1 Introduction

Let p be chosen as a fixed odd prime number. In this paper \(\mathbb{Z}_{p}\), \(\mathbb{Q}_{p}\), \(\mathbb{C}\) and \(\mathbb{C}_{p}\) will, respectively, denote the ring of p-adic rational integers, the field of p-adic rational numbers, the complex numbers, and the completion of an algebraic closure of \(\mathbb{Q}_{p}\).

Let \(v_{p}\) be a normalized exponential valuation of \(\mathbb{C}_{p}\) by

$$ \vert p\vert _{p}=p^{-v_{p} ( p ) }=\frac {1}{p}. $$

When one talks of a q-extension, q is variously considered as an indeterminate, a complex number \(q\in\mathbb{C}\) or a p-adic number \(q\in \mathbb{C}_{p}\). If \(q\in\mathbb{C}\), we assume that \(\vert q\vert <1\). If \(q\in \mathbb{C}_{p}\), we assume that \(\vert 1-q\vert _{p}<1\) (see, for details, [1–16]).

The following measure is defined by Kim: for any positive integer n and \(0\leq a< p^{n}\),

$$ \mu_{q} \bigl( a+p^{n} \mathbb{Z}_{p} \bigr) = ( -q ) ^{a}\frac{ ( 1+q ) }{1+q^{p^{n}}}, $$

which can be extended to a measure on \(\mathbb{Z}_{p}\) (for details, see [5–11]).

Extended q-Euler polynomials (also known as weighted q-Euler polynomials) are defined by

$$ \widetilde{E}_{n,q}^{ ( \alpha ) } ( x ) =\int_{\mathbb{Z}_{p}} \biggl( \frac{1-q^{\alpha ( x+\xi ) }}{1-q^{\alpha }} \biggr) ^{n}\, d\mu_{q} ( \xi ) $$
(1)

for \(n\in \mathbb{Z}_{+}:= \{ 0,1,2,3,\ldots \} \). We note that

$$ \lim_{q\rightarrow1}\widetilde{E}_{n,q}^{ ( \alpha ) } ( x ) =E_{n} ( x ), $$

where \(E_{n} ( x ) \) are nth Euler polynomials, which are defined by the rule

$$ \sum_{n=0}^{\infty}E_{n} ( x ) \frac {t^{n}}{n!}=e^{tx}\frac{2}{e^{t}+1}, \quad \vert t\vert < \pi $$

(for details, see [13]). In the case \(x=0\) in (1), then we have \(\widetilde{E}_{n,q}^{ ( \alpha ) } ( 0 ) :=\widetilde{E}_{n,q}^{ ( \alpha ) }\), which are called extended q-Euler numbers (or weighted q-Euler numbers).

Extended q-Euler numbers and polynomials have the following explicit formulas:

$$\begin{aligned}& \widetilde{E}_{n,q}^{ ( \alpha ) } = \frac{1+q}{ ( 1-q^{\alpha} ) ^{n}}\sum _{l=0}^{n}\binom{n}{l} ( -1 ) ^{l}\frac{1}{1+q^{\alpha l+1}}, \end{aligned}$$
(2)
$$\begin{aligned}& \widetilde{E}_{n,q}^{ ( \alpha ) } ( x ) = \frac{1+q}{ ( 1-q^{\alpha} ) ^{n}}\sum _{l=0}^{n}\binom{n}{l} ( -1 ) ^{l}\frac{q^{\alpha lx}}{1+q^{\alpha l+1}}, \end{aligned}$$
(3)
$$\begin{aligned}& \widetilde{E}_{n,q}^{ ( \alpha ) } ( x ) = \sum _{l=0}^{n}\binom{n}{l}q^{\alpha lx} \widetilde{E}_{l,q}^{ ( \alpha ) } \biggl( \frac{1-q^{\alpha x}}{1-q^{\alpha}} \biggr) ^{n-l}. \end{aligned}$$
(4)

Moreover, for \(d\in \mathbb{N}\) with \(d\equiv1\ ( \operatorname{mod}2 ) \),

$$ \widetilde{E}_{n,q}^{ ( \alpha ) } ( x ) = \biggl( \frac{1+q}{1+q^{d}} \biggr) \biggl( \frac{1-q^{\alpha d}}{1-q^{\alpha}} \biggr) ^{n}\sum _{a=0}^{d-1} ( -1 ) ^{a}\widetilde{E}_{n,q}^{ ( \alpha ) } \biggl( \frac{x+a}{d} \biggr) ; $$
(5)

see [13].

