Poly-Dedekind sums associated with poly-Bernoulli functions

Abstract

Apostol considered generalized Dedekind sums by replacing the first Bernoulli function appearing in Dedekind sums by any Bernoulli functions and derived a reciprocity relation for them. Recently, poly-Dedekind sums were introduced by replacing the first Bernoulli function appearing in Dedekind sums by any type 2 poly-Bernoulli functions of arbitrary indices and were shown to satisfy a reciprocity relation. In this paper, we consider other poly-Dedekind sums that are obtained by replacing the first Bernoulli function appearing in Dedekind sums by any poly-Bernoulli functions of arbitrary indices. We derive a reciprocity relation for these poly-Dedekind sums.

1 Introduction

The sawtooth function, denoted by $( (x) )$, is defined by

$$ \bigl( (x) \bigr)=\textstyle\begin{cases} x-[x]-\frac{1}{2}, & \text{if }x\notin \mathbb{Z}, \\ 0, & \text{if }x\in \mathbb{Z}, \end{cases}\displaystyle \quad (\mbox{see [1--5]}), $$

(1)

where $[x]$ denotes the greatest integer function not exceeding x.

The Dedekind sums are defined by

$$\begin{aligned} S(h,m) &= \sum_{\mu =1}^{m-1} \biggl( \biggl(\frac{\mu }{m} \biggr) \biggr) \biggl( \biggl(\frac{h\mu }{m} \biggr) \biggr) \\ &= \sum_{\mu =1}^{m-1} \biggl( \frac{\mu }{m}-\frac{1}{2} \biggr) \biggl( \biggl( \frac{h\mu }{m} \biggr) \biggr) \\ &=\sum_{\mu =1}^{m-1} \frac{\mu }{m} \biggl( \biggl(\frac{h\mu }{m} \biggr) \biggr), \end{aligned}$$

(2)

where h is any integer and m is a positive integer (see [9–11, 17, 19, 20]).

It is well known that the Bernoulli polynomials are defined by

$$ \frac{t}{e^{t}-1}e^{xt}=\sum_{n=0}^{\infty }B_{n}(x) \frac{t^{n}}{n!}\quad \bigl( \vert t \vert < 2 \pi \bigr), (\text{see [1--13, 17, 19, 20]}). $$

(3)

When $x=0$, $B_{n}=B_{n}(0)$, ($n\ge 0$) are called the Bernoulli numbers.

From (3), we note that

$$B_n(x)=\sum _{l=0}^n\left(\begin{array}{c}n\\ l\end{array}\right)B_{n-l}{x}^l(n\ge 0),(\text{see [7–13]}).$$

(4)

By (3), we easily get

$$ \sum_{l=0}^{n-1}l^{m}= \frac{1}{m+1} \bigl(B_{m+1}(n)-B_{m+1} \bigr),\quad (n\in \mathbb{N}, m\ge 0), (\text{see [13]}), $$

(5)

and

$$ d^{n-1}\sum_{l=0}^{d-1}B_{n} \biggl(\frac{x+i}{d} \biggr)=B_{n}(x),\quad (n\ge 0, d\in \mathbb{N}), (\text{see [10, 13]}). $$

(6)

The modified Hardy’s polyexponential function of index k is defined by

$$ \operatorname{Ei}_{k}(x)=\sum_{n=1}^{\infty } \frac{x^{n}}{n^{k}(n-1)!}\quad (k\in \mathbb{Z}), (\text{see [7]}). $$

(7)

Note that $\operatorname{Ei}_{1}(x)=e^{x}-1$.

Recently, the type 2 poly-Bernoulli polynomials of index k are defined by

$$ \frac{\operatorname{Ei}_{k}(\log (1+t))}{e^{t}-1}e^{xt}=\sum_{n=0}^{\infty }B_{n}^{(k)}(x) \frac{t^{n}}{n!} \quad (k\in \mathbb{Z}). $$

(8)

When $x=0$, $B_{n}^{(k)}=B_{n}^{(k)}(0)$, ($n\ge 0$) are called the type 2 poly-Bernoulli numbers of index k. Note that $B_{n}^{(1)}(x)=B_{n}(x)$, ($n\ge 0$).

