Introduction

Let \(p\) be a prime number with \(p\equiv 1\) (mod 2). Throughout this paper, \({{\mathbb {Z}}_{p}}\), \(\mathbb {Q}_{p}\) and \(\mathbb {C}_{p}\) will, respectively, denote the ring of \(p\)-adic integers, the field of \(p\)-adic rational numbers and the completion of algebraic closure of \(\mathbb {Q}_{p}\). Let \(\left| \cdot \right| _{p}\) be the normalized \(p\)-adic absolute value with \(\left| p\right| _{p}=\frac{1}{p}\) and let \(q\) be an indeterminate in \(\mathbb {C}_{p}\) with \(\left| 1-q\right| _{p}<p^{-\frac{1}{p-1}}\). For a continuous function \(f\,:\,{{\mathbb {Z}}_{p}}\rightarrow \mathbb {C}_{p}\), the fermionic \(p\)-adic \(q\)-integral on \({{\mathbb {Z}}_{p}}\) is defined by Kim to be [9–17]

$$\begin{aligned} I_{-q}\left( f\right)&=\int _{{{\mathbb {Z}}_{p}}}f\left( x\right) {\rm d}\mu _{-q}\left( x\right) \\&=\lim _{N\rightarrow \infty }\frac{1}{\left[ p^{N}\right] _{-q}}\sum _{x=0}^{p^{N}-1}f\left( x\right) \left( -q\right) ^{x}, \end{aligned}$$
(1)

See [9, 17]. where \(\left[ x\right] _{-q}=\frac{1-\left( -q\right) ^{x}}{1+q}\) and \(\left[ x\right] _{q}=\frac{1-q^{x}}{1-q}\).

From (1), we have

$$\begin{aligned} qI_{-q}\left( f_{1}\right) +I_{-q}\left( f\right) =\left[ 2\right] _{q}f\left( 0\right) , \end{aligned}$$
(2)

where \(f_{1}\left( x\right) =f\left( x+1\right) \).

In general, one derives that

$$\begin{aligned} q^{n}I_{-q}\left( f_{n}\right) +\left( -1\right) ^{n-1}I_{-q}\left( f\right) =\left[ 2\right] _{q}\sum _{l=0}^{n-1}\left( -1\right) ^{n-1-l}q^{l}f\left( l\right) , \end{aligned}$$
(3)

where \(f_{n}\left( x\right) =f\left( x+n\right) \), \(\left( n\ge 1\right) \).

So, for \(n\equiv 1\) (mod 2),

$$\begin{aligned} q^{n}I_{-q}\left( f_{n}\right) +I_{-q}\left( f\right) =\left[ 2\right] _{q}\sum _{l=0}^{n-1}\left( -1\right) ^{l}q^{l}f\left( l\right) ; \end{aligned}$$
(4)

for \(n\equiv 0\) (mod 2),

$$\begin{aligned} q^{n}I_{-q}\left( f_{n}\right) -I_{-q}\left( f\right) =-\left[ 2\right] _{q}\sum _{l=0}^{n-1}\left( -1\right) ^{l}q^{l}f\left( l\right) . \end{aligned}$$
(5)

In particular, for \(q=1\), we have

$$\begin{aligned} I_{-1}\left( f\right) =\int _{{{\mathbb {Z}}_{p}}}f\left( x\right) {\rm d}\mu _{-1}\left( x\right) =\lim _{N\rightarrow \infty }\sum _{x=0}^{p^{N}-1}f\left( x\right) \left( -1\right) ^{x}, \end{aligned}$$
(6)

and

$$\begin{aligned} I_{-1}\left( f_{1}\right) +I_{-1}\left( f\right) =2f\left( 0\right) . \end{aligned}$$
(7)

As is well known, the ordinary Euler polynomials are defined by the generating function to be

$$\begin{aligned} \frac{2}{{\rm e}^{t}+1}{\rm e}^{xt}=\sum _{n=0}^{\infty }E_{n}\left( x\right) \frac{t^{n}}{n!} \end{aligned}$$
(8)

(see [1–24]). When \(x=0\), \(E_{n}=E_{n}\left( 0\right) \) are called the Euler numbers.

From (5) and (6), we can easily derive

$$\begin{aligned} \int _{{{\mathbb {Z}}_{p}}}{\rm e}^{\left( x+y\right) t}{\rm d}\mu _{-1}\left( y\right) =\frac{2}{{\rm e}^{t}+1}{\rm e}^{xt}=\sum _{n=0}^{\infty }E_{n}\left( x\right) \frac{t^{n}}{n!}. \end{aligned}$$
(9)

By (7), we get

$$\begin{aligned} E_{0}=1,\quad \left( E+1\right) ^{n}+E_{n}=2\delta _{0,n},\quad \end{aligned}$$
(10)

[11, 16, 17] with the usual convention about replacing \(E^{n}\) by \(E_{n}\).

From (9), we note that

$$\begin{aligned} E_{n}\left( x\right)&=\int _{{{\mathbb {Z}}_{p}}}\left( x+y\right) ^{n}{\rm d}\mu _{-1}\left( y\right) =\sum _{l=0}^{n}\left(\begin{array}{l}{n}\\{l}\end{array}\right)x^{n-l}\int _{{{\mathbb {Z}}_{p}}}y^{l}{\rm d}\mu _{-1}\left( y\right) \\ &=\sum _{l=0}^{n}\left(\begin{array}{l}{n}\\ {l}\end{array}\right)x^{n-l}E_{l}.\nonumber \end{aligned}$$
(11)

In light of (10), the Carlitz-type \(q\)-Euler numbers are given by

$$\begin{aligned} q\left( q\mathcal {E}_{q}+1\right) ^{n}+\mathcal {E}_{n,q}=\left[ 2\right] _{q}\delta _{n,0,}\quad \left( n\ge 0\right) , \end{aligned}$$
(12)

with the usual convention about replacing \(\mathcal {E}_{q}^{n}\) by \(\mathcal {E}_{n,q}\) (see [10, 12, 15]).

