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# Extended q-Dedekind-type Daehee-Changhee sums associated with extended q-Euler polynomials

## Abstract

In the present paper, we aim to specify a p-adic continuous function for an odd prime inside a p-adic q-analog of the extended Dedekind-type sums of higher order according to extended q-Euler polynomials (or weighted q-Euler polynomials) which is derived from a fermionic p-adic q-deformed integral on $$\mathbb{Z}_{p}$$.

## 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, ).

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 ).

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 ). 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 .

For any positive integer h, k and m, Dedekind-type DC sums are given by Kim in [5, 6], and  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  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  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 , modified q-Genocchi polynomials with weight α , and weighted q-Genocchi polynomials .

Recently, weighted q-Bernoulli numbers and polynomials were first defined by Kim in . Next, many mathematicians, by utilizing Kim’s paper , 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, 1113].

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 $$\mathbb{Z}_{p}$$.

## 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 . These results seem to be interesting for further studies as in [5, 7] and .

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## Acknowledgements

The authors thank the reviewers for their helpful comments and suggestions, which have improved the quality of the paper.

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Correspondence to Serkan Araci.

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The authors declare that they have no competing interests.

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All authors contributed equally to this work. All authors read and approved the revised manuscript.

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