Open Access

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

Advances in Difference Equations20152015:272

https://doi.org/10.1186/s13662-015-0610-8

Received: 22 June 2015

Accepted: 15 August 2015

Published: 4 September 2015

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 \(\mathbb{Z}_{p}\).

Keywords

Dedekind sums q-Dedekind-type sums p-adic q-integralextended q-Euler numbers and polynomials

MSC

11S8011B68

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, [116]).

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 [511]).
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, 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 \(\mathbb{Z}_{p}\).

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].

Declarations

Acknowledgements

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

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Economics, Faculty of Economics, Administrative and Social Sciences, Hasan Kalyoncu University
(2)
Department of Mathematics, Faculty of Science and Arts, Kırklareli University

References

  1. Araci, S, Acikgoz, M, Park, KH: A note on the q-analogue of Kim’s p-adic log gamma-type functions associated with q-extension of Genocchi and Euler numbers with weight α. Bull. Korean Math. Soc. 50(2), 583-588 (2013) MATHMathSciNetView ArticleGoogle Scholar
  2. Araci, S, Erdal, D, Seo, JJ: A study on the fermionic p-adic q-integral representation on \(\mathbb{Z}_{p}\) associated with weighted q-Bernstein and q-Genocchi polynomials. Abstr. Appl. Anal. 2011, Article ID 649248 (2011) MathSciNetView ArticleGoogle Scholar
  3. Araci, S, Acikgoz, M, Seo, JJ: Explicit formulas involving q-Euler numbers and polynomials. Abstr. Appl. Anal. 2012, Article ID 298531 (2012). doi:10.1155/2012/298531 MathSciNetView ArticleGoogle Scholar
  4. Araci, S, Acikgoz, M, Esi, A: A note on the q-Dedekind-type Daehee-Changhee sums with weight α arising from modified q-Genocchi polynomials with weight α. J. Assam Acad. Math. 5, 47-54 (2012) MathSciNetGoogle Scholar
  5. Kim, T: A note on p-adic q-Dedekind sums. C. R. Acad. Bulgare Sci. 54, 37-42 (2001) MATHMathSciNetGoogle Scholar
  6. Kim, T: Note on q-Dedekind-type sums related to q-Euler polynomials. Glasg. Math. J. 54, 121-125 (2012) MATHMathSciNetView ArticleGoogle Scholar
  7. Kim, T: Note on Dedekind type DC sums. Adv. Stud. Contemp. Math. 18, 249-260 (2009) MATHGoogle Scholar
  8. Kim, T: The modified q-Euler numbers and polynomials. Adv. Stud. Contemp. Math. 16, 161-170 (2008) MATHGoogle Scholar
  9. Kim, T: q-Volkenborn integration. Russ. J. Math. Phys. 9, 288-299 (2002) MATHMathSciNetGoogle Scholar
  10. Kim, T: On a q-analogue of the p-adic log gamma functions and related integrals. J. Number Theory 76, 320-329 (1999) MATHMathSciNetView ArticleGoogle Scholar
  11. Kim, T: On the weighted q-Bernoulli numbers and polynomials. Adv. Stud. Contemp. Math. 21(2), 207-215 (2011) MATHMathSciNetGoogle Scholar
  12. Rim, SH, Jeong, J: A note on the modified q-Euler numbers and polynomials with weight α. Int. Math. Forum 6(65), 3245-3250 (2011) MATHMathSciNetGoogle Scholar
  13. Ryoo, CS: A note on the weighted q-Euler numbers and polynomials. Adv. Stud. Contemp. Math. 21, 47-54 (2011) MATHMathSciNetGoogle Scholar
  14. Seo, JJ, Araci, S, Acikgoz, M: q-Dedekind-type Daehee-Changhee sums with weight α associated with modified q-Euler polynomials with weight α. J. Chungcheong Math. Soc. 27(1), 1-8 (2014) MathSciNetView ArticleGoogle Scholar
  15. Simsek, Y: q-Dedekind type sums related to q-zeta function and basic L-series. J. Math. Anal. Appl. 318, 333-351 (2006) MATHMathSciNetView ArticleGoogle Scholar
  16. Şen, E, Acikgoz, M, Araci, S: A note on the modified q-Dedekind sums. Notes Number Theory Discrete Math. 19(3), 60-65 (2013) MATHGoogle Scholar

Copyright

© Araci and Özer 2015