It is well known that the polylogarithmic function of index k is defined by

$$ \operatorname{Li}_{k}(x) =\sum_{n=1}^{\infty } \frac{x^{n}}{n^{k}},\quad (k \in \mathbb{Z}), \vert x \vert < 1, (\text{see [6, 9, 12]}). $$

(9)

Note that $\operatorname{Li}_{1}(x)=-\log (1-x)$.

In [6, 7, 12], the poly-Bernoulli polynomials of index k are defined by the generating function

$$ \frac{\operatorname{Li}_{k}(1-e^{-t})}{e^{t}-1}e^{xt}=\sum_{n=0}^{\infty } \beta _{n}^{(k)}(x)\frac{t^{n}}{n!}. $$

(10)

When $x=0$, $\beta _{n}^{(k)}= \beta _{n}^{(k)}(0)$ are called the poly-Bernoulli numbers of index k.

From (10), we note that

$$\sum _{l=0}^n\left(\begin{array}{c}n\\ l\end{array}\right){\beta }_{n-l}^{(k)}{x}^l={\beta }_n^{(k)}(x),(n\ge 0),(\text{see [6, 7, 12]}).$$

(11)

The fractional part of x is defined by

$$\begin{aligned}& \langle x\rangle =x-[x]. \end{aligned}$$

The Bernoulli functions are defined by

$$\begin{aligned}& \overline{B}_{n}(x)=B_{n} \bigl(\langle x\rangle \bigr), \quad (n\ge 0), (\text{see [1, 2]}). \end{aligned}$$

From (2), we have

$$\begin{aligned} S(h,m) &= \sum_{\mu =1}^{m-1} \frac{\mu }{m} \biggl(\frac{h\mu }{m}- \biggl[\frac{h\mu }{m} \biggr]-\frac{1}{2} \biggr) \\ &= \sum_{\mu =1}^{m-1} \biggl( \frac{\mu }{m}-\frac{1}{2} \biggr) \biggl( \frac{h\mu }{m}- \biggl[\frac{h\mu }{m} \biggr]-\frac{1}{2} \biggr) \\ &= \sum_{\mu =1}^{m-1}\overline{B}_{1} \biggl(\frac{\mu }{m} \biggr) \overline{B}_{1} \biggl( \frac{h\mu }{m} \biggr), \end{aligned}$$

(12)

where h, m are relatively prime positive integers.

Apostol considered the generalized Dedekind sums, which are given by

$$ S_{p}(h,m)=\sum_{\mu =1}^{m-1} \frac{\mu }{m}\overline{B}_{p} \biggl( \frac{h\mu }{m} \biggr), $$

(13)

and showed in [1, 2] that they satisfy the reciprocity relation

$$(p+1)(h{m}^p{S}_p(h,m)+m{h}^p{S}_p(m,h))=p{B}_{p+1}+\sum _{s=0}^{p+1}\left(\begin{array}{c}p+1\\ s\end{array}\right){(-1)}^s{B}_s{B}_{p+1-s}{h}^s{m}^{p+1-s}.$$

As one generalization of Apostol’s generalized Dedekind sums, the poly-Dedekind sums associated with the type 2 poly-Bernoulli functions of index k

$$ S_{P}^{(k)}(h,m)=\sum_{\mu =1}^{m-1} \frac{\mu }{m}\overline{B}_{p}^{(k)} \biggl( \frac{h\mu }{m} \biggr) $$

(14)

were recently introduced (see [13]) and, among other things, a reciprocity relation for them was derived.

In this paper, as another generalization of Apostol’s generalized Dedekind sums, we consider the poly-Dedekind sums defined by

$$ T_{p}^{(k)}(h,m)= \sum_{\mu =1}^{m-1} \frac{\mu }{m}\overline{\beta }_{p}^{(k)} \biggl( \frac{h\mu }{m} \biggr), $$

where $\overline{\beta }_{p}^{(k)}(x)=\beta _{p}^{(k)}(\langle x \rangle )$ are the poly-Bernoulli functions of index k (see (10)). Note here that $T_{p}^{(1)}(h,m)=S_{p}(h,m)$. We show the following reciprocity relation for the poly-Dedekind sums given by (see Theorem 7)