The \(q\)-Euler polynomials are defined by (see [10, 16])

$$\begin{aligned} \mathcal {E}_{n,q}\left( x\right)&=\left( q^{x}\mathcal {E}_{q}+\left[ x\right] _{q}\right) ^{n}\\&=\sum _{l=0}^{n}\left(\begin{array}{l}{n}\\ {l}\end{array}\right) q^{lx}\mathcal {E}_{l,q}\left[ x\right] _{q}^{n-l}. \end{aligned}$$
(13)

From (2), we derive

$$\begin{aligned} \sum _{n=0}^{\infty }\mathcal {E}_{n,q}\frac{t^{n}}{n!}&= \int _{{{\mathbb {Z}}_{p}}}{\rm e}^{\left[ y\right] _{q}t}{\rm d}\mu _{-q}\left( y\right) \\&= \left[ 2\right] _{q}\sum _{m=0}^{\infty }\left( -1\right) ^{m}q^{m}{\rm e}^{\left[ m\right] _{q}t},\nonumber \end{aligned}$$
(14)

and

$$\begin{aligned} \sum _{n=0}^{\infty }\mathcal {E}_{n,q}\left( x\right) \frac{t^{n}}{n!}&=\int _{{{\mathbb {Z}}_{p}}}{\rm e}^{\left[ x+y\right] _{q}t}{\rm d}\mu _{-q}\left( y\right) \\&=\left[ 2\right] _{q}\sum _{m=0}^{\infty }\left( -1\right) ^{m}q^{m}{\rm e}^{\left[ m+x\right] _{q}t}.\nonumber \end{aligned}$$
(15)

By (13) and (15), we get

$$\begin{aligned} \mathcal {E}_{n,q}\left( x+y\right)&=\sum _{l=0}^{n}\left(\begin{array}{l}{n}\\ {l}\end{array}\right) q^{lx}\mathcal {E}_{l,q}\left( y\right) \left[ x\right] _{q}^{n-l}\\&=\sum _{l=0}^{n}\left(\begin{array}{l}{n}\\ {l}\end{array}\right) q^{\left( n-l\right) x}\mathcal {E}_{n-l,q}\left( y\right) \left[ x\right] _{q}^{l}.\nonumber \end{aligned}$$
(16)

Explicit expressions for Carlitz-type \(q\)-Euler numbers can be obtained, for example, from their generating functions :

$$\begin{aligned} \mathcal {E}_{n,q}\left( x\right) =\frac{\left[ 2\right] _{q}}{\left( 1-q\right) ^{n}}\sum _{l=0}^{n}\left(\begin{array}{l}{n}\\ {l}\end{array}\right)\left( -1\right) ^{l}q^{lx}\frac{1}{1+q^{l+1}}, \end{aligned}$$
(17)

and

$$\begin{aligned} \mathcal {E}_{n,q}=\frac{\left[ 2\right] _{q}}{\left( 1-q\right) ^{n}}\sum _{l=0}^{n}\left(\begin{array}{l}{n}\\ {l}\end{array}\right)\left( -1\right) ^{l}\frac{1}{1+q^{l+1}}. \end{aligned}$$
(18)

In [10, 14, 15], Kim introduced the polynomials \(\mathcal {E}_{n,q}^{\left( h,k\right) }\left( x\right) \) in terms of the following multiple fermionic \(p\)-adic \(q\)-integral on \({{\mathbb {Z}}_{p}}\) :

$$\begin{aligned}{\mathcal{E}}_{n,q}^{\left( h,k\right) }\left( x\right)& =\int _{{{\mathbb {Z}}_{p}}}\cdots \int _{{{\mathbb {Z}}_{p}}}q^{\sum _{l=1}^{k}\left( h-l\right) y_{l}}\\ &\quad\times\left[ x+y_{1}+\cdots +y_{k}\right] _{q}^{n}{\rm d}\mu _{-q}\left( y_{1}\right) \cdots {\rm d}\mu _{-q}\left( y_{k}\right) . \end{aligned}$$
(19)

In particular, if \(k=1\), \(\mathcal {E}_{n,q}^{\left( h,1\right) }\left( x\right) \) will be simply denoted by \(\mathcal {E}_{n,q}^{\left( h\right) }\left( x\right) \) so that

$$\begin{aligned} \mathcal {E}_{n,q}^{\left( h\right) }\left( x\right) =\int _{{{\mathbb {Z}}_{p}}}q^{\left( h-1\right) y}\left[ x+y\right] _{q}^{n}{\rm d}\mu _{-q}\left( y\right) . \end{aligned}$$
(20)

One can derive the following explicit expression of \(\mathcal {E}_{n,q}^{\left( h,k\right) }\left( x\right) \) :

$$\begin{aligned} \mathcal {E}_{n,q}^{\left( h,k\right) }\left( x\right) =\frac{\left[ 2\right] _{q}^{k}}{\left( 1-q\right) ^{n}}\sum _{l=0}^{n}\left(\begin{array}{l}{n}\\ {l}\end{array}\right)\left( -1\right) ^{n-l}q^{lx}\frac{1}{\left( -q^{l+h};q^{-1}\right) _{k}}, \end{aligned}$$
(21)

where \(\left( x:q\right) _{n}=\left( 1-x\right) \left( 1-xq\right) \cdots \left( 1-xq^{n-1}\right) \) (see [10, 14, 15]).

The following simple facts will be used over and over again :

$$\begin{aligned} \left[ a+b\right] _{q}=\left[ a\right] _{q}+q^{a}\left[ b\right] _{q}. \end{aligned}$$
(22)

From (22), one can show

$$\begin{aligned} \left[ a+b+c\right] _{q}=\left[ a\right] _{q}+q^{a}\left[ b\right] _{q}+q^{a+b}\left[ c\right] _{q}, \end{aligned}$$
(23)
$$\begin{aligned} \left[ ab\right] _{q}=\left[ a\right] _{q}\left[ b\right] _{q^{a}}. \end{aligned}$$

In this paper, we give several identities of symmetry in three variables related to Carlitz-type \(q\)-Euler polynomials and alternating \(q\)-power sums which are derived from the triple fermionic \(p\)-adic \(q\)-integral on \({{\mathbb {Z}}_{p}}\). These and most of their identities are new, since there have been results only about identities of symmetry in two variables.

Symmetric identities of \(q\)-Euler polynomials

First, we will consider the following triple integral which is obviously invariant under any permutations of \(w_{1},w_{2},w_{3}\). This simple observation is the philosophy that underlies this paper.