For any positive integer h, k and m, Dedekind-type DC sums are given by Kim in [5, 6], and [7] as follows:

$$ S_{m} ( h,k ) =\sum_{M=1}^{k-1} ( -1 ) ^{M-1}\frac{M}{k}\overline{E}_{m} \biggl( \frac{hM}{k} \biggr), $$

where \(\overline{E}_{m} ( x ) \) are mth periodic Euler functions.

Kim [6] derived some interesting properties for Dedekind-type DC sums and considered a p-adic continuous function for an odd prime number to contain a p-adic q-analog of the higher order Dedekind-type DC sums \(k^{m}S_{m+1} ( h,k ) \). Simsek [15] gave a q-analog of Dedekind-type sums and derived interesting properties. Furthermore, Araci et al. studied Dedekind-type sums in accordance with modified q-Euler polynomials with weight α [14], modified q-Genocchi polynomials with weight α [4], and weighted q-Genocchi polynomials [16].

Recently, weighted q-Bernoulli numbers and polynomials were first defined by Kim in [11]. Next, many mathematicians, by utilizing Kim’s paper [11], have introduced various generalization of some known special polynomials such as Bernoulli polynomials, Euler polynomials, Genocchi polynomials, and so on, which are called weighted q-Bernoulli, weighted q-Euler, and weighted q-Genocchi polynomials in [1, 2, 11–13].

By the same motivation of the above knowledge, we give a weighted p-adic q-analog of the higher order Dedekind-type DC sums \(k^{m}S_{m+1} ( h,k ) \) which are derived from a fermionic p-adic q-deformed integral on \(\mathbb{Z}_{p}\).

2 Extended q-Dedekind-type sums associated with extended q-Euler polynomials

Let w be the Teichmüller character \((\operatorname {mod}p)\). For \(x\in \mathbb{Z}_{p}^{\ast}:= \mathbb{Z}_{p}/p \mathbb{Z}_{p}\), set

$$ \langle x:q \rangle=w^{-1} ( x ) \biggl( \frac {1-q^{x}}{1-q} \biggr) . $$

Let a and N be positive integers with \(( p,a ) =1\) and \(p\mid N \). We now consider

$$ \widetilde{C}_{q}^{ ( \alpha ) } \bigl( s,a,N:q^{N} \bigr) =w^{-1} ( a ) \bigl\langle a:q^{\alpha} \bigr\rangle ^{s}\sum_{j=0}^{\infty} \binom{s}{j}q^{\alpha aj} \biggl( \frac {1-q^{\alpha N}}{1-q^{\alpha a}} \biggr) ^{j}\widetilde{E}_{j,q^{N}}^{ ( \alpha ) }. $$

In particular, if \(m+1\equiv0\ (\operatorname{mod}p-1)\), then

$$\begin{aligned} \widetilde{C}_{q}^{ ( \alpha ) } \bigl( m,a,N:q^{N} \bigr) =& \biggl( \frac{1-q^{\alpha a}}{1-q^{\alpha}} \biggr) ^{m}\sum _{j=0}^{m}\binom{m}{j}q^{\alpha aj} \widetilde{E}_{j,q^{N}}^{ ( \alpha ) } \biggl( \frac{1-q^{\alpha N}}{1-q^{\alpha a}} \biggr) ^{j} \\ =& \biggl( \frac{1-q^{\alpha N}}{1-q^{\alpha}} \biggr) ^{m}\int_{\mathbb{Z}_{p}} \biggl( \frac{1-q^{\alpha N ( \xi+\frac{a}{N} ) }}{1-q^{\alpha N}} \biggr) ^{m}\, d\mu_{q^{N}} ( \xi ) . \end{aligned}$$

Thus, \(\widetilde{C}_{q}^{ ( \alpha ) } ( m,a,N:q^{N} ) \) is a continuous p-adic extension of

$$ \biggl( \frac{1-q^{\alpha N}}{1-q^{\alpha}} \biggr) ^{m}\widetilde{E} _{m,q^{N}}^{ ( \alpha ) } \biggl( \frac{a}{N} \biggr) . $$

Let \([ \cdot ] \) be the Gauss symbol and let \(\{ x \} =x- [ x ] \). Thus, we are now ready to introduce the q-analog of the higher order Dedekind-type DC sums \(\widetilde{J}_{m,q}^{ ( \alpha ) } ( h,k:q^{l} ) \) by the rule

$$ \widetilde{J}_{m,q}^{ ( \alpha ) } \bigl( h,k:q^{l} \bigr) = \sum_{M=1}^{k-1} ( -1 ) ^{M-1} \biggl( \frac{1-q^{\alpha M}}{1-q^{\alpha k}} \biggr) \int_{ \mathbb{Z}_{p}} \biggl( \frac{1-q^{\alpha ( l\xi+l \{ \frac {hM}{k} \} ) }}{1-q^{\alpha l}} \biggr) ^{m}\, d\mu_{q^{l}} ( \xi ). $$

