1 Introduction

We attribute a substantial study of Bernoulli polynomials and Euler polynomials to Carlitz. He used an inductive method to define these special polynomials (see [1]).

In this paper we use the p-adic q-integral to define the special polynomials. It is Kim who has established the theory of the p-adic q-integral (see [2]). The p-adic q-integral provides fruitful and essential results in studying special polynomials such as Bernoulli, Euler and Catalan polynomials. One of such results is the symmetry of the special polynomials.

Throughout this paper p is a fixed odd prime number. We use the notations \({\mathbb{Z}}_{p}\), \({\mathbb{Q}}_{p}\) and \({\mathbb{C}}_{p}\) to express the ring of p-adic integers, the field of p-adic rational numbers and the completion of algebraic closure of \({\mathbb{Q}}_{p}\), respectively. The p-adic norm \(|\cdot|_{p}\) is normalized as \(|p|_{p}=\frac{1}{p}\). For \(q, x \in{\mathbb{C}}_{p}\) with \(|q-1|_{p}< p^{-\frac{1}{p-1}}\). We define the q-analogue of a number x to be \([x]_{q}=\frac{1-q^{x}}{1-q}\). Note that \(\lim_{q\rightarrow1}[x]_{q}=x\).

Let \(\mathit{UD}(\mathbb{Z}_{p})\) be the space of uniformly differentiable functions on \(\mathbb{Z}_{p}\) and \(f \in\mathit{UD}(\mathbb{Z}_{p})\). Kim has introduced the fermionic p-adic q-integral \(I_{-q}(f)\) on \(\mathbb{Z}_{p}\) (see [1, 3–10]).

$$ I_{-q}(f) = \int_{\mathbb{Z}_{p}} f(x)\,du_{-q}(x)= \lim_{N\rightarrow \infty} \frac{1}{[p^{N}]_{-q}} \sum_{x=0}^{p^{N}-1}f(x) (-q)^{x}. $$
(1.1)

The q-Euler polynomials \(\mathcal{E}_{n,q}(x)\) are defined by the generating function

$$ \int_{\mathbb{Z}_{p}} e^{[x+y]_{q} t} \,d\mu_{-q}(y) = \sum _{n=0} ^{\infty}\mathcal{E}_{n,q}(x) \frac{t^{n}}{n!}\quad \bigl( \text{see [2, 10--18]}\bigr). $$
(1.2)

Also the modified q-Euler polynomials \(E_{n, q}(x)\) are defined by the generating function

$$ \int_{\mathbb{Z}_{p}} e^{[x+y]_{q} t} \,d\mu_{-1}(y) = \sum _{n=0} ^{\infty} E_{n,q}(x) \frac{t^{n}}{n!}\quad \bigl( \text{see [19, 20]}\bigr). $$
(1.3)

Note that \(\mathcal{E}_{n,q}(x) = \int_{\mathbb{Z}_{p}} [x+y]_{q}^{n} \,d\mu _{-q}(y)\) and \(E_{n,q}(x) = \int_{\mathbb{Z}_{p}} [x+y]_{q}^{n} \,d\mu_{-1}(y)\).

For v and w odd positive integers, put \(p(x, u, v, w) = \sum_{i=0}^{v-1} (-1)^{i} \int_{\mathbb{Z}_{p}} e^{ [u x + vy + w i ]_{q} t} \,d\mu_{-1}(y)\). Then

$$ \begin{aligned}[b] p(x, u, v, w) = & \sum _{i=0}^{v-1} (-1)^{i} \int_{\mathbb{Z}_{p}} e^{ [u x + v y + w i ]_{q} t} \,d\mu_{-1}(y) \\ = & \sum_{i=0}^{v-1} (-1)^{i} \lim_{N \rightarrow\infty} \sum_{y=0}^{p^{N} -1} e^{ [u x + v y + w i ]_{q} t} (-1)^{y} \\ = & \sum_{i=0}^{v-1} (-1)^{i} \lim_{N \rightarrow\infty} \sum_{y=0}^{p^{N} -1} \sum_{k=0}^{w-1} e^{ [ux + v(k+wy)+ wi ]_{q} t} (-1)^{k+wy} \\ = & \sum_{i=0}^{v-1} \sum _{k=0}^{w-1} (-1)^{i+k} \lim _{N \rightarrow \infty} \sum_{y=0}^{p^{N} -1} e^{ [ux+vwy + vk+wi ]_{q} t} (-1)^{y}. \end{aligned} $$
(1.4)