Let

$$\begin{aligned} I&= \int _{{\mathbb {Z}}_{p}^{3}}{\rm e}^{\left[ w_{2}w_{3}x_{1}+w_{1}w_{3}x_{2}+w_{1}w_{2}x_{3}+w_{1}w_{2}w_{3}\left( y_{1}+y_{2}+y_{3}\right) \right] _{q}t}\\&\quad \times\, {\rm d}\mu _{-q^{w_{2}w_{3}}}\left( x_{1}\right) {\rm d}\mu _{-q^{w_{1}w_{3}}}\left( x_{2}\right) {\rm d}\mu _{-q^{w_{1}w_{2}}}\left( x_{3}\right).\nonumber \end{aligned}$$
(24)

It is not difficult to show that

$$\begin{aligned}&\left[ w_{2}w_{3}x_{1}+w_{1}w_{3}x_{2}+w_{1}w_{2}x_{3}+w_{1}w_{2}w_{3}\left( y_{1}+y_{2}+y_{3}\right) \right] _{q}\\ &\quad=\left[ w_{2}w_{3}\right] _{q}\left[ x_{1}+w_{1}y_{1}\right] _{q^{w_{2}w_{3}}}+q^{w_{2}w_{3}\left( x_{1}+w_{1}y_{1}\right) }\left[ w_{1}w_{3}\right] _{q}\left[ x_{2}+w_{2}y_{2}\right] _{q^{w_{1}w_{3}}}\nonumber \\&\quad\quad+q^{w_{2}w_{3}\left( x_{1}+w_{1}y_{1}\right) +w_{1}w_{3}\left( x_{2}+w_{2}y_{2}\right) }\left[ w_{1}w_{2}\right] _{q}\left[ x_{3}+w_{3}y_{3}\right] _{q^{w_{1}w_{2}}}.\nonumber \end{aligned}$$
(25)

So the integrand of \(I\) is

$$\begin{aligned}&{\rm e}^{\left[ w_{2}w_{3}x_{1}+w_{1}w_{3}x_{2}+w_{1}w_{2}x_{3}+w_{1}w_{2}w_{3}\left( y_{1}+y_{2}+y_{3}\right) \right] _{q}t}\\ &={\rm e}^{\left[ w_{2}w_{3}\right] _{q}\left[ x_{1}+w_{1}y_{1}\right] _{q^{w_{2}w_{3}}}t}{\rm e}^{q^{w_{2}w_{3}\left( x_{1}+w_{1}y_{1}\right) }\left[ w_{1}w_{3}\right] _{q}\left[ x_{2}+w_{2}y_{2}\right] _{q^{w_{1}w_{3}}}t}\nonumber \\&\quad \times {\rm e}^{q^{w_{2}w_{3}\left( x_{1}+w_{1}y_{1}\right) +w_{1}w_{3}\left( x_{2}+w_{2}y_{2}\right) }\left[ w_{1}w_{2}\right] _{q}\left[ x_{3}+w_{3}y_{3}\right] _{q^{w_{1}w_{2}}}t}\nonumber \\ &=\sum _{n=0}^{\infty }\left\{ \sum _{k+l+m=n}\left(\begin{array}{l}{n}\\ {k,l,m}\end{array}\right)\left[ w_{2}w_{3}\right] _{q}^{k}\left[ w_{1}w_{3}\right] _{q}^{l}\left[ w_{1}w_{2}\right] _{q}^{m}q^{w_{1}w_{2}w_{3}\left( l+m\right) y_{1}}\right. \\&\quad \times q^{w_{1}w_{2}w_{3}my_{2}}q^{w_{2}w_{3}\left( l+m\right) x_{1}}\left[ x_{1}+w_{1}y_{1}\right] _{q^{w_{2}w_{3}}}^{k}q^{w_{1}w_{3}mx_{2}}\left[ x_{2}+w_{2}y_{2}\right] _{q^{w_{1}w_{3}}}^{l}\\& \quad \times\left. \left[ x_{3}+w_{3}y_{3}\right] _{q^{w_{1}w_{2}}}^{m}\right\} \frac{t^{n}}{n!}. \end{aligned}$$
(26)

Thus the integral in (24) is

$$\begin{aligned} I=&\sum _{n=0}^{\infty }\left\{ \sum _{k+l+m=n}\left(\begin{array}{l}{n}\\ {k,l,m}\end{array}\right)\left[ w_{2}w_{3}\right] _{q}^{k}\left[ w_{1}w_{3}\right] _{q}^{l}\left[ w_{1}w_{2}\right] _{q}^{m}q^{w_{1}w_{2}w_{3}\left( l+m\right) y_{1}}\right. \\&\,\times q^{w_{1}w_{2}w_{3}my_{2}}\int _{{{\mathbb {Z}}_{p}}}q^{w_{2}w_{3}\left( l+m\right) x_{1}}\left[ x_{1}+w_{1}y_{1}\right] _{q^{w_{2}w_{3}}}^{k}{\rm d}\mu _{-q^{w_{2}w_{3}}}\left( x_{1}\right) \nonumber \\&\,\times \int _{{{\mathbb {Z}}_{p}}}q^{w_{1}w_{3}mx_{2}}\left[ x_{2}+w_{2}y_{2}\right] _{q^{w_{1}w_{3}}}^{l}{\rm d}\mu _{-q^{w_{1}w_{3}}}\left( x_{2}\right) \nonumber \\&\left. \times \int _{{{\mathbb {Z}}_{p}}}\left[ x_{3}+w_{3}y_{3}\right] _{q^{w_{1}w_{2}}}^{m}{\rm d}\mu _{-q^{w_{1}w_{2}}}\left( x_{3}\right) \right\} \frac{t^{n}}{n!}\nonumber \\ =&\sum _{n=0}^{\infty }\left\{ \sum _{k+l+m=n}\left(\begin{array}{l}{n}\\ {k,l,m}\end{array}\right)\left[ w_{2}w_{3}\right] _{q}^{k}\left[ w_{1}w_{3}\right] _{q}^{l}\left[ w_{1}w_{2}\right] _{q}^{m}q^{w_{1}w_{2}w_{3}\left( l+m\right) y_{1}}\right. \nonumber \\&\left. \times q^{w_{1}w_{2}w_{3}my_{2}}\mathcal {E}_{k,q^{w_{2}w_{3}}}^{\left( l+m+1\right) }\left( w_{1}y_{1}\right) \mathcal {E}_{l,q^{w_{1}w_{3}}}^{\left( m+1\right) }\left( w_{2}y_{2}\right) \mathcal {E}_{m,q^{w_{1}w_{2}}}\left( w_{3}y_{3}\right) \right\} \frac{t^{n}}{n!}.\nonumber \end{aligned}$$
(27)

Thus, we get the following theorem.