$$\begin{array}{rl}& h{m}^p{T}_p^{(k)}(h,m)+m{h}^p{T}_p^{(k)}(m,h)\\ & =\sum _{\mu =0}^{m-1}\sum _{j=0}^p\sum _{\nu =0}^{h-1}\sum _{l=1}^{p-j+1}\frac{{(mh)}^{j-1}l!S_2(p-j+1,l)}{(p-j+1)l^k}\\ & \times \left(\begin{array}{c}p\\ j\end{array}\right){(-1)}^{p-j+1-l}((\mu h)m^{p-j}+(m\nu )h^{p-j}){\bar{B}}_j(\frac{\nu }{h}+\frac{\mu }{m}).\end{array}$$

For $k=1$, this reciprocity relation for the poly-Dedekind sums reduces to that for Apostol’s generalized Dedekind sums given by (see Corollary 8)

$$\begin{aligned} &hm^{p}S_{p}(h,m)+mh^{p}S_{p}(m,h) \\ &\quad = \sum_{\mu =0}^{m-1}\sum _{\nu =0}^{h-1}(mh)^{p-1}(\mu h+m\nu ) \overline{B}_{p} \biggl(\frac{\nu }{h}+\frac{\mu }{m} \biggr). \end{aligned}$$

We recommend the readers to look at the articles [15, 16, 18, 21] and the more recent one [14], which are related to the present paper. In Sect. 2, we derive various facts about the poly-Bernoulli polynomials that will be needed in the next section. In Sect. 3, we define the poly-Dedekind sums associated with the poly-Bernoulli functions and demonstrate a reciprocity relation for them.

2 Poly-Dedekind sums associated with poly-Bernoulli functions

Let n be a nonnegative integer. Then the Stirling numbers of the second kind are defined by

$$\begin{aligned}& x^{n}=\sum_{k=0}^{n}S_{2}(n,k) (x)_{k},\quad (n\ge 0), ( \text{see [1--14, 17, 19]}), \end{aligned}$$

where $(x)_{0}=1$, $(x)_{n}=x(x-1)\cdots (x-n+1)$, ($n\ge 1$).

From (9) and (10), we note that

$$ \frac{\operatorname{Li}_{k}(1-e^{-t})}{e^{t}-1}=\sum_{n=0}^{\infty } \beta _{n}^{(k)} \frac{t^{n}}{n!}. $$

(15)

Thus, by (15), we get

$$\begin{aligned} \operatorname{Li}_{k} \bigl(1-e^{-t} \bigr) &= \Biggl( \sum_{l=0}^{\infty } \beta _{l}^{(k)} \frac{t^{l}}{l!} \Biggr) \bigl(e^{t}-1 \bigr) \\ &= \sum_{n=0}^{\infty } \bigl(\beta _{n}^{(k)}(1)-\beta _{n}^{(k)} \bigr)\frac{t^{n}}{n!}. \end{aligned}$$

(16)

On the other hand,

$$\begin{aligned} \operatorname{Li}_{k} \bigl(1-e^{-t} \bigr) &= \sum _{m=1}^{\infty } \frac{1}{m^{k}} \bigl(1-e^{-t} \bigr)^{m} = \sum _{m=1}^{\infty } \frac{(-1)^{m}m!}{m^{k}} \frac{1}{m!} \bigl(e^{-t}-1 \bigr)^{m} \\ &= \sum_{m=1}^{\infty }\frac{(-1)^{m}m!}{m^{k}} \sum_{n=m}^{\infty }S_{2}(n,m) (-1)^{n} \frac{t^{n}}{n!} \\ &= \sum_{m=1}^{\infty } \Biggl(\sum _{m=1}^{n} \frac{(-1)^{n-m}m!}{m^{k}}S_{2}(n,m) \Biggr)\frac{t^{n}}{n!}. \end{aligned}$$

(17)

Therefore, by (16) and (17), we obtain the following theorem.

Theorem 1

For $n\in \mathbb{N}$, we have

$$\begin{aligned}& \beta _{n}^{(k)}(1)-\beta _{n}^{(k)}= \sum_{m=1}^{n} \frac{(-1)^{n-m}m!}{m^{k}}S_{2}(n,m). \end{aligned}$$

From Theorem 1, we note that

$$\begin{aligned}& \beta _{0}^{(k)}=1,\qquad \beta _{1}^{(k)}=-1+ \frac{1}{2^{k}},\qquad \beta _{2}^{(k)}=1- \frac{3}{2^{k}}+\frac{2}{3^{k}}, \ldots . \end{aligned}$$

Taking $k=1$ in Theorem 1 gives us the following corollary.

