Reciprocity of degenerate poly-Dedekind-type DC sums

ABSTRACT

Dedekind-type DC sums and their properties are defined in terms of Euler functions. Ma et al. recently introduced poly-Dedekind-type DC sums and demonstrated that they satisfy a reciprocity relation. In this paper, we introduce the degenerate poly-Euler polynomials and numbers, and we also consider the reciprocity relations of degenerate poly-Dedekind-type DC sums. Equivalently, several properties and identities of degenerate poly-Euler functions are obtained.

KEYWORDS:

• degenerate poly-Dedekind DC sums
• degenerate polylogarithm function
• degenerate poly-Euler polynomials

MATHEMATICS SUBJECT CLASSIFICATIONS:

• 11A15
• 11F20
• 11B68

1. Introduction

Dedekind sums is a sum introduced by Dedekind when studying η functions. Moreover, the study of Dedekind sums and similar sums plays an important role in analytic mathematics, combinatorics, and algebraic number theory, and is closely related to the transformation formula of Dedekind functions and special value problems. Hence, Dedekind sums is defined by the relation: (1) $s(h,k)=\sum _{a=1}^{k-1}\left(\left(\frac{a}{k}\right)\right)\left(\left(\frac{ha}{k}\right)\right),$(1) sawtooth function $(())$: $\mathbb{R}\to \mathbb{R}$ is defined by $((x))=\{\begin{array}{ll}x-[x]-\frac{1}{2},& \text{if}x\text{is not an integer},\\ 0,& \text{otherwise}x\text{is an integer},\end{array}$$[x]$ being the largest integer $\le x$ (see [1–6]).

From (1(1) $s(h,k)=\sum _{a=1}^{k-1}\left(\left(\frac{a}{k}\right)\right)\left(\left(\frac{ha}{k}\right)\right),$(1) ), we know the reciprocity law of the classical Dedekind sums $s(h,k)+s(k,h)=-\frac{1}{4}+\frac{1}{12}(\frac{h}{k}+\frac{k}{h}+\frac{1}{hk}),$where $(h,k)=1$ and $h,k\in \mathbb{N}:=1,2,3,\dots$, and ${\mathbb{N}}_0:=\mathbb{N}\cup \{0\}$.

In [7], for $\mathrm{\Re }(s)>1$ and $a\ne 0,-1,-2,\dots$, Apostol defined the Hurwitz zeta functions (2) $\zeta (s,a)=\sum _{n=0}^{\mathrm{\infty }}(n+a{)}^{-s},$(2) and the generalized Dedekind sums was given by (3) $s_p(h,k)=ip!(2\pi ik{)}^{-p}\sum _{\mu =1}^{k-1}cot\frac{\pi h\mu }{k}\zeta (p,\frac{\mu }{k}),$(3) where $\zeta (s,a)$ is the Hurwitz zeta function.

Furthermore, he deduced the reciprocity law which using complex integration (4) $\begin{array}{rl}& \frac{i(p+1)!}{(2\pi i{)}^p}\{h\sum _{\mu =1}^{k-1}cot\frac{\pi h\mu }{k}\zeta (p,\frac{\mu }{k})+k\sum _{\nu =1}^{h-1}cot\frac{\pi k\nu }{h}\zeta (p,\frac{\nu }{h})\}\\ & =p{B}_{p+1}+\sum _{s=0}^{p+1}(\genfrac{}{}{0ex}{}{p+1}{s})B_s{B}_{p+1-s}{h}^s{k}^{p+1-s},(\mathrm{odd}p>1).\end{array}$(4) Tak$\acute{a}$cs [6] introduced the generalized Dedekind sums by (5) $s_r(a,b|x,y)=\sum _{j=0}^{|b|-1}{P}_r(\frac{a(j+y)}{b}+x)P_1\left(\frac{j+y}{b}\right),$(5) where r is nonnegative integer, a, b are integers with $b\ne 0$, and x, y are real numbers, $P_r(x)(0\le x<1)$ is the rth Bernoulli polynomials. When $x=y=0,r=1$, $s_1(a,b|0,0)-\frac{1}{4}=s(a,b),$where $s(a,b)$ is Dedekind sums.

Carlitz [8] generalized Dedekind sums by (6) $s_n(h,k:x,y)=\sum _{a=0}^{k-1}{\overline{B}}_n(h\frac{a+y}{k}+x){\overline{B}}_1\left(\frac{a+y}{k}\right),$(6) where n, h, k are positive integers and x, y are arbitrary real numbers, ${\overline{B}}_n(x)$ is the nth Bernoulli polynomials. Note that for odd n and $x=y=0,s_n(h,k:x,y)$ reduces to $s_n(h,k)$.