Note that the last line is invariant under the transposition \((v, w)\). So we obtain the ’basic’ symmetry of the generating functions

$$ p(x, u, v, w)=p(x, u, w, v). $$
(1.5)

In this paper we use four weights \(w_{1}\), \(w_{2} \), \(w_{3} \), \(w_{4}\) instead of u, v, w. Then the symmetric group \(S_{4}\) of degree 4 naturally acts on the generating functions. We shall describe the group action more generally. Let \(p=p(x, w_{1} , \ldots, w_{n})\) be generating functions with a variable x and n weights \(w_{1} , \ldots, w_{n}\) involved in the definition of p. For \(\sigma\in S_{n}\), define

$$ \sigma \bigl( p(x, w_{1} , \ldots, w_{n}) \bigr) = p \bigl( x, w_{\sigma}(1) , \ldots, w_{\sigma}(n) \bigr) . $$
(1.6)

Then it is obvious that (i) \(\mathit{id}(p) = p\) and (ii) \((\sigma\tau)(p) = (\sigma(\tau(p)))\) hold. Hence this is a group action naturally.

In this paper we will consider a set X of 315 generating functions, \(p=p(x, w_{1} , \ldots, w_{4})\), and let \(S_{4}\) act on this set. There will be 19 orbits in X under the action of \(S_{4}\), and 4 non-isomorphic subgroups of \(S_{4}\) as the isotropy groups.

Throughout this paper \(w_{1}\), \(w_{2} \), \(w_{3} \), \(w_{4}\) are odd positive integers. Consider the generating functions \(p(x)\), where

$$ p(x) = \sum_{i=0}^{w_{j} w_{k} -1} (-1)^{i} \int_{\mathbb{Z}_{p}} e^{ [(w_{a} w_{b} +w_{c} w_{d} ) x + w_{j} w_{k} y + w_{l} w_{m} i ]_{q} t} \,d\mu_{-1}(y), $$
(1.7)

where \(1 \leq a, b, c, d, j, k, l, m \leq4 \), \(a\neq b\), \(c\neq d\), \(j \neq k\), \(l\neq m \) and \(w_{a} w_{b} \neq w_{c} w_{d} \). That is, we use square-free quadratic forms of \(w's\).

By the abuse of notation, we denote \(p(x)\) by \(p(ab+cd, jk, lm)\). That is,

$$ \begin{aligned} p(ab+cd, jk, lm) = \sum _{i=0}^{w_{j} w_{k} -1} (-1)^{i} \int_{\mathbb {Z}_{p}} e^{ [(w_{a} w_{b} +w_{c} w_{d} ) x + w_{j} w_{k} y + w_{l} w_{m} i ]_{q} t} \,d\mu_{-1}(y). \end{aligned} $$
(1.8)

Then, due to the ‘basic’ symmetry in (1.5), \(p(ab+cd, jk, lm)=p(ab+cd, lm, jk)\). Let X be the set of all \(p(ab+cd, jk, lm)\). Then there are \(15 \times6\) generating functions of the form \(p(ab+cd, jk, jk)\) and \(15 \times15\) generating functions of the form \(p(ab+cd, jk, lm)\), where \(w_{j} w_{k} \neq w_{l} w_{m}\). So X is a set of 315 generating functions.

Let the symmetric group \(S_{4}\) act on X naturally. That is, for any \(\sigma\in S_{4} \),

$$ \sigma \bigl( p(ab+cd, jk, lm) \bigr) = p \bigl( \sigma(a) \sigma (b) + \sigma(c) \sigma(d), \sigma(j) \sigma(k), \sigma(l) \sigma (m) \bigr) . $$
(1.9)

We will find each orbit and the isotropy subgroup of \(S_{4} \) of an element in the orbit. Note that we have the basic counting of group action such that the product of the cardinality of an orbit and that of the corresponding isotropy subgroup is \(\vert S_{4} \vert = 24\).