Corollary 2

For $n\in \mathbb{N}$, we have

$$\begin{aligned}& \sum_{m=1}^{n}(-1)^{n-m}(m-1)!S_{2}(n,m)= \delta _{n,1}, \end{aligned}$$

where $\delta _{n,k}$ is the Kronecker symbol.

The three identities in the following lemma can be shown just as in Theorem 3, Corollary 4, and Theorem 5 of [13], and hence their proofs are left to the reader as exercises.

Lemma 3

For $s,p\in \mathbb{N}$, we have

$$\begin{array}{c}\sum _{\nu =0}^p\left(\begin{array}{c}p\\ \nu \end{array}\right)\frac{{\beta }_{\nu }^{(k)}}{p-\nu +2}=\left(\begin{array}{c}p+1\\ s\end{array}\right)\frac{{\beta }_{p-s+1}^{(k)}(1)}{p+1}+\frac{s-1}{p+1}\left(\begin{array}{c}p+2\\ s\end{array}\right)\frac{{\beta }_{p-s+2}^{(k)}(1)}{p+2},\hfill \\ \sum _{\nu =0}^{p-s+1}\left(\begin{array}{c}p\\ \nu \end{array}\right)\left(\begin{array}{c}p-\nu +2\\ s\end{array}\right)\frac{{\beta }_{\nu }^{(k)}}{p-\nu +2}\hfill \\ =\left(\begin{array}{c}p+1\\ s\end{array}\right)\frac{{\beta }_{p-s+1}^{(k)}(1)}{p+1}+\frac{s-1}{p+1}\left(\begin{array}{c}p+2\\ s\end{array}\right)\frac{{\beta }_{p-s+2}^{(k)}(1)}{p+2}-\frac{1}{s}\left(\begin{array}{c}p\\ s-2\end{array}\right){\beta }_{p-s+2}^{(k)},\hfill \end{array}$$

and

$$\sum _{s=0}^p\left(\begin{array}{c}p\\ s\end{array}\right){\beta }_s^{(k)}\frac{1}{p+2-s}=\frac{{\beta }_{p+1}^{(k)}(1)}{p+1}-\frac{{\beta }_{p+2}^{(k)}(1)}{(p+1)(p+2)}+\frac{{\beta }_{p+2}^{(k)}}{(p+1)(p+2)}.$$

As a further generalization of Apostol’s Dedekind sums, we study poly-Dedekind sums associated with poly-Bernoulli functions of index k, which are given by

$$ T_{p}^{(k)}(h,m)=\sum_{\mu =1}^{m-1} \frac{\mu }{m}\overline{\beta }_{p}^{(k)} \biggl( \frac{h\mu }{m} \biggr), $$

(18)

where $h,m,p\in \mathbb{N}$, $k\in \mathbb{Z}$, and $\overline{\beta }_{p}^{(k)}(x)=\beta _{p}^{(k)} (\langle x\rangle )$ are the poly-Bernoulli functions of index k.

Note that

$$\begin{aligned}& T_{p}^{(1)}(h,m)=\sum_{\mu =1}^{m-1} \frac{\mu }{m}\overline{B}_{p} \biggl(\frac{h\mu }{m} \biggr)=S_{p}(h,m). \end{aligned}$$

The two identities in Lemma 4 can be proved in the same way as in Proposition 6 and Theorem 7 in [13], while the identity in Lemma 5 can be shown just as in Theorem 8 in [13]. Therefore their proofs are left to the reader.