Kim [9] defined q-Bernoulli polynomials ${\beta }_m(x:q)(x\in {\mathbb{Z}}_p,m\in \mathbb{N})$, and constructed a p-adic continuous function for an odd prime to contain a p-adic q-analogue of higher-order Dedekind sums $k^m{S}_{m+1}(h,k)$ and introduced p-adic q-Dedekind sums (7) $S_{m,q}(h,k:q^l)=\sum _{M=1}^{k-1}\frac{1-q^M}{1-q^k}{\int }_{{\mathbb{Z}}_p}{q}^{-lx}{\left(\frac{1-q^{l(x+\{\frac{hm}{k}\})}}{1-q^l}\right)}^m{d}_{{\mu }_{q^l}}(x),$(7) where h, k are fixed integers with $(h,k)=(p,k)=1$, $\{x\}$ is the fractional part of x.

Bayad and Simsek [10] defined p-adic q-twisted Dedekind-type sums (8) $s_{\xi }^{(h)}(m,a,b:q)=\sum _{c=0}^{b-1}\frac{c{q}^{hc}{\xi }^c}{b}{\int }_{{\mathbb{Z}}_p}{q}^{hx}{\varphi }_{\xi }(x){(x+\left\{\frac{ca}{b}\right\})}^m{d}_{{\mu }_1}(x),$(8) where $h,a,b$ are positive integers with $(a,b)=1$, p is an odd prime such that $p\nmid b$, $\{t\}$ is the fractional part of t.

Jackson [11] first defined the q-derivative operator ${\mathfrak{D}}_q$. In recent years, the q-derivatives have been widely used in other polynomials, such as, Khan et al. [12] defined a number of subclasses of q-starlike functions, which associated with Janowski functions, and some coefficient inequalities with them. Shi et al. [13] defined a new subclass of Janowski-type multivalent q-starlike functions. Zhang et al. [14] defined a new subclass of bi-univalent functions and obtained some interesting results. By using q-Poisson distribution, Khan et al. [15] defined function classes and derived some new subclasses and some useful results.

Berndt [16] defined Dedekind sums with characters by (9) $s(h,k;\chi )=\sum _{a=0}^{kf-1}\chi (a){\overline{B}}_{1,\chi }\left(\frac{ha}{k}\right){\overline{B}}_1\left(\frac{a}{kf}\right),$(9) where $(h,k)=1$, χ is a primitive character of conductor f, and ${\overline{B}}_{n,\chi }(x)$ is character Bernoulli function, for 0<x<1, ${\overline{B}}_{n,\chi }(x)=B_{n,\chi }(x)$, where $B_{n,\chi }(x)$ denotes character Bernoulli polynomials.

Kim et al. [2] introduced the poly-Dedekind sums (10) $S_p^{(k)}(h,m)=\sum _{\mu =1}^{m-1}\frac{\mu }{m}{\overline{B}}_p^{(k)}\left(\frac{h\mu }{m}\right),$(10) where $h,m,p\in \mathbb{N},k\in \mathbb{Z}$, and ${\overline{B}}_p^{(k)}=B_p^{(k)}$ are the type 2 poly-Bernoulli functions.

Cenkci et al. [1] defined the degenerate Dedekind sums (11) $s_n(h,k:x,y|\lambda )=\sum _{a=0}^{|k|-1}{\overline{\beta }}_n(\lambda ,h\frac{a+y}{k}+x){\overline{\beta }}_1(\lambda ,\frac{a+y}{k}),$(11) where h, k is arbitrary integers with $(h,k)=d$, ${\overline{\beta }}_n(\lambda ,x)$ is the nth degenerate Bernoulli polynomial and the reciprocity law are introduced by $\begin{array}{rl}& (n+1)\{h{k}^n{s}_n(h,k:x,y|\lambda )+k{h}^n{s}_n(k,h:y,x|\lambda )\}\\ & =\sum _{j=0}^{n+1}(\genfrac{}{}{0ex}{}{n+1}{j})h^j{\overline{\beta }}_j(k\lambda ,y)k^{n+1-j}{\overline{\beta }}_{(n+1-j)}(h\lambda ,x)\\ & +\frac{\lambda }{2}hk{d}^n(n+1)(h+k+2n-2){\overline{\beta }}_n(\frac{hk}{d}\lambda ,\frac{hy+kx}{d})\\ & +n{d}^{n+1}{\overline{\beta }}_{n+1}(\frac{hk}{d}\lambda ,\frac{hy+kx}{d}),\end{array}$where d stands for the greatest common divisor of h and k.