Theorem 1

Let \(w_{1},w_{2},w_{3}\) be any positive integers, \(n\) any nonnegative integer. Then the following expression is invariant under any permutations of \(w_{1},w_{2},w_{3}\) , so that it gives us six symmetries:

$$\begin{aligned}&\sum _{k+l+m=n}\left(\begin{array}{l}{n}\\ {k,l,m}\end{array}\right)\left[ w_{2}w_{3}\right] _{q}^{k}\left[ w_{1}w_{3}\right] _{q}^{l}\left[ w_{1}w_{2}\right] _{q}^{m}q^{w_{1}w_{2}w_{3}\left( l+m\right) y_{1}}\\&\times q^{w_{1}w_{2}w_{3}my_{2}}\mathcal {E}_{k,q^{w_{2}w_{3}}}^{\left( l+m+1\right) }\left( w_{1}y_{1}\right) \mathcal {E}_{l,q^{w_{1}w_{3}}}^{\left( m+1\right) }\left( w_{2}y_{2}\right) \mathcal {E}_{m,q^{w_{1}w_{2}}}\left( w_{3}y_{3}\right) \\ =&\sum _{k+l+m=n}\left(\begin{array}{l}{n}\\ {k,l,m}\end{array}\right)\left[ w_{1}w_{3}\right] _{q}^{k}\left[ w_{2}w_{3}\right] _{q}^{l}\left[ w_{1}w_{2}\right] _{q}^{m}q^{w_{1}w_{2}w_{3}\left( l+m\right) y_{1}}\\&\times q^{w_{1}w_{2}w_{3}my_{2}}\mathcal {E}_{k,q^{w_{1}w_{3}}}^{\left( l+m+1\right) }\left( w_{2}y_{1}\right) \mathcal {E}_{l,q^{w_{2}w_{3}}}^{\left( m+1\right) }\left( w_{1}y_{2}\right) \mathcal {E}_{m,q^{w_{1}w_{2}}}\left( w_{3}y_{3}\right) \\ =&\sum _{k+l+m=n}\left(\begin{array}{l}{n}\\ {k,l,m}\end{array}\right)\left[ w_{1}w_{3}\right] _{q}^{k}\left[ w_{1}w_{2}\right] _{q}^{l}\left[ w_{2}w_{3}\right] _{q}^{m}q^{w_{1}w_{2}w_{3}\left( l+m\right) y_{1}}\\&\times q^{w_{1}w_{2}w_{3}my_{2}}\mathcal {E}_{k,q^{w_{1}w_{3}}}^{\left( l+m+1\right) }\left( w_{2}y_{1}\right) \mathcal {E}_{l,q^{w_{1}w_{2}}}^{\left( m+1\right) }\left( w_{3}y_{2}\right) \mathcal {E}_{m,q^{w_{2}w_{3}}}\left( w_{1}y_{3}\right) \\ =&\sum _{k+l+m=n}\left(\begin{array}{l}{n}\\ {k,l,m}\end{array}\right)\left[ w_{2}w_{3}\right] _{q}^{k}\left[ w_{1}w_{2}\right] _{q}^{l}\left[ w_{1}w_{3}\right] _{q}^{m}q^{w_{1}w_{2}w_{3}\left( l+m\right) y_{1}}\\&\times q^{w_{1}w_{2}w_{3}my_{2}}\mathcal {E}_{k,q^{w_{2}w_{3}}}^{\left( l+m+1\right) }\left( w_{1}y_{1}\right) \mathcal {E}_{l,q^{w_{1}w_{2}}}^{\left( m+1\right) }\left( w_{3}y_{2}\right) \mathcal {E}_{m,q^{w_{1}w_{3}}}\left( w_{2}y_{3}\right) \\ =&\sum _{k+l+m=n}\left(\begin{array}{l}{n}\\ {k,l,m}\end{array}\right)\left[ w_{1}w_{2}\right] _{q}^{k}\left[ w_{2}w_{3}\right] _{q}^{l}\left[ w_{1}w_{3}\right] _{q}^{m}q^{w_{1}w_{2}w_{3}\left( l+m\right) y_{1}}\\&\times q^{w_{1}w_{2}w_{3}my_{2}}\mathcal {E}_{k,q^{w_{1}w_{2}}}^{\left( l+m+1\right) }\left( w_{3}y_{1}\right) \mathcal {E}_{l,q^{w_{2}w_{3}}}^{\left( m+1\right) }\left( w_{1}y_{2}\right) \mathcal {E}_{m,q^{w_{1}w_{3}}}\left( w_{2}y_{3}\right) \\ =&\sum _{k+l+m=n}\left(\begin{array}{l}{n}\\ {k,l,m}\end{array}\right)\left[ w_{1}w_{2}\right] _{q}^{k}\left[ w_{1}w_{3}\right] _{q}^{l}\left[ w_{2}w_{3}\right] _{q}^{m}q^{w_{1}w_{2}w_{3}\left( l+m\right) y_{1}}\\&\times q^{w_{1}w_{2}w_{3}my_{2}}\mathcal {E}_{k,q^{w_{1}w_{2}}}^{\left( l+m+1\right) }\left( w_{3}y_{1}\right) \mathcal {E}_{l,q^{w_{1}w_{3}}}^{\left( m+1\right) }\left( w_{2}y_{2}\right) \mathcal {E}_{m,q^{w_{2}w_{3}}}\left(w_{1}y_{3}\right) . \end{aligned}$$

We define, for nonnegative integers \(n,m,w\), \(K_{n,m}\left( w|q\right) \) as

$$\begin{aligned} K_{n,m}\left( w|q\right) =\sum _{i=0}^{w}\left( -1\right) ^{i}q^{ni}\left[ i\right] _{q}^{m}. \end{aligned}$$
(28)

In particular, for \(w=0\), or \(m=0\), we have

$$\begin{aligned} K_{n,m}\left( 0|q\right) ={\left\{ \begin{array}{ll} 1, & \quad \text{ if } m=0\\ 0, & \quad \text{ if } m>0, \end{array}\right. } \end{aligned}$$
(29)

and

$$\begin{aligned} K_{n,0}\left( w|q\right) =\left[ w+1\right] _{-q^{n}}. \end{aligned}$$
(30)