Lemma 4

Let p be an odd positive integer ≥3, and $m\in \mathbb{N}$. Then we have

$$\begin{array}{c}{m}^p{T}_p^{(k)}(1,m)\hfill \\ =\sum _{\nu =0}^p\left(\begin{array}{c}p\\ \nu \end{array}\right)\frac{{\beta }_{\nu }^{(k)}}{p+2-\nu }{m}^{p+1}+\sum _{i=1}^{p-1}\sum _{\nu =0}^{p+1-i}\left(\begin{array}{c}p\\ \nu \end{array}\right)\left(\begin{array}{c}p+2-\nu \\ i\end{array}\right)\frac{{\beta }_{\nu }^{(k)}}{p+2-\nu }{B}_i{m}^{p+1-i}+B_{p+1}\hfill \end{array}$$

and

$$\begin{array}{rl}& (p+1)m^p{T}_p^{(k)}(1,m)\\ & =\sum _{i=0}^{p+1}\left(\begin{array}{c}p+1\\ i\end{array}\right)B_i{m}^{p+1-i}{\beta }_{p+1-i}^{(k)}(1)\\ & +\frac{1}{p+2}\sum _{i=0}^{p+1}\left(\begin{array}{c}p+2\\ i\end{array}\right)(i-1)B_i{m}^{p+1-i}({\beta }_{p+2-i}^{(k)}(1)-{\beta }_{p+2-i}^{(k)}).\end{array}$$

Lemma 5

For $m,n,h\in \mathbb{N}$ with $(h,m)=1$, and p any positive odd integer ≥3, we have

$$\begin{array}{rl}& \sum _{s=0}^{p+1}\left(\begin{array}{c}p+1\\ s\end{array}\right)B_s{\beta }_{p+1-s}^{(k)}(1){(mh)}^{p+1-s}\\ & =m^p\sum _{\mu =0}^{m-1}\sum _{s=0}^{p+1}\left(\begin{array}{c}p+1\\ s\end{array}\right)h^s{\beta }_s^{(k)}\left(\frac{\mu }{m}\right)B_{p+1-s}(h-\left[\frac{h\mu }{m}\right]).\end{array}$$

For $d\in \mathbb{N}$, we observe that

$$\begin{array}{rl}\sum _{n=0}^{\mathrm{\infty }}{\beta }_n^{(k)}(x)\frac{t^n}{n!}& =\frac{{Li}_k(1-e^{-t})}{e^t-1}{e}^{xt}=\frac{{Li}_k(1-e^{-t})}{e^{dt}-1}\sum _{i=0}^{d-1}{e}^{(i+x)t}\\ & =\frac{1}{dt}{Li}_k(1-e^{-t})\sum _{i=0}^{d-1}\frac{dt}{e^{dt}-1}{e}^{(\frac{i+x}{d})dt}\\ & =\sum _{j=0}^{\mathrm{\infty }}{d}^{j-1}\sum _{i=0}^{d-1}{B}_j\left(\frac{x+i}{d}\right)\frac{t^j}{j!}\frac{1}{t}\sum _{l=1}^{\mathrm{\infty }}\frac{l!}{l^k}\frac{1}{l!}{(1-e^{-t})}^l\\ & =\sum _{j=0}^{\mathrm{\infty }}{d}^{j-1}\sum _{i=0}^{d-1}{B}_j\left(\frac{x+i}{d}\right)\frac{t^j}{j!}\frac{1}{t}\sum _{l=1}^{\mathrm{\infty }}\frac{{(-1)}^{l}l!}{l^k}\sum _{m=l}^{\mathrm{\infty }}{S}_2(m,l)\frac{{(-t)}^m}{m!}\\ & =\sum _{j=0}^{\mathrm{\infty }}{d}^{j-1}\sum _{i=0}^{d-1}{B}_j\left(\frac{x+i}{d}\right)\frac{t^j}{j!}\sum _{m=0}^{\mathrm{\infty }}\frac{1}{m+1}\sum _{l=1}^{m+1}\frac{l!{(-1)}^{l+m-1}}{l^k}{S}_2(m+1,l)\frac{t^m}{m!}\\ & =\sum _{n=0}^{\mathrm{\infty }}(\sum _{j=0}^n\sum _{i=0}^{d-1}\sum _{l=1}^{n-j+1}\left(\begin{array}{c}n\\ j\end{array}\right)d^{j-1}{B}_j\left(\frac{x+i}{d}\right)\frac{l!{(-1)}^{n-j+1-l}}{(n-j+1)l^k}{S}_2(n-j+1,l))\frac{t^n}{n!}.\end{array}$$

(19)

Therefore, by (19), we obtain the following theorem.