2 Quadratic symmetry of the modified q-Euler polynomials with the isotropy group \(D_{4}\)

We use the notation in (1.8) and put \(f_{1} = p(12+34, 12 , 34)\), \(f_{2} = p(13+24, 13, 24 )\) and \(f_{3} = p(14+24, 14, 23)\). Then, for \(\tau= (2, 3, 4)\) and \(\tau^{2} = (2, 4, 3)\),

$$ \begin{gathered} \begin{aligned}\tau(f_{1}) & = \tau \bigl( p(12+34, 12, 34) \bigr) = p \bigl( \tau (1) \tau(2) + \tau(3) \tau(4), \tau(1) \tau(2), \tau(3) \tau(4) \bigr) \\ & = p(13+42, 13, 42) = p(13+24, 13, 24) = f_{2},\end{aligned} \\ \tau^{2} (f_{1}) = \tau^{2} \bigl( p(12+34, 12, 34) \bigr) = p(14+23, 14, 23) = f_{3}. \end{gathered} $$
(2.1)

Hence \(f_{1} \), \(f_{2} \), \(f_{3} \) are in an orbit under the action of \(S_{4}\) on X.

Now consider a 4-cycle \(\sigma_{1} = (1, 3, 2, 4)\) and a 2-cycle \(\sigma _{2} = (1, 2)\). It is obvious that \(\sigma_{2}\) fixes \(f_{1}\). Note that \(w_{1} w_{2} + w_{3} w_{4} = w_{3} w_{4} + w_{1} w_{2}\). By the basic symmetry in (1.5), we obtain

$$ \begin{aligned}[b] \sigma_{1} \bigl( p(12+34, 12, 34) \bigr) & = p \bigl( \sigma_{1} (1) \sigma_{1} (2) + \sigma_{1} (3) \sigma_{1} (4), \sigma_{1} (1) \sigma_{1} (2), \sigma_{1} (3) \sigma_{1} (4) \bigr) \\ &= p(34+12, 34, 12) = p(12+34, 12, 34). \end{aligned} $$
(2.2)

That is, \(\sigma_{1}\) fixes \(f_{1}\) also. Hence the dihedral group \(D_{4} = \langle (1, 3, 2, 4), (1,2) \rangle \) is a subgroup of the isotropy subgroup of \(f_{1}\). However, we find three elements \(f_{1}\), \(f_{2}\), \(f_{3}\) in the orbit of \(f_{1}\) and \(\vert D_{4} \vert = 8\). Thus \(X_{1} = \{ f_{1}, f_{2}, f_{3} \} \) is the orbit of \(f_{1}\) and \(D_{4} = \langle (1, 3, 2, 4), (1,2) \rangle \) is the isotropy subgroup of \(S_{4}\) fixing \(f_{1}\).

Further, we can check that the cyclic group \(\langle (2, 3, 4) \rangle \) acts transitively on \(X_{1}\). That is,

$$ \begin{gathered} (2, 3, 4) (f_{1}) = (2, 3, 4) \bigl( p(12+34, 12, 34) \bigr) = p(13+42, 13, 42) = f_{2}, \\ (2, 4, 3) (f_{1}) = (2, 4, 3) \bigl( p(12+34, 12, 34) \bigr) = p(14+23, 14, 23) = f_{3}. \end{gathered} $$
(2.3)

Theorem 2.1

Let X be the set of all generating functions of the form

$$p(ab+cd, jk, lm) = \sum_{i=0}^{w_{j} w_{k} -1}(-1)^{i} \int_{\mathbb{Z}_{p}} e^{ [(w_{a} w_{b} +w_{c} w_{d} ) x + w_{j} w_{k} y + w_{l} w_{m} i ]_{q} t} \,d\mu_{-1}(y) $$

and \(S_{4}\) act on X by permuting \(w_{1} \), \(w_{2} \), \(w_{3} \), \(w_{4}\). Take \(f_{1} = p(12+34, 12 , 34) \in X\). Then the isotropy subgroup of \(f_{1}\) is the dihedral group \(D_{4} = \langle (1, 3, 2, 4), (1,2) \rangle \) and the orbit of \(f_{1}\) is \(X_{1} = \{ f_{1}, f_{2} = p(13+24, 13, 24 ), f_{3} = p(14+23, 14, 23) \}\). Moreover, the cyclic group \(\langle (2, 3, 4) \rangle \) acts transitively on \(X_{1}\).

Remark

Note that the isotropy subgroup of \(f_{2} = p(13+24, 13, 24 )\) is \(D_{4} ^{1} = \langle (1, 2, 3, 4), (1,3) \rangle \) and the isotropy subgroup of \(f_{3} = p(14+23, 14, 23)\) is \(D_{4} ^{2} = \langle (1, 2, 4, 3), (1,4) \rangle \).