We now apply the formula of (3) as follows :

$$\begin{aligned} q^{n}I_{-q}\left( f_{n}\right) +\left( -1\right) ^{n-1}I_{-q}\left( f\right) =\left[ 2\right] _{q}\sum _{l=0}^{n-1}\left( -1\right) ^{n-1-l}q^{l}f\left( l\right) , \end{aligned}$$
(31)

with

$$ \begin{aligned} f(x)={\rm e}^{\left[ w_{1}w_{2}x\right] _{q}t}, ''q=q^{w_{1}w_{2}}{''}, n=w_{3}. \end{aligned}$$

From (35), we have

$$\begin{aligned}&\left( q^{w_{1}w_{2}}\right) ^{w_{3}}\int _{{{\mathbb {Z}}_{p}}}{\rm e}^{\left[ w_{1}w_{2}\left( x+w_{3}\right) \right] _{q}t}{\rm d}\mu _{-q^{w_{1}w_{2}}}\left( x\right)\\ &\qquad +\left( -1\right) ^{w_{3}-1}\int _{{{\mathbb {Z}}_{p}}}{\rm e}^{\left[ w_{1}w_{2}x\right] _{q}t}{\rm d}\mu _{-q^{w_{1}w_{2}}}\left( x\right) \nonumber \\ &\quad =\left[ 2\right] _{q^{w_{1}w_{2}}}\sum _{l=0}^{w_{3}-1}\left( -1\right) ^{w_{3}-1-l}q^{w_{1}w_{2}l}{\rm e}^{\left[ w_{1}w_{2}l\right] _{q}t}\nonumber \\ &\quad =\left( -1\right) ^{w_{3}-1}\left[ 2\right] _{q^{w_{1}w_{2}}}\sum _{m=0}^{\infty }K_{1,m}\left( w_{3}-1|q^{w_{1}w_{2}}\right) \frac{\left( \left[ w_{1}w_{2}\right] _{q}t\right) ^{m}}{m!}\nonumber \\ &\quad=\sum _{m=0}^{\infty }\left\{ \left[ w_{1}w_{2}\right] _{q}^{m}\right. \nonumber \\&\qquad\times \left. \int _{{{\mathbb {Z}}_{p}}}\left( q^{w_{1}w_{2}w_{3}}\left[ x_{1}+w_{3}\right] _{q^{w_{1}w_{2}}}^{m}+\left( -1\right) ^{w_{3}-1}\left[ x\right] _{q^{w_{1}w_{2}}}^{m}\right) {\rm d}\mu _{-q^{w_{1}w_{2}}}\left( x\right) \right\} \frac{t^{m}}{m!}.\nonumber \end{aligned}$$
(32)

Thus, by (32), we obtain the following lemma.

Lemma 2

Let \(w_{1},w_{2}\) be any positive integers.

  1. (i)

    For \(w_{3}\equiv 1\) (mod 2), we have

    $$\begin{aligned}&q^{w_{1}w_{2}w_{3}}\int _{{{\mathbb {Z}}_{p}}}{\rm e}^{\left[ w_{1}w_{2}\left( x+w_{3}\right) \right] _{q}t}{\rm d}\mu _{-q^{w_{1}w_{2}}}\left( x\right) +\int _{{{\mathbb {Z}}_{p}}}{\rm e}^{\left[ w_{1}w_{2}x\right] _{q}t}{\rm d}\mu _{-q^{w_{1}w_{2}}}\left( x\right) \\ &\quad=\sum _{m=0}^{\infty }\left\{ \left[ w_{1}w_{2}\right] _{q}^{m}\int _{{{\mathbb {Z}}_{p}}}\left( q^{w_{1}w_{2}w_{3}}\left[ x+w_{3}\right] _{q^{w_{1}w_{2}}}^{m}+\left[ x\right] _{q^{w_{1}w_{2}}}^{m}\right) {\rm d}\mu _{-q^{w_{1}w_{2}}}\left( x\right) \right\} \frac{t^{m}}{m!}\\ &\quad=\left[ 2\right] _{q^{w_{1}w_{2}}}\sum _{i=0}^{w_{3}-1}\left( -1\right) ^{i}q^{w_{1}w_{2}i}{\rm e}^{\left[ w_{1}w_{2}i\right] _{q}t}\\ &\quad=\left[ 2\right] _{q^{w_{1}w_{2}}}\sum _{m=0}^{\infty }K_{1,m}\left( w_{3}-1|q^{w_{1}w_{2}}\right) \frac{\left( \left[ w_{1}w_{2}\right] _{q}t\right) ^{m}}{m!}. \end{aligned}$$
  2. (ii)

    For \(w_{3}\equiv 0\) (mod 2), we have

    $$\begin{aligned}&q^{w_{1}w_{2}w_{3}}\int _{{{\mathbb {Z}}_{p}}}{\rm e}^{\left[ w_{1}w_{2}\left( x+w_{3}\right) \right] _{q}t}{\rm d}\mu _{-q^{w_{1}w_{2}}}\left( x\right) -\int _{{{\mathbb {Z}}_{p}}}{\rm e}^{\left[ w_{1}w_{2}x\right] _{q}t}{\rm d}\mu _{-q^{w_{1}w_{2}}}\left( x\right) \\ =&\sum _{m=0}^{\infty }\left\{ \left[ w_{1}w_{2}\right] _{q}^{m}\int _{{{\mathbb {Z}}_{p}}}\left( q^{w_{1}w_{2}w_{3}}\left[ x+w_{3}\right] _{q^{w_{1}w_{2}}}^{m}-\left[ x\right] _{q^{w_{1}w_{2}}}^{m}\right) {\rm d}\mu _{-q^{w_{1}w_{2}}}\left( x\right) \right\} \frac{t^{m}}{m!}\\ =&-\left[ 2\right] _{q^{w_{1}w_{2}}}\sum _{i=0}^{w_{3}-1}\left( -1\right) ^{i}q^{w_{1}w_{2}i}{\rm e}^{\left[ w_{1}w_{2}i\right] _{q}t}\\ =&-\left[ 2\right] _{q^{w_{1}w_{2}}}\sum _{m=0}^{\infty }K_{1,m}\left( w_{3}-1|q^{w_{1}w_{2}}\right) \frac{\left( \left[ w_{1}w_{2}\right] _{q}t\right) ^{m}}{m!}. \end{aligned}$$

Consider the following sum of triple integrals.