Theorem 6

For $k\in \mathbb{Z}$, $d\in \mathbb{N}$, and $n\ge 0$, we have

$${\beta }_n^{(k)}(x)=\sum _{j=0}^n\sum _{i=0}^{d-1}\sum _{l=1}^{n-j+1}\left(\begin{array}{c}n\\ j\end{array}\right)d^{j-1}{B}_j\left(\frac{x+i}{d}\right)\frac{l!{(-1)}^{n-j+1-l}}{(n-j+1)l^k}{S}_2(n-j+1,l).$$

By (18), Lemmas 3–5, and Theorem 6, we get

$$\begin{array}{rl}& h{m}^p{T}_p^{(k)}(h,m)+m{h}^p{T}_p^{(k)}(m,h)\\ & =h{m}^p\sum _{\mu =0}^{m-1}\frac{\mu }{m}{\bar{\beta }}_p^{(k)}\left(\frac{h\mu }{m}\right)+m{h}^p\sum _{\nu =0}^{h-1}\left(\frac{\mu }{h}\right){\bar{\beta }}_p^{(k)}\left(\frac{m\nu }{h}\right)\\ & =h{m}^p\sum _{\mu =0}^{m-1}\frac{\mu }{m}\sum _{j=0}^p{h}^{j-1}\left(\begin{array}{c}p\\ j\end{array}\right)\sum _{\nu =0}^{h-1}\sum _{l=1}^{p-j+1}\frac{l!{(-1)}^{p-j+1-l}}{(p-j+1)l^k}{S}_2(p-j+1,l){\bar{B}}_j(\frac{\mu }{m}+\frac{\nu }{h})\\ & +m{h}^p\sum _{\nu =0}^{h-1}\frac{\nu }{h}\sum _{j=0}^p{m}^{j-1}\left(\begin{array}{c}p\\ j\end{array}\right)\sum _{\mu =0}^{m-1}\sum _{l=1}^{p-j+1}\frac{l!{(-1)}^{p-j+1-l}}{(p-j+1)l^k}{S}_2(p-j+1,l){\bar{B}}_j(\frac{\nu }{h}+\frac{\mu }{m})\\ & =\sum _{\mu =0}^{m-1}\frac{\mu }{m}\sum _{j=0}^p{m}^{p-j}{(mh)}^j\left(\begin{array}{c}p\\ j\end{array}\right)\sum _{\nu =0}^{h-1}\sum _{l=1}^{p-j+1}{\bar{B}}_j(\frac{\mu }{m}+\frac{\nu }{h})\frac{l!S_2(p-j+1,l)}{(p-j+1)l^k}{(-1)}^{p-j+1-l}\\ & +\sum _{\nu =0}^{h-1}\frac{\nu }{h}\sum _{j=0}^p{h}^{p-j}{(mh)}^j\left(\begin{array}{c}p\\ j\end{array}\right)\sum _{\mu =0}^{m-1}\sum _{l=1}^{p-j+1}{\bar{B}}_j(\frac{\nu }{h}+\frac{\mu }{m})\frac{l!S_2(p-j+1,l)}{(p-j+1)l^k}{(-1)}^{p-j+1-l}\\ & =\sum _{\mu =0}^{m-1}\sum _{j=0}^p\sum _{\nu =0}^{h-1}\sum _{l=1}^{p-j+1}(\mu h){(mh)}^{-1}{m}^{p-j}{(mh)}^j\left(\begin{array}{c}p\\ j\end{array}\right)\\ & \times {\bar{B}}_j(\frac{\mu }{m}+\frac{\nu }{h})\frac{l!S_2(p-j+1,l)}{(p-j+1)l^k}{(-1)}^{p-j+1-l}\\ & +\sum _{\mu =0}^{m-1}\sum _{j=0}^p\sum _{\nu =0}^{h-1}\sum _{l=1}^{p-j+1}(m\nu ){(mh)}^{-1}{h}^{p-j}{(mh)}^j\left(\begin{array}{c}p\\ j\end{array}\right)\\ & \times {\bar{B}}_j(\frac{\nu }{h}+\frac{\mu }{m})\frac{l!S_2(p-j+1,l)}{(p-j+1)l^k}{(-1)}^{p-j+1-l}\\ & =\sum _{\mu =0}^{m-1}\sum _{j=0}^p\sum _{\nu =0}^{h-1}\sum _{l=1}^{p-j+1}\frac{{(mh)}^{j-1}l!S_2(p-j+1,l)}{(p-j+1)l^k}\\ & \times \left(\begin{array}{c}p\\ j\end{array}\right){(-1)}^{p-j+1-l}((\mu h)m^{p-j}+(m\nu )h^{p-j}){\bar{B}}_j(\frac{\nu }{h}+\frac{\mu }{m}).\end{array}$$