Now we need to transform the q-analogue in the definition of \(f_{1} = p(12+34, 12, 34 )\) to express it in the form of Euler polynomials.

$$ \begin{aligned} [b]&\bigl[ (w_{1} w_{2} + w_{3} w_{4} )x + w_{1} w_{2} y + w_{3} w_{4} i \bigr]_{q} \\&\quad = \frac{1-q^{(w_{1} w_{2} + w_{3} w_{4} )x + w_{1} w_{2} y + w_{3} w_{4} i}}{1-q} \\ &\quad = \biggl( \frac{1-q^{w_{1} w_{2}}}{1-q} \biggr) \biggl( \frac {1-q^{(w_{1} w_{2} + w_{3} w_{4} )x + w_{1} w_{2} y + w_{3} w_{4} i }}{1-q^{w_{1} w_{2}}} \biggr) \\ &\quad = \biggl( \frac{1-q^{w_{1} w_{2}}}{1-q} \biggr) \biggl( \frac{ 1-q^{w_{1} w_{2} \{(1+ \frac{w_{3} w_{4}}{w_{1} w_{2}})x + y + \frac {w_{3}w_{4}}{w_{1}w_{2}}i \}} }{1-q^{w_{1} w_{2}}} \biggr) \\ &\quad = [w_{1} w_{2} ]_{q} \biggl[ \biggl( 1+ \frac{w_{3} w_{4}}{w_{1} w_{2}} \biggr)x+y +\frac{w_{3} w_{4}}{w_{1} w_{2}}i \biggr]_{q^{w_{1} w_{2}}}. \end{aligned} $$
(2.4)

Hence

$$ \begin{aligned}[b] f_{1} & = p(12+34, 12 , 34) \\ & = \sum_{i=0}^{w_{1} w_{2} -1} (-1)^{i} \int_{\mathbb{Z}_{p}} e^{ [(w_{1} w_{2} +w_{3} w_{4} ) x + w_{1} w_{2} y + w_{3} w_{4} i ]_{q} t} \,d\mu _{-1}(y) \\ & = \sum_{n=0}^{\infty}\sum _{i=0}^{w_{1} w_{2} -1} (-1)^{i} [w_{1} w_{2} ]_{q} ^{n} E_{n, q^{w_{1} w_{2}}} \biggl( \biggl( 1+ \frac{w_{3} w_{4}}{w_{1} w_{2}} \biggr)x+\frac{w_{3} w_{4}}{w_{1} w_{2}}i \biggr) \frac{t^{n}}{n!}. \end{aligned} $$
(2.5)

Theorem 2.2

For \(q \in{\mathbb{C}}_{p}\) with \(|q-1|_{p}< p^{-\frac{1}{p-1}}\) and \(n \geq0\), the modified q-Euler polynomials

$$ \sum_{i=0}^{w_{1} w_{2} -1} (-1)^{i} [w_{1} w_{2} ]_{q} ^{n} E_{n, q^{w_{1} w_{2}}} \biggl( \biggl( 1+ \frac{w_{3} w_{4}}{w_{1} w_{2}} \biggr)x+ \frac{w_{3} w_{4}}{w_{1} w_{2}}i \biggr) $$
(2.6)

are invariant under any permutation in the dihedral group \(D_{4} = \langle (1, 3, 2, 4), (1,2) \rangle \).

3 Quadratic symmetry of the modified q-Euler polynomials with the isotropy group \(V_{4}\)

In this section we investigate two orbits each consisting of six generating functions.

Let \(X_{2}\) be the set of six generating functions \(g_{1} = p(13+24, 12 , 34)\), \(g_{2} = p(14+23, 12 , 34)\), \(g_{3} = p(12+34, 13 , 24)\), \(g_{4} = p(14+24, 13 , 24)\), \(g_{5} = p(12+34, 14 , 23)\) and \(g_{6} = p(13+24, 14 , 23)\) as in the notation of (1.8). That is, each \(g_{i}\) is a generating function of the form

$$ p(ab+cd, ac, bd) = \sum_{i=0}^{w_{a} w_{d} -1}(-1)^{i} \int_{\mathbb{Z}_{p}} e^{ [(w_{a} w_{b} +w_{c} w_{d} ) x + w_{a} w_{c} y + w_{b} w_{d} i ]_{q} t} \,d\mu_{-1}(y), $$
(3.1)

where \(1 \leq a, b, c, d \leq4\) and a, b, c, d are distinct.