$$\begin{aligned} I_{1}&=q^{w_{1}w_{2}w_{3}}\int _{{\mathbb {Z}}_{p}^{3}}{\rm e}^{\left[ w_{2}w_{3}x_{1}+w_{1}w_{3}x_{2}+w_{1}w_{2}x_{3}+w_{1}w_{2}w_{3}\left( y_{1}+y_{2}+1\right) \right] _{q}t}\\&\quad \times {\rm d}\mu _{-q^{w_{2}w_{3}}}\left( x_{1}\right) {\rm d}\mu _{-q^{w_{1}w_{3}}}\left( x_{2}\right) {\rm d}\mu _{-q^{w_{1}w_{2}}}\left( x_{3}\right) \nonumber \\&\quad+\int _{{\mathbb {Z}}_{p}^{3}}{\rm e}^{\left[ w_{2}w_{3}x_{1}+w_{1}w_{3}x_{2}+w_{1}w_{2}x_{3}+w_{1}w_{2}w_{3}\left( y_{1}+y_{2}\right) \right] _{q}t}\nonumber \\&\quad \times {\rm d}\mu _{-q^{w_{2}w_{3}}}\left( x_{1}\right) {\rm d}\mu _{-q^{w_{1}w_{3}}}\left( x_{2}\right) {\rm d}\mu _{-q^{w_{1}w_{2}}}\left( x_{3}\right) ,\nonumber \end{aligned}$$
(33)

which is obviously invariant under any permutations of \(w_{1}\), \(w_{2}\), \(w_{3}\).

For simplicity, we put

$$\begin{aligned} a=a\left( x_{1}\right) =q^{w_{2}w_{3}\left( x_{1}+w_{1}y_{1}\right) },\quad b=b\left( x_{2}\right) =q^{w_{1}w_{3}\left( x_{2}+w_{2}y_{2}\right) }. \end{aligned}$$
(34)

Then, from (33), we note that

$$\begin{aligned} I_{1}=&\sum _{k,l=0}^{\infty }\left[ w_{2}w_{3}\right] _{q}^{k}\left[ w_{1}w_{3}\right] _{q}^{l}\frac{t^{k+l}}{k!l!}\int _{\mathbb {Z}_{p}^{2}}a^{l}\left[ x_{1}+w_{1}y_{1}\right] _{q^{w_{2}w_{3}}}^{k}\left[ x_{2}+w_{2}y_{2}\right] _{q^{w_{1}w_{3}}}^{l}\\&\times \left\{ \sum _{m=0}^{\infty }\frac{\left[ w_{1}w_{2}\right] _{q}^{m}\left( abt\right) ^{m}}{m!}\right. \nonumber \\&\times \left. \int _{{{\mathbb {Z}}_{p}}}\left( q^{w_{1}w_{2}w_{3}}\left[ x_{3}+w_{3}\right] _{q^{w_{1}w_{2}}}^{m}+\left[ x_{3}\right] _{q^{w_{1}w_{2}}}^{m}\right) {\rm d}\mu _{-q^{w_{1}w_{2}}}\left( x_{3}\right) \right\} \nonumber \\&\times {\rm d}\mu _{-q^{w_{2}w_{3}}}\left( x_{1}\right) {\rm d}\mu _{-q^{w_{1}w_{3}}}\left( x_{2}\right) .\nonumber \end{aligned}$$
(35)

Assume now that \(w_{3}\equiv 1\) (mod 2). Then, by (i) of Lemma 2, we get

$$\begin{aligned} I_{1}=&\left[ 2\right] _{q^{w_{1}w_{2}}}\sum _{k,l,m=0}^{\infty }\left[ w_{2}w_{3}\right] _{q}^{k}\left[ w_{1}w_{3}\right] _{q}^{l}\left[ w_{1}w_{2}\right] _{q}^{m}\frac{t^{k+l+m}}{k!l!m!}\\&\times K_{1,m}\left( w_{3}-1|q^{w_{1}w_{2}}\right) \int _{{{\mathbb {Z}}_{p}}}a^{l+m}\left[ x_{1}+w_{1}y_{1}\right] _{q^{w_{2}w_{3}}}^{k}{\rm d}\mu _{-q^{w_{2}w_{3}}}\left( x_{1}\right) \nonumber \\&\times \int _{{{\mathbb {Z}}_{p}}}b^{m}\left[ x_{2}+w_{2}y_{2}\right] _{q^{w_{1}w_{3}}}^{l}{\rm d}\mu _{-q^{w_{1}w_{3}}}\left( x_{2}\right) .\nonumber \end{aligned}$$
(36)

Recovering \(a=q^{w_{2}w_{3}\left( x_{1}+w_{1}y_{1}\right) }\), \(b=q^{w_{1}w_{3}\left( x_{2}+w_{2}y_{2}\right) }\), \(I_{1}\) can be rewritten as

$$\begin{aligned} I_{1}=&\left[ 2\right] _{q^{w_{1}w_{2}}}\sum _{k,l,m=0}^{\infty }\left[ w_{2}w_{3}\right] _{q}^{k}\left[ w_{1}w_{3}\right] _{q}^{l}\left[ w_{1}w_{2}\right] _{q}^{m}\frac{t^{k+l+m}}{k!l!m!}\\&\times K_{1,m}\left( w_{3}-1|q^{w_{1}w_{2}}\right) q^{w_{1}w_{2}w_{3}\left( l+m\right) y_{1}}q^{w_{1}w_{2}w_{3}my_{2}}\nonumber \\&\times \int _{{{\mathbb {Z}}_{p}}}q^{w_{2}w_{3}\left( l+m\right) x_{1}}\left[ x_{1}+w_{1}y_{1}\right] _{q^{w_{2}w_{3}}}^{k}{\rm d}\mu _{-q^{w_{2}w_{3}}}\left( x_{1}\right) \nonumber \\&\times \int _{{{\mathbb {Z}}_{p}}}q^{w_{1}w_{3}mx_{2}}\left[ x_{2}+w_{2}y_{2}\right] _{q^{w_{1}w_{3}}}^{l}{\rm d}\mu _{-q^{w_{1}w_{3}}}\left( x_{2}\right) \nonumber \\ =&\sum _{n=0}^{\infty }\left\{ \left[ 2\right] _{q^{w_{1}w_{2}}}\sum _{k+l+m=n}\left(\begin{array}{l}{n}\\ {k,l,m}\end{array}\right)\left[ w_{2}w_{3}\right] _{q}^{k}\left[ w_{1}w_{3}\right] _{q}^{l}\left[ w_{1}w_{2}\right] _{q}^{m}\right. \nonumber \\&\,\times K_{1,m}\left( w_{3}-1|q^{w_{1}w_{2}}\right) q^{w_{1}w_{2}w_{3}\left( l+m\right) y_{1}}q^{w_{1}w_{2}w_{3}my_{2}}\nonumber \\& \times \left.\mathcal {E}_{k,q^{w_{2}w_{3}}}^{\left( l+m+1\right) }\left( w_{1}y_{1}\right) \mathcal {E}_{l,q^{w_{1}w_{3}}}^{\left( m+1\right) }\left( w_{2}y_{2}\right) \right\} \frac{t^{n}}{n!}.\nonumber \end{aligned}$$
(37)