Kim [17] introduced the Dedekind-type DC sums (12) $T_p(h,k)=2\sum _{u=1}^{k-1}(-1{)}^{u-1}\frac{u}{k}{\overline{E}}_p\left(\frac{hu}{k}\right),(h\in {\mathbb{Z}}_{+}),$(12) and proved reciprocity relation to Dedekind-type DC sums by using Euler polynomials, (13) $\begin{array}{rl}& k^p{T}_p(h,k)+h^p{T}_p(k,h)\\ & =2\sum _{\stackrel{u=0}{u-[\frac{hu}{k}]\equiv 1\mathrm{mod}2}}^{k-1}{(kh(E+\frac{u}{k})+k(E+h-[\frac{hu}{k}]))}^p+(hE+kE{)}^p+(p+2)E_P,\end{array}$(13) where h, k are relative prime positive integers.

Simsek [18] introduced the trigonometric function representation of Dedekind-type DC sums and the reciprocity law to Dedekind-type DC sums. Ma et al. [4] introduced a new type of degenerate poly-Euler polynomials and numbers using the polylogarithm functions and degenerate polylogarithm functions, and introduced the poly-Dedekind type DC sums $T_p^{(k)}(h,m)$ and derived the reciprocity relations to poly-Dedekind DC sums. In particular, we are interested in degenerate poly-Euler polynomials and numbers. The goal of this paper is to show the degenerate poly-Dedekind type DC sums and demonstrate it satisfy a reciprocity relation by using the degenerate polylogarithm functions. In this paper, we introduce the degenerate poly-Euler polynomials and numbers, give some equations and properties. Meanwhile, we give the reciprocity relation to the degenerate poly-Dedekind-type DC sums.

Now, we give several definitions and properties. At the first, the poly-Dedekind-type DC sums (see [4]) was given by (14) $T_p^{(k)}(h,m)=2\sum _{\mu =1}^{m-1}(-1{)}^{\mu }\frac{\mu }{m}{\overline{E}}_p^{(k)}\left(\frac{h\mu }{m}\right),$(14) where $h,m,p\in \mathbb{N}$, ${\overline{E}}_p^{(k)}(x)$ are poly-Euler functions, ${\overline{E}}_p^{(k)}(x)=E_p^{(k)}(x-[x])$.

In [1–9,16–30], the Euler polynomials are usually defined by (15) $\frac{2}{e^t+1}{e}^{xt}=\sum _{n=0}^{\mathrm{\infty }}{E}_n(x)\frac{t^n}{n!}.$(15) From [19], Carlitz introduced the ordinary degenerate Euler polynomials (16) $\frac{2}{e_{\lambda }(t)+1}{e}_{\lambda }^x(t)=\sum _{n=0}^{\mathrm{\infty }}{E}_{n,\lambda }(x)\frac{t^n}{n!}.$(16) The poly-Euler polynomials are defined by the generating function [29] (17) $\frac{L{i}_k(1-e^{-2t})}{t(e^t+1)}{e}^{xt}=\sum _{n=0}^{\mathrm{\infty }}{E}_n^{(k)}(x)\frac{t^n}{n!},$(17) when x = 0, $E_n^{(k)}=E_n^{(k)}(0)$ are called the poly-Euler numbers. In particular, $E_n^{(1)}(x)=E_n(x)(n\ge 0)$ are called Euler polynomials.

The degenerate exponential function is defined by (18) $e_{\lambda }^x(t)=\sum _{n=0}^{\mathrm{\infty }}(x{)}_{n,\lambda }\frac{t^n}{n!},(\lambda \in \mathbb{R}(\mathrm{or}\mathbb{C})),$(18) where $(x{)}_{0,\lambda }=1$, $(x{)}_{n,\lambda }=x(x-\lambda )\cdots (x-(n-1)\lambda )$, $(n\ge 1)$ [17,20,22–24].

From (18(18) $e_{\lambda }^x(t)=\sum _{n=0}^{\mathrm{\infty }}(x{)}_{n,\lambda }\frac{t^n}{n!},(\lambda \in \mathbb{R}(\mathrm{or}\mathbb{C})),$(18) ), we note that (19) $e_{\lambda }^x(t)=(1+\lambda t{)}^{\frac{x}{\lambda }},$(19) when x = 1, $e_{\lambda }(t)=(1+\lambda t{)}^{\frac{1}{\lambda }}=e_{\lambda }^1(t)$ [20,22–24].