Note that the action of \(S_{3} = \langle (1, 2), (1, 2, 3) \rangle \) on \(X_{2} = \{ g_{1} , \ldots, g_{6} \}\) is well defined.

$$ \begin{gathered} (1, 2) (g_{1}) = (1, 2) \bigl( p(13+24, 12, 34) \bigr) = p(23+14, 21, 34) = g_{2}, \\ (1, 3) (g_{1}) = (1, 3) \bigl( p(13+24, 12, 34) \bigr) = p(31+24, 32, 14) = g_{6}, \\ (2, 3) (g_{1}) = (2, 3) \bigl( p(13+24, 12, 34) \bigr) = p(12+34, 13, 24) = g_{3}, \\ (1, 2, 3) (g_{1}) = (1, 2, 3) \bigl( p(13+24, 12, 34) \bigr) = p(21+34, 23, 14) = g_{5}, \\ (1, 3, 2) (g_{1}) = (1, 3, 2) \bigl( p(13+24, 12, 34) \bigr) = p(32+14, 31, 24) = g_{4}. \end{gathered} $$
(3.2)

Consider the normal subgroup \(V_{4} = \{ (1), \gamma= (1, 2)(3, 4), \delta= (1,3)(2,4), \gamma\delta= (1,4)(2,3) \}\) of \(S_{4}\).

$$ \begin{gathered} \gamma(g_{1}) = (1, 2) (3,4) \bigl( p(13+24, 12, 34) \bigr) = p(24+13, 21, 34) = g_{1}, \\ \delta(g_{1}) = (1, 3) (2,4) \bigl( p(13+24, 12, 34) \bigr) = p(31+42, 34, 12) = g_{1}. \end{gathered} $$
(3.3)

Hence \(V_{4}\) fixes \(g_{1}\). Since \(\vert S_{3} \vert \times \vert V_{4} \vert = \vert S_{4} \vert \), we obtain the following theorem.

Theorem 3.1

Let X be the set of all generating functions of the form

$$p(ab+cd, jk, lm) = \sum_{i=0}^{w_{j} w_{k} -1}(-1)^{i} \int_{\mathbb{Z}_{p}} e^{ [(w_{a} w_{b} +w_{c} w_{d} ) x + w_{j} w_{k} y + w_{l} w_{m} i ]_{q} t} \,d\mu_{-1}(y) $$

and \(S_{4}\) act on X by permuting \(w_{1} \), \(w_{2} \), \(w_{3} \), \(w_{4}\). Take \(g_{1} = p(13+24, 12 , 34) \in X\). Then the orbit of \(g_{1}\) is \(X_{2} = \{ g_{1}, \ldots, g_{6} \}\) and the isotropy subgroup of \(g_{1}\) is \(V_{4} = \{ (1), (1, 2)(3, 4), (1,3)(2,4), (1,4)(2,3) \}\). Moreover, \(S_{3}\) acts transitively on \(X_{2}\).

Remark

The same computation in (3.3) shows that the isotropy group of each \(g_{i}\) is the normal subgroup \(V_{4} = \{ (1), (1, 2)(3, 4), (1,3)(2,4), (1,4)(2,3) \}\) of \(S_{4}\).

Next we adapt the computation in (2.4) for the change of the q-analogue in \(g_{1} = p(13+24, 12, 34 )\) and obtain the corresponding modified q-Euler polynomials.

$$\begin{aligned}& \bigl[ (w_{1} w_{3} + w_{2} w_{4} )x + w_{1} w_{2} y + w_{3} w_{4} i \bigr]_{q} = [w_{1} w_{2} ]_{q} \biggl[ \biggl( \frac{w_{3}}{w_{2}}+ \frac {w_{4}}{w_{1}} \biggr)x+y +\frac{w_{3} w_{4}}{w_{1} w_{2}}i \biggr]_{q^{w_{1} w_{2}}} , \end{aligned}$$
(3.4)
$$\begin{aligned}& \begin{aligned}[b] g_{1} & = p(13+24, 12 , 34) \\ & = \sum_{i=0}^{w_{1} w_{2} -1} (-1)^{i} \int_{\mathbb{Z}_{p}} e^{ [(w_{1} w_{3} +w_{2} w_{4} ) x + w_{1} w_{2} y + w_{3} w_{4} i ]_{q} t} \,d\mu _{-1}(y) \\ & = \sum_{n=0}^{\infty}\sum _{i=0}^{w_{1} w_{2} -1} (-1)^{i} [w_{1} w_{2} ]_{q} ^{n} E_{n, q^{w_{1} w_{2}}} \biggl( \biggl( \frac{w_{3}}{w_{2}}+ \frac{w_{4}}{w_{1}} \biggr)x +\frac{w_{3} w_{4}}{w_{1} w_{2}}i \biggr) \frac{t^{n}}{n!}. \end{aligned} \end{aligned}$$
(3.5)

