terms of permutations and multinomial distributions. These formulas extend the shuffle framework and serve as key tools for generating further identities, as demonstrated in Section 5.

Finally, by applying these new shuffle formulas, we derived nontrivial combinatorial identities. In particular, we obtained symmetric decompositions of binomial coefficients and general summation formulas involving multiple parameters, including extensions of classical identities such as the hockey-stick identity.

The combinatorial identities obtained in this paper suggest potential applications in the study of special functions, structured binomial sums, and generating functions arising from nested summations. In particular, the framework developed here may be useful for deriving closed-form expressions for multi-parameter sums that are difficult to handle by elementary methods.

Moreover, these methods may provide new insights into the algebraic structures underlying such identities, and could be further explored in connection with problems in enumerative combinatorics and related areas.

These results demonstrate that shuffle product algebra is not only a fundamental tool in the study of multiple zeta values, but also provides a systematic framework for deriving and interpreting combinatorial identities. The interplay between algebraic structures and combinatorial formulas revealed here suggests several directions for further research.

Finally, we remark that the present work focuses mainly on shuffle product structures involving two types of letters at a time, such as the pairs $(x,z)$ and $(y,z)$, together with their interplay via the standard substitution $t\mapsto 1-u$.

It would be natural to further investigate the combinatorial and analytic structures associated with words involving all three letters $x,y,z$ simultaneously. Such a setting is expected to lead to more intricate expressions involving mixtures of zeta-type values and combinatorial coefficients, and may give rise to a richer class of zeta-like values and a more comprehensive theory of shuffle products in this three-letter setting. We leave this direction for future research.

More broadly, the above perspective suggests several further directions for investigation. Possible directions for further research include extending the present approach to more general classes of iterated integrals and investigating higher-dimensional analogues of the shuffle identities developed here. It would also be of interest to explore connections with other combinatorial structures, such as symmetric functions and partition identities, within this extended shuffle framework.