For $k\in \mathbb{Z}$, Kim–Kim introduced the degenerate polylogarithm function [20] defined by (20) $L{i}_{k,\lambda }(x)=\sum _{n=1}^{\mathrm{\infty }}\frac{(-\lambda {)}^{n-1}(1{)}_{n,1/\lambda }}{(n-1)!n^k}{x}^n,(|x|<1).$(20)

When k = 1, $L{i}_{1,\lambda }(x)=\sum _{n=1}^{\mathrm{\infty }}\frac{(-\lambda {)}^{n-1}(1{)}_{n,1/\lambda }}{n!}{x}^n=-{\log }_{\lambda }(1-x),$where ${\log }_{\lambda }(x)$ is the inverse to $e_{\lambda }(x)$.

The type 2 degenerate Stirling numbers are defined by [20,21] $(x{)}_{n,\lambda }=\sum _{k=0}^n{S}_{2,\lambda }(n,k)(x{)}_k,(n\ge 0),$so we have (21) $\frac{1}{k!}(e_{\lambda }(t)-1{)}^k=\sum _{n=k}^{\mathrm{\infty }}{S}_{2,\lambda }(n,k)\frac{t^n}{n!},(n\ge 0).$(21) Ma et al. [29] introduced the degenerate poly-Euler polynomials, by using the degenerate polylogrithm functions as follows: (22) $\frac{L{i}_{k,\lambda }(1-e_{\lambda }(-2t))}{t(e_{\lambda }(t)+1)}{e}_{\lambda }^x(t)=\sum _{n=0}^{\mathrm{\infty }}{E}_{n,\lambda }^{(k)}(x)\frac{t^n}{n!},$(22) note that $E_{n,\lambda }^{(1)}(x)=E_{n,\lambda }(x)(n\ge 0)$, when x = 0, $E_{n,\lambda }^{(k)}=E_{n,\lambda }^{(k)}(0)$ are called the degenerate poly-Euler numbers.

From (22(22) $\frac{L{i}_{k,\lambda }(1-e_{\lambda }(-2t))}{t(e_{\lambda }(t)+1)}{e}_{\lambda }^x(t)=\sum _{n=0}^{\mathrm{\infty }}{E}_{n,\lambda }^{(k)}(x)\frac{t^n}{n!},$(22) ), we note that $E_{n,\lambda }(1)+E_{n,\lambda }=2{\delta }_{0,n}$, $(n\ge 0)$, where ${\delta }_{n,k}$ is the Kronecker's symbol.

By (22(22) $\frac{L{i}_{k,\lambda }(1-e_{\lambda }(-2t))}{t(e_{\lambda }(t)+1)}{e}_{\lambda }^x(t)=\sum _{n=0}^{\mathrm{\infty }}{E}_{n,\lambda }^{(k)}(x)\frac{t^n}{n!},$(22) ), we have (23) $\begin{array}{rl}\sum _{n=0}^{\mathrm{\infty }}{E}_{n,\lambda }^{(k)}(x)\frac{t^n}{n!}& =\frac{L{i}_{k,\lambda }(1-e_{\lambda }(-2t))}{t(e_{\lambda }(t)+1)}{e}_{\lambda }^x(t)\\ & =\left(\sum _{m=0}^{\mathrm{\infty }}{E}_{m,\lambda }^{(k)}\frac{t^m}{m!}\right)\left(\sum _{l=0}^{\mathrm{\infty }}(x{)}_{l,\lambda }\frac{t^l}{l!}\right)\\ & =\sum _{n=0}^{\mathrm{\infty }}\left(\sum _{l=0}^n(\genfrac{}{}{0ex}{}{n}{l})E_{n-l,\lambda }^{(k)}(x{)}_{l,\lambda }\right)\frac{t^n}{n!}.\end{array}$(23) Comparing both sides of (23(23) $\begin{array}{rl}\sum _{n=0}^{\mathrm{\infty }}{E}_{n,\lambda }^{(k)}(x)\frac{t^n}{n!}& =\frac{L{i}_{k,\lambda }(1-e_{\lambda }(-2t))}{t(e_{\lambda }(t)+1)}{e}_{\lambda }^x(t)\\ & =\left(\sum _{m=0}^{\mathrm{\infty }}{E}_{m,\lambda }^{(k)}\frac{t^m}{m!}\right)\left(\sum _{l=0}^{\mathrm{\infty }}(x{)}_{l,\lambda }\frac{t^l}{l!}\right)\\ & =\sum _{n=0}^{\mathrm{\infty }}\left(\sum _{l=0}^n(\genfrac{}{}{0ex}{}{n}{l})E_{n-l,\lambda }^{(k)}(x{)}_{l,\lambda }\right)\frac{t^n}{n!}.\end{array}$(23) ), we can derive the following identity: (24) $E_{n,\lambda }^{(k)}(x)=\sum _{l=0}^n(\genfrac{}{}{0ex}{}{n}{l})E_{n-l,\lambda }^{(k)}(x{)}_{l,\lambda }=\sum _{l=0}^n(\genfrac{}{}{0ex}{}{n}{l})E_{l,\lambda }^{(k)}(x{)}_{n-l,\lambda },(n\ge 0).$(24)