Theorem 3.2

For \(q \in{\mathbb{C}}_{p}\) with \(|q-1|_{p}< p^{-\frac{1}{p-1}}\) and \(n \geq0\), the modified q-Euler polynomials

$$ \sum_{i=0}^{w_{1} w_{2} -1} (-1)^{i} [w_{1} w_{2} ]_{q} ^{n} E_{n, q^{w_{1} w_{2}}} \biggl( \biggl( \frac{w_{3}}{w_{2}}+ \frac{w_{4}}{w_{1}} \biggr)x+\frac{w_{3} w_{4}}{w_{1} w_{2}}i \biggr) $$
(3.6)

are invariant under any permutation in the normal subgroup \(V_{4} = \{ (1), (1, 2)(3, 4), (1,3)(2,4), (1,4)(2,3) \}\) of \(S_{4}\).

Now we consider the set \(X_{3}\) of six polynomials \(h_{1} = p(12+34, 12 , 12)\), \(h_{2} = p(13+24, 13, 13)\), \(h_{3} = p(14+23, 14 , 14)\), \(h_{4} = p(12+34, 34, 34)\), \(h_{5} = p(13+24, 24, 24)\) and \(h_{6} = p(14+23, 23, 23)\). That is, each \(h_{i}\) is a polynomial of the form

$$p(ab+cd, ab, ab) = \sum_{i=0}^{w_{a} w_{b} -1}(-1)^{i} \int_{\mathbb{Z}_{p}} e^{ [(w_{a} w_{b} +w_{c} w_{d} ) x + w_{a} w_{b} y + w_{a} w_{b} i ]_{q} t} \,d\mu_{-1}(y), $$

where \(1 \leq a, b, c, d \leq4\) and a, b, c, d are distinct.

Now we have the action of \(S_{4}\) to collect all of \(h_{i}\) in an orbit.

$$ \begin{gathered} (2, 3) (h_{1}) = (2, 3) \bigl( p(12+34, 12, 12) \bigr) = p(13+24, 13, 13) = h_{2}, \\ (2, 4) (h_{1}) = (2, 3) \bigl( p(12+34, 12, 12) \bigr) = p(14+32, 14, 14) = h_{3}, \\ (1, 3) (2, 4) (h_{1}) = (1, 3) (2, 4) \bigl( p(12+34, 12, 12) \bigr) = \bigl( p(34+12, 12) \bigr) = h_{4}, \\ (1, 2, 4) (h_{1}) = (1, 2, 4) \bigl( p(12+34, 12, 12) \bigr) = \bigl( p(24+31, 24, 24) \bigr) = h_{5}, \\ (1, 2, 3) (h_{1}) = (1, 2, 3) \bigl( p(12+34, 12, 12) \bigr) = \bigl( p(23+14, 23, 23) \bigr) = h_{6}. \end{gathered} $$
(3.7)

It is obvious that each permutation in the subgroup \(\widehat{V_{4}} = \langle (1, 2), (3, 4) \rangle \) of \(S_{4}\) fixes \(h_{1}\). As \(\vert \hat{V_{4}} \vert \times6 = 24\), we conclude the following.

Theorem 3.3

Let X be the set of all generating functions of the form

$$p(ab+cd, jk, lm) = \sum_{i=0}^{w_{j} w_{k} -1}(-1)^{i} \int_{\mathbb{Z}_{p}} e^{ [(w_{a} w_{b} +w_{c} w_{d} ) x + w_{j} w_{k} y + w_{l} w_{m} i ]_{q} t} \,d\mu_{-1}(y) $$

and \(S_{4}\) act on X by permuting \(w_{1} \), \(w_{2} \), \(w_{3} \), \(w_{4}\). Take \(h_{1} = p(12+34, 12, 12) \in X\). Then the orbit of \(h_{1}\) is \(X_{3} = \{ h_{1}, \ldots, h_{6} \}\) and the isotropy subgroup of \(h_{1}\) is \(\widehat{V_{4}} = \langle (1, 2), (3, 4) \rangle \) in \(S_{4}\).