As the expression in (37) is invariant under any permutations of \(w_{1},w_{2},w_{3}\) and it is equal to (37) provided that \(w_{3}\equiv 1\) (mod 2), we see that the expression in the curly bracket of (37) is invariant under any permutations of \(w_{1},w_{2},w_{3}\), when \(w_{1}\equiv w_{2}\equiv w_{3}\equiv 1\) (mod 2). Instead of the sum of triple integrals in (33), we now consider their difference, namely,

$$\begin{aligned}&q^{w_{1}w_{2}w_{3}}\int _{{\mathbb {Z}}_{p}^{3}}{\rm e}^{\left[ w_{2}w_{3}x_{1}+w_{1}w_{3}x_{2}+w_{1}w_{2}x_{3}+w_{1}w_{2}w_{3}\left( y_{1}+y_{2}+1\right) \right] _{q}t}\\&\times {\rm d}\mu _{-q^{w_{2}w_{3}}}\left( x_{1}\right) {\rm d}\mu _{-q^{w_{1}w_{3}}}\left( x_{2}\right) {\rm d}\mu _{-q^{w_{1}w_{2}}}\left( x_{3}\right) \nonumber \\&\quad -\int _{{\mathbb {Z}}_{p}^{3}}{\rm e}^{\left[ w_{2}w_{3}x_{1}+w_{1}w_{3}x_{2}+w_{1}w_{2}x_{3}+w_{1}w_{2}w_{3}\left( y_{1}+y_{2}\right) \right] _{q}t}\nonumber \\&\times {\rm d}\mu _{-q^{w_{2}w_{3}}}\left( x_{1}\right) {\rm d}\mu _{-q^{w_{1}w_{3}}}\left( x_{2}\right) {\rm d}\mu _{-q^{w_{1}w_{2}}}\left( x_{3}\right) ,\nonumber \end{aligned}$$
(38)

which is invariant under any permutations of \(w_{1},w_{2},w_{3}\). Proceeding analogously to the above and using (ii) of Lemma 2, we see that (38) is equal to the negative of the expression in (37), provided that \(w_{3}\equiv 0\) (mod 2). Thus, we see that the expression in curly bracket of (37) is invariant under any permutations of \(w_{1},w_{2},w_{3}\), when \(w_{1}\equiv w_{2}\equiv w_{3}\equiv 0\) (mod \(2\)).

Thus, we obtain the following theorem.

Theorem 3

Let \(w_{1},w_{2},w_{3}\) be positive integers satisfying either \(w_{1}\equiv w_{2}\equiv w_{3}\equiv 1\) (mod 2) or \(w_{1}\equiv w_{2}\equiv w_{3}\equiv 0\) (mod 2). Then, for any nonnegative integer \(n\) , the following expressions

$$\begin{aligned}&\left[ 2\right] _{q^{w_{\sigma \left( 1\right) \sigma \left( 2\right) }}}\sum _{k+l+m=n}\left(\begin{array}{l}{n}\\ {k,l,m}\end{array}\right)\mathcal {E}_{k,q^{w_{\sigma \left( 2\right) }w_{\sigma \left( 3\right) }}}^{\left( l+m+1\right) }\left( w_{\sigma \left( 1\right) }y_{1}\right) \\&\times \mathcal {E}_{l,q^{w_{\sigma \left( 1\right) }w_{\sigma \left( 3\right) }}}^{\left( m+1\right) }\left( w_{\sigma \left( 2\right) }y_{2}\right) K_{1,m}\left( w_{\sigma \left( 3\right) }-1|q^{w_{\sigma \left( 1\right) }w_{\sigma \left( 2\right) }}\right) \\&\times q^{w_{1}w_{2}w_{3}\left( l+m\right) y_{1}}q^{w_{1}w_{2}w_{3}my_{2}}\left[ w_{\sigma \left( 2\right) }w_{\sigma \left( 3\right) }\right] _{q}^{k}\\&\times \left[ w_{\sigma \left( 1\right) }w_{\sigma \left( 3\right) }\right] _{q}^{l}\left[ w_{\sigma \left( 1\right) }w_{\sigma \left( 2\right) }\right] _{q}^{m} \end{aligned}$$

are all the same for any \(\sigma \in S_{3}\).

Remark 1 We can obtain many interesting identities by letting \(w_{3}=1\) or \(w_{2}=w_{3}=1\), in view of (29). However, writing those down requires much space and so we omit it.

With the same

$$\begin{aligned} a=q^{w_{2}w_{3}\left( x_{1}+w_{1}y_{1}\right) },\quad b=q^{w_{1}w_{3}\left( x_{2}+w_{2}y_{2}\right) }, \end{aligned}$$

\(I_{1}\) can be written as

$$\begin{aligned} I_{1}=&\int _{{\mathbb {Z}}_{p}^{2}}{\rm e}^{\left[ w_{2}w_{3}\right] _{q}\left[ x_{1}+w_{1}y_{1}\right] _{q^{w_{2}w_{3}}}t}{\rm e}^{\left[ w_{1}w_{3}\right] _{q}\left[ x_{2}+w_{2}y_{2}\right] _{q^{w_{1}w_{3}}}at}\\&\times {\rm d}\mu _{-q^{w_{2}w_{3}}}\left( x_{1}\right) {\rm d}\mu _{-q^{w_{1}w_{3}}}\left( x_{2}\right) \nonumber \\&\times \int _{{{\mathbb {Z}}_{p}}}\left( q^{w_{1}w_{2}w_{3}}{\rm e}^{\left[ w_{1}w_{2}\left( x_{3}+w_{3}\right) \right] _{q}abt}+{\rm e}^{\left[ w_{1}w_{2}x_{3}\right] _{q}abt}\right) {\rm d}\mu _{-q^{w_{1}w_{2}}}\left( x_{3}\right) .\nonumber \end{aligned}$$
(39)