(20)

Therefore, by (20), we obtain the following reciprocity theorem for the poly-Dedekind sums associated with poly-Bernoulli functions with index k.

Theorem 7

For $m,h,p\in \mathbb{N}$ and $k\in \mathbb{Z}$, we have

$$\begin{array}{rl}& h{m}^p{T}_p^{(k)}(h,m)+m{h}^p{T}_p^{(k)}(m,h)\\ & =\sum _{\mu =0}^{m-1}\sum _{j=0}^p\sum _{\nu =0}^{h-1}\sum _{l=1}^{p-j+1}\frac{{(mh)}^{j-1}l!S_2(p-j+1,l)}{(p-j+1)l^k}\\ & \times \left(\begin{array}{c}p\\ j\end{array}\right){(-1)}^{p-j+1-l}((\mu h)m^{p-j}+(m\nu )h^{p-j}){\bar{B}}_j(\frac{\nu }{h}+\frac{\mu }{m}).\end{array}$$

In case of $k = 1$, by making use of Corollary 2, we obtain the following reciprocity relation for the generalized Dedekind sums defined by Apostol.

Corollary 8

For $m,h,p \in \mathbb{N}$, we have

$$\begin{aligned} hm^{p}T_{p}^{(1)}(h,m)+mh^{p}T_{p}^{(1)}(m,h) &=mh^{p}S_{p}(h,m)+mh^{p}S_{p}(m,h) \\ &= \sum_{\mu =0}^{m-1}\sum _{\nu =0}^{h-1}(mh)^{p-1} (\mu h+m \nu ) \overline{B}_{p} \biggl(\frac{\nu }{h}+\frac{\mu }{m} \biggr). \end{aligned}$$

3 Conclusion

The quantity called the Dedekind sum,

$$\begin{aligned}& S(h,m) = \sum_{\mu =1}^{m-1} \frac{\mu }{m}\overline{B}_{1} \biggl( \frac{h\mu }{m} \biggr), \end{aligned}$$

occurs in the transformation behavior of the logarithm of the Dedekind eta-function under substitutions from the modular group. It was shown by Dedekind that they satisfy the following reciprocity relation:

$$ S(h,m)+S(m,h)=\frac{1}{12} \biggl(\frac{h}{m}+ \frac{1}{hm}+\frac{m}{h} \biggr)-\frac{1}{4} $$

if h and m are relatively prime positive integers.

Apostol considered the generalized Dedekind sums

$$\begin{aligned}& S_{p}(h,m)=\sum_{\mu =1}^{m-1} \frac{\mu }{m}\overline{B}_{p} \biggl( \frac{h\mu }{m} \biggr) \end{aligned}$$

and derived a reciprocity relation for them. Recently, as one generalization of the generalized Dedekind sums, the poly-Dedekind sums

$$\begin{aligned}& S_{P}^{(k)}(h,m)=\sum_{\mu =1}^{m-1} \frac{\mu }{m}\overline{B}_{p}^{(k)} \biggl( \frac{h\mu }{m} \biggr), \end{aligned}$$

associated with the type 2 poly-Bernoulli functions of arbitrary indices, were introduced and were shown to satisfy a reciprocity relation. In this paper, as another generalization of the generalized Dedekind sums, we considered the poly-Dedekind sums

$$\begin{aligned}& T_{p}^{(k)}(h,m)= \sum_{\mu =1}^{m-1} \frac{\mu }{m}\overline{\beta }_{p}^{(k)} \biggl( \frac{h\mu }{m} \biggr), \end{aligned}$$

associated with the poly-Bernoulli functions of arbitrary indices, and derived a reciprocity relation for these poly-Dedekind sums.