Remark

The subgroup \(\widehat{V_{4}} = \langle (1, 2), (3, 4) \rangle \) is not normal in \(S_{4}\). Further, none of four subgroups of order 6 in \(S_{4}\) act transitively on \(X_{3}\).

A slight change of the formula in (2.4) makes \(h_{1} = p(13+24, 12, 34 )\) to be modified q-Euler polynomials.

$$\begin{aligned}& \bigl[ (w_{1} w_{2} + w_{3} w_{4} )x + w_{1} w_{2} y + w_{1} w_{2} i \bigr]_{q} = [w_{1} w_{2} ]_{q} \biggl[ \biggl( 1+ \frac{w_{3} w_{4}}{w_{1} w_{2}} \biggr)x+y +i \biggr]_{q^{w_{1} w_{2}}}, \end{aligned}$$
(3.8)
$$\begin{aligned}& \begin{aligned} [b]h_{1} & = p(12+34, 12 , 12) \\ & = \sum_{i=0}^{w_{1} w_{2} -1} (-1)^{i} \int_{\mathbb{Z}_{p}} e^{ [(w_{1} w_{2} +w_{3} w_{4} ) x + w_{1} w_{2} y + w_{1} w_{2} i ]_{q} t} \,d\mu _{-1}(y) \\ & = \sum_{n=0}^{\infty}\sum _{i=0}^{w_{1} w_{2} -1} (-1)^{i} [w_{1} w_{2} ]_{q} ^{n} E_{n, q^{w_{1} w_{2}}} \biggl( \biggl( 1+ \frac{w_{3} w_{4}}{w_{1} w_{2}} \biggr)x + i \biggr) \frac{t^{n}}{n!}. \end{aligned} \end{aligned}$$
(3.9)

Theorem 3.4

For \(q \in{\mathbb{C}}_{p}\) with \(|q-1|_{p}< p^{-\frac{1}{p-1}}\) and \(n \geq0\), the modified q-Euler polynomials

$$ \sum_{i=0}^{w_{1} w_{2} -1} (-1)^{i} [w_{1} w_{2} ]_{q} ^{n} E_{n, q^{w_{1} w_{2}}} \biggl( \biggl( 1+ \frac{w_{3} w_{4}}{w_{1} w_{2}} \biggr)x+ i \biggr) $$
(3.10)

are invariant under any permutation in the subgroup \(\widehat{V_{4}} = \langle (1, 2), (3, 4) \rangle \) of \(S_{4}\).

4 Less symmetric modified q-Euler polynomials

When \(S_{4}\) acts on the set X of 315 generating functions listed in (1.8), there are seven orbits having 12 generating functions and nine orbits having 24 generating functions. That is, these are the generating functions whose isotropy subgroup is a cyclic group of order 2 or a trivial group. In this section we just illustrate a typical one for each case.

Let \(X_{4}\) be the set of all generating functions of the form

$$ p(ac+ad, ab, ab) = \sum_{i=0}^{w_{a} w_{b} -1}(-1)^{i} \int_{\mathbb{Z}_{p}} e^{ [(w_{a} w_{c} +w_{a} w_{d} ) x + w_{a} w_{b} y + w_{a} w_{b} i ]_{q} t} \,d\mu_{-1}(y), $$
(4.1)

where \(1 \leq a, b, c, d \leq4\) and a, b, c, d are distinct.