Let \(w_{3}\equiv 1\) (mod 2). Then, from (i) of Lemma 2, the inner integral in (39) is

$$\begin{aligned} \left[ 2\right] _{q^{w_{1}w_{2}}}\sum _{i=0}^{w_{3}-1}\left( -1\right) ^{i}q^{w_{1}w_{2}i}{\rm e}^{\left[ w_{1}w_{2}i\right] _{q}abt}. \end{aligned}$$
(40)

So (40) is equal to

$$\begin{aligned} I_{1}=&\left[ 2\right] _{q^{w_{1}w_{2}}}\sum _{i=0}^{w_{3}-1}\left( -1\right) ^{i}q^{w_{1}w_{2}i}\int _{{\mathbb {Z}}_{p}^{2}}{\rm e}^{\left[ w_{2}w_{3}\right] _{q}\left[ x_{1}+w_{1}y_{1}\right] _{q^{w_{2}w_{3}}}t}\\&\times {\rm e}^{\left[ w_{1}w_{3}\right] _{q}\left[ x_{2}+w_{2}y_{2}+\frac{w_{2}}{w_{3}}i\right] _{q^{w_{1}w_{3}}}at}{\rm d}\mu _{-q^{w_{2}w_{3}}}\left( x_{1}\right) {\rm d}\mu _{-q^{w_{1}w_{3}}}\left( x_{2}\right) \nonumber \\ =&\left[ 2\right] _{q^{w_{1}w_{2}}}\sum _{i=0}^{w_{3}-1}\left( -1\right) ^{i}q^{w_{1}w_{2}i}\sum _{k,l=0}\frac{t^{k+l}}{k!l!}q^{w_{1}w_{2}w_{3}ly_{1}}\left[ w_{2}w_{3}\right] _{q}^{k}\left[ w_{1}w_{3}\right] _{q}^{l}\nonumber \\&\times \int _{{{\mathbb {Z}}_{p}}}q^{w_{2}w_{3}lx_{1}}\left[ x_{1}+w_{1}y_{1}\right] _{q^{w_{2}w_{3}}}^{k}{\rm d}\mu _{-q^{w_{2}w_{3}}}\left( x_{1}\right) \nonumber \\&\times \int _{{{\mathbb {Z}}_{p}}}\left[ x_{2}+w_{2}y_{2}+\frac{w_{2}}{w_{3}}i\right] _{q^{w_{1}w_{3}}}^{l}{\rm d}\mu _{-q^{w_{1}w_{3}}}\left( x_{2}\right) \nonumber \\ =&\sum _{n=0}^{\infty }\left\{ \left[ 2\right] _{q^{w_{1}w_{2}}}\sum _{k=0}^{n}\left(\begin{array}{l}{n}\\ {k}\end{array}\right)\mathcal {E}_{k,q^{w_{2}w_{3}}}^{\left( n-k+1\right) }\left( w_{1}y_{1}\right) q^{w_{1}w_{2}w_{3}\left( n-k\right) y_{1}}\left[ w_{2}w_{3}\right] _{q}^{k}\left[ w_{1}w_{3}\right] _{q}^{n-k}\right. \nonumber \\&\left. \times \sum _{i=0}^{w_{3}-1}\left( -1\right) ^{i}q^{w_{1}w_{2}i}\mathcal {E}_{l,q^{w_{1}w_{3}}}\left( w_{2}y_{2}+\frac{w_{2}}{w_{3}}i\right) \right\} \frac{t^{n}}{n!}.\nonumber \end{aligned}$$
(41)

Recalling that \(I_{1}\) is invariant under any permutations of \(w_{1},w_{2},w_{3}\) and it is equal to (41) for \(w_{3}\equiv 1\) (mod 2), we see that the expression in the curly bracket of (41) is invariant under any permutations of \(w_{1},w_{2},w_{3}\), when \(w_{1}\equiv w_{2}\equiv w_{3}\equiv 1\) (mod 2). Also, starting from (39), using (ii) of Lemma 2, and proceeding analogous to the above, we see that the expression in the curly bracket of (41) is also invariant under any permutations of \(w_{1},w_{2},w_{3}\), when \(w_{1}\equiv w_{2}\equiv w_{3}\equiv 0\) (mod 2). Thus, we have the following theorem.

Theorem 4

Let \(w_{1},w_{2},w_{3}\) be positive integers satisfying either \(w_{1}\equiv w_{2}\equiv w_{3}\equiv 1\) (mod 2) or \(w_{1}\equiv w_{2}\equiv w_{3}\equiv 0\) (mod 2). Then, for any nonnegative integer \(n\) , the following expressions

$$\begin{aligned}&\left[ 2\right] _{q^{w_{\sigma \left( 1\right) }w_{\sigma \left( 2\right) }}}\sum _{k=0}^{n}\left(\begin{array}{l}{n}\\ {k}\end{array}\right)\mathcal {E}_{k,q^{w_{\sigma \left( 2\right) }w_{\sigma \left( 3\right) }}}^{\left( n-k+1\right) }\left( w_{\sigma \left( 1\right) }y_{1}\right) q^{w_{1}w_{2}w_{3}\left( n-k\right) y_{1}}\left[ w_{\sigma \left( 2\right) }w_{\sigma \left( 3\right) }\right] _{q}^{k}\\&\times \left[ w_{\sigma \left( 1\right) }w_{\sigma \left( 3\right) }\right] _{q}^{n-k}\sum _{i=0}^{w_{\sigma \left( 3\right) }-1}\left( -1\right) ^{i}q^{w_{\sigma \left( 1\right) }w_{\sigma \left( 2\right) }i}\mathcal {E}_{l,q^{w_{\sigma \left( 1\right) }w_{\sigma \left( 3\right) }}}\left( w_{\sigma \left( 2\right) }y_{2}+\frac{w_{\sigma \left( 2\right) }}{w_{\sigma \left( 3\right) }}i\right) \end{aligned}$$

are all the same for any \(\sigma \in S_{3}\).

Remark 2 In view of (29), by specializing \(w_{3}=1\) or \(w_{2}=w_{3}=1\), we can obtain many interesting identities. However, we will omit those, as this requires much space.