If \(m+1\equiv0\ ( \operatorname{mod}p-1 ) \),

$$\begin{aligned}& \biggl( \frac{1-q^{\alpha k}}{1-q^{\alpha}} \biggr) ^{m+1}\sum _{M=1}^{k-1} ( -1 ) ^{M-1} \biggl( \frac {1-q^{\alpha M}}{1-q^{\alpha k}} \biggr) \int_{\mathbb{Z}_{p}} \biggl( \frac{1-q^{\alpha k ( \xi+\frac{hM}{k} ) }}{1-q^{\alpha k}} \biggr) ^{m}\, d\mu_{q^{k}} ( \xi ) \\& \quad = \sum_{M=1}^{k-1} ( -1 ) ^{M-1} \biggl( \frac{1-q^{\alpha M}}{1-q^{\alpha}} \biggr) \biggl( \frac{1-q^{\alpha k}}{1-q^{\alpha }} \biggr) ^{m}\int_{\mathbb{Z} _{p}} \biggl( \frac{1-q^{\alpha k ( \xi+\frac{hM}{k} ) }}{1-q^{\alpha k}} \biggr) ^{m}\, d\mu_{q^{k}} ( \xi ), \end{aligned}$$

where \(p\mid k\), \(( hM,p ) =1\) for each M. By (1), we easily state the following:

$$\begin{aligned}& \biggl( \frac{1-q^{\alpha k}}{1-q^{\alpha}} \biggr) ^{m+1}\widetilde {J}_{m,q}^{ ( \alpha ) } \bigl( h,k:q^{k} \bigr) \\& \quad =\sum_{M=1}^{k-1} \biggl( \frac{1-q^{\alpha M}}{1-q^{\alpha}} \biggr) \biggl( \frac{1-q^{\alpha k}}{1-q^{\alpha}} \biggr) ^{m} ( -1 ) ^{M-1} \\& \qquad {}\times\int_{ \mathbb{Z}_{p}} \biggl( \frac{1-q^{\alpha k ( \xi+\frac{hM}{k} ) }}{1-q^{\alpha k}} \biggr) ^{m}\, d\mu_{q^{k}} ( \xi ) \\& \quad =\sum_{M=1}^{k-1} ( -1 ) ^{M-1} \biggl( \frac{1-q^{\alpha M}}{1-q^{\alpha}} \biggr) \widetilde{C}_{q}^{ ( \alpha ) } \bigl( m, ( hM ) _{k}:q^{k} \bigr), \end{aligned}$$
(6)

where \((hM)_{k}\) denotes the integer x such that \(0\leq x< n\) and \(x\equiv \alpha\ ( \operatorname{mod}k ) \).

It is not difficult to indicate the following:

$$\begin{aligned}& \int_{\mathbb{Z} _{p}} \biggl( \frac{1-q^{\alpha ( x+\xi ) }}{1-q^{\alpha }} \biggr) ^{k} \, d\mu_{q} ( \xi ) \\& \quad = \biggl( \frac{1-q^{\alpha m}}{1-q^{\alpha}} \biggr) ^{k}\frac {1+q}{1+q^{m}}\sum_{i=0}^{m-1} ( -1 ) ^{i}\int _{\mathbb{Z}_{p}} \biggl( \frac{1-q^{\alpha m ( \xi+\frac{x+i}{m} ) }}{ 1-q^{\alpha m}} \biggr) ^{k}\, d \mu_{q^{m}} ( \xi ) . \end{aligned}$$
(7)

On account of (6) and (7), we easily see that

$$\begin{aligned}& \biggl( \frac{1-q^{\alpha N}}{1-q^{\alpha}} \biggr) ^{m}\int_{\mathbb{Z}_{p}} \biggl( \frac{1-q^{\alpha N ( \xi+\frac{a}{N} ) }}{1-q^{\alpha N}} \biggr) ^{m}\, d\mu_{q^{N}} ( \xi ) \\& \quad =\frac{1+q^{N}}{1+q^{Np}}\sum_{i=0}^{p-1} ( -1 ) ^{i} \biggl( \frac{1-q^{\alpha Np}}{1-q^{\alpha}} \biggr) ^{m}\int _{\mathbb{Z}_{p}} \biggl( \frac{1-q^{\alpha pN ( \xi+\frac{a+iN}{pN} ) }}{1-q^{\alpha pN}} \biggr) ^{m}\, d \mu_{q^{pN}} ( \xi ) . \end{aligned}$$
(8)