That is, \(X_{4}\) consists of the following 12 polynomials:

$$ \begin{gathered} f_{1} = p(13+14, 12, 12),\qquad f_{2} = p(23+24, 12, 12), \\ f_{3} = p(12+14, 13, 13),\qquad f_{4} = p(32+34, 13, 13), \\ f_{5} = p(12+13, 14, 14), \qquad f_{6} = p(42+43, 14, 14), \\ f_{7} = p(21+24, 23, 23), \qquad f_{8} = p(31+34, 23, 23), \\ f_{9} = p(21+23, 24, 24), \qquad f_{10} = p(41+43, 24, 24), \\ f_{11} = p(31+32, 34, 34), \qquad f_{12} = p(41+42, 34, 34). \end{gathered} $$
(4.2)

We can check that the alternating group \(A_{4}\) acts on \(X_{4}\) transitively.

$$ \begin{gathered} (1, 2, 3)f_{1} = f_{7},\qquad (1, 3, 2)f_{1} = f_{4}, \\ (1, 2, 4)f_{1} = f_{9},\qquad (1, 4, 2)f_{1} = f_{6}, \\ (1, 3, 4)f_{1} = f_{8},\qquad (1, 4, 3)f_{1} = f_{10}, \\ (2, 3, 4)f_{1} = f_{3},\qquad (2, 4, 3)f_{1} = f_{5}, \\ (1, 2) (3, 4)f_{1} = f_{2},\qquad (1, 3) (2, 4))f_{1} = f_{11}, \\ (1, 4) (2, 3))f_{1} = f_{12}. \end{gathered} $$
(4.3)

As \((3, 4)f_{1} =(3, 4) ( p(13+14, 12, 12) ) = p(14+13, 12, 12) = f_{1}\), the isotropy subgroup of \(f_{1}\) is \(\langle (3, 4) \rangle \) and \(X_{4}\) is the orbit of f. Note that \(\langle (3, 4) \rangle \) consists of one even permutation \((1)\) and one odd permutation \((3, 4)\). So, for any \(p(ac+ad, ab, ab) \in X_{4}\), there is an even permutation τ such that \(\tau(f_{1}) = p(ac+ad, ab, ab)\). Hence the alternating subgroup \(A_{4}\) of \(S_{4}\) acts transitively on \(X_{4}\).

Theorem 4.1

Let X be the set of all generating functions of the form

$$p(ab+cd, jk, lm) = \sum_{i=0}^{w_{j} w_{k} -1}(-1)^{i} \int_{\mathbb{Z}_{p}} e^{ [(w_{a} w_{b} +w_{c} w_{d} ) x + w_{j} w_{k} y + w_{l} w_{m} i ]_{q} t} \,d\mu_{-1}(y) $$

and \(S_{4}\) act on X by permuting \(w_{1} \), \(w_{2} \), \(w_{3} \), \(w_{4}\). Take \(f_{1} = p(13+14, 12 , 12) \in X\). Then the orbit of \(f_{1}\) is \(X_{4}\) consisting of 12 polynomials as above and the isotropy subgroup of \(f_{1}\) is \(\langle (3, 4) \rangle \). Moreover, \(A_{4}\) acts transitively on \(X_{4}\).

Finally, we illustrate one of the nine orbits each consisting of 24 generating functions.

Let \(X_{5}\) be the set of all generating functions of the form

$$ p(ab+ac, ab, ab) = \sum_{i=0}^{w_{a} w_{b} -1}(-1)^{i} \int_{\mathbb{Z}_{p}} e^{ [(w_{a} w_{b} +w_{a} w_{c} ) x + w_{a} w_{b} y + w_{a} w_{b} i ]_{q} t} \,d\mu_{-1}(y), $$
(4.4)

where \(1 \leq a, b, c\leq4\) and a, b, c are distinct. Note that there are six choices for the coefficient \(w_{a} w_{b}\) of y and four choices for the coefficient \(w_{a} w_{b} + w_{a} w_{c}\) of x. Hence \(X_{5}\) consists of 24 generating functions. We use \(g=p(12+13, 12, 12) \in X_{5}\), then we can see that the action of \(S_{4}\) on \(X_{5}\) is transitive.

5 Conclusion

In this paper we investigate the symmetric property of the Euler polynomials. If we use n weights \(w_{1}, \ldots, w_{n}\) in the definition of the generating functions of Euler polynomials, then the symmetric group \(S_{n}\) naturally acts on a prescribed set of generating functions of the Euler polynomials. This paper uses four weights \(w_{1}, \ldots, w_{4}\) in a quadratic form. We find that the dihedral group \(D_{4}\) (or the Klein 4 group \(V_{4}\)) is an isotropy group of some generating function. As a result, the corresponding Euler polynomial is fixed by \(D_{4}\) (or \(V_{4}\)). The results in the paper extend to other special polynomials such as degenerate Euler polynomials and Catalan polynomials. It is a further problem to find special polynomials that are invariant under the alternating group \(A_{n}\).