Because of (6), (7), and (8), we develop the p-adic integration as follows:

$$ \widetilde{C}_{q}^{ ( \alpha ) } \bigl( s,a,N:q^{N} \bigr) =\frac{1+q^{N}}{1+q^{Np}}\sum_{\stackrel{0\leq i\leq p-1}{a+iN\neq0\ (\operatorname {mod}p)}} ( -1 ) ^{i} \widetilde{C}_{q}^{ ( \alpha ) } \bigl( s, ( a+iN ) _{pN},p^{N}:q^{pN} \bigr) . $$

So,

$$\begin{aligned} \widetilde{C}_{q}^{ ( \alpha ) } \bigl( m,a,N:q^{N} \bigr) =& \biggl( \frac{1-q^{\alpha N}}{1-q^{\alpha}} \biggr) ^{m}\int_{\mathbb{Z}_{p}} \biggl( \frac{1-q^{\alpha N ( \xi+\frac{a}{N} ) }}{1-q^{\alpha N}} \biggr) ^{m}\, d\mu_{q^{N}} ( \xi ) \\ &{}- \biggl( \frac{1-q^{\alpha Np}}{1-q^{\alpha}} \biggr) ^{m}\int _{\mathbb{Z}_{p}} \biggl( \frac{1-q^{\alpha pN ( \xi+\frac{a+iN}{pN} ) }}{1-q^{\alpha pN}} \biggr) ^{m}\, d \mu_{q^{pN}} ( \xi ), \end{aligned}$$

where \(( p^{-1}a ) _{N}\) denotes the integer x with \(0\leq x< N\), \(px\equiv a\ ( \operatorname{mod}N ) \) and m is integer with \(m+1\equiv0\ (\operatorname{mod}p-1)\). Therefore, we have

$$\begin{aligned}& \sum_{M=1}^{k-1} ( -1 ) ^{M-1} \biggl( \frac{1-q^{\alpha M}}{1-q^{\alpha}} \biggr) \widetilde{C}_{q}^{ ( \alpha ) } \bigl( m,hM,k:q^{k} \bigr) \\& \quad = \biggl( \frac{1-q^{\alpha k}}{1-q^{\alpha}} \biggr) ^{m+1}\widetilde {J}_{m,q}^{ ( \alpha ) } \bigl( h,k:q^{k} \bigr) - \biggl( \frac{1-q^{\alpha k}}{1-q^{\alpha}} \biggr) ^{m+1} \\& \qquad {}\times\biggl( \frac{1-q^{\alpha kp}}{1-q^{\alpha k}} \biggr) \widetilde{J}_{m,q}^{ ( \alpha ) } \bigl( \bigl( p^{-1}h \bigr) ,k:q^{pk} \bigr), \end{aligned}$$

where \(p\nmid k\) and \(p\nmid hm\) for each M. Thus, we give the following definition, which seems interesting for further studying the theory of Dedekind sums.

Definition 1

Let h, k be positive integer with \(( h,k ) =1\), \(p\nmid k\). For \(s\in \mathbb{Z}_{p}\), we define a p-adic Dedekind-type DC sums as follows:

$$ \widetilde{J}_{p,q}^{ ( \alpha ) } \bigl( s:h,k:q^{k} \bigr) =\sum_{M=1}^{k-1} ( -1 ) ^{M-1} \biggl( \frac{1-q^{\alpha M}}{1-q^{\alpha}} \biggr) \widetilde{C}_{q}^{ ( \alpha ) } \bigl( m,hM,k:q^{k} \bigr) . $$

As a result of the above definition, we state the following theorem.

Theorem 2.1

For \(m+1\equiv0\ (\operatorname{mod}p-1)\) and \(( p^{-1}a ) _{N}\) denotes the integer x with \(0\leq x< N\), \(px\equiv a\ ( \operatorname {mod}N ) \), then we have

$$\begin{aligned} \widetilde{J}_{p,q}^{ ( \alpha ) } \bigl( s:h,k:q^{k} \bigr) =& \biggl( \frac{1-q^{\alpha k}}{1-q^{\alpha}} \biggr) ^{m+1}\widetilde {J}_{m,q}^{ ( \alpha ) } \bigl( h,k:q^{k} \bigr) \\ &{}- \biggl( \frac{1-q^{\alpha k}}{1-q^{\alpha}} \biggr) ^{m+1} \biggl( \frac{1-q^{\alpha kp}}{1-q^{\alpha k}} \biggr) \widetilde{J}_{m,q}^{ ( \alpha ) } \bigl( \bigl( p^{-1}h \bigr) ,k:q^{pk} \bigr) . \end{aligned}$$

In the special case \(\alpha=1\), our applications in theory of Dedekind sums resemble Kim’s results in [6]. These results seem to be interesting for further studies as in [5, 7] and [15].