2. Degenerate poly-Dedekind-type DC sums and reciprocity laws

In this section, we introduce the degenerate poly-Dedekind-type DC sums $T_{p,\lambda }^{(k)}(h,m)$ and derive the $E_{n,\lambda }^{(k)}(x)$ and reciprocity relations $m^p{T}_{p,\lambda }^{(k)}(h,m)+h^p{T}_{p,\lambda }^{(k)}(m,h)$.

From (14(14) $T_p^{(k)}(h,m)=2\sum _{\mu =1}^{m-1}(-1{)}^{\mu }\frac{\mu }{m}{\overline{E}}_p^{(k)}\left(\frac{h\mu }{m}\right),$(14) ), we give the degenerate poly-Dedekind type DC sums as (25) $T_{p,\lambda }^{(k)}(h,m)=2\sum _{\mu =1}^{m-1}(-1{)}^{\mu }\frac{\mu }{m}{\overline{E}}_{p,\lambda }^{(k)}\left(\frac{h\mu }{m}\right),$(25) where $h,m,p\in \mathbb{N}$, ${\overline{E}}_{p,\lambda }^{(k)}(x)$ are called degenerate poly-Euler functions, ${\overline{E}}_{p,\lambda }^{(k)}(x)=E_{p,\lambda }^{(k)}(x-[x])$.

Note that $T_{p,\lambda }^{(1)}(h,m)=2\sum _{\mu =1}^{m-1}(-1{)}^{\mu }\frac{\mu }{m}{\overline{E}}_{p,\lambda }^{(1)}\left(\frac{h\mu }{m}\right)=2\sum _{\mu =1}^{m-1}(-1{)}^{\mu }\frac{\mu }{m}{\overline{E}}_{p,\lambda }\left(\frac{h\mu }{m}\right).$

Theorem 2.1

For $n\in \mathbb{N}$, we have (26) $n{E}_{n-1,\lambda }^{(k)}(1)+n{E}_{n-1,\lambda }^{(k)}=\sum _{m=1}^n\frac{(-1{)}^{n-1}{2}^n{\lambda }^{m-1}(1{)}_{m,1/\lambda }}{m^{k-1}}{S}_{2,\lambda }(n,m).$(26)

Proof.

From (22(22) $\frac{L{i}_{k,\lambda }(1-e_{\lambda }(-2t))}{t(e_{\lambda }(t)+1)}{e}_{\lambda }^x(t)=\sum _{n=0}^{\mathrm{\infty }}{E}_{n,\lambda }^{(k)}(x)\frac{t^n}{n!},$(22) ), we note that (27) $\frac{L{i}_{k,\lambda }(1-e_{\lambda }(-2t))}{t(e_{\lambda }(t)+1)}=\sum _{n=0}^{\mathrm{\infty }}{E}_{n,\lambda }^{(k)}\frac{t^n}{n!}.$(27) By (27(27) $\frac{L{i}_{k,\lambda }(1-e_{\lambda }(-2t))}{t(e_{\lambda }(t)+1)}=\sum _{n=0}^{\mathrm{\infty }}{E}_{n,\lambda }^{(k)}\frac{t^n}{n!}.$(27) ), we have (28) $\begin{array}{rl}L{i}_{k,\lambda }(1-e_{\lambda }(-2t))& =\frac{L{i}_{k,\lambda }(1-e_{\lambda }(-2t))}{t(e_{\lambda }(t)+1)}t(e_{\lambda }(t)+1)\\ & =\frac{L{i}_{k,\lambda }(1-e_{\lambda }(-2t))}{t(e_{\lambda }(t)+1)}t{e}_{\lambda }(t)+\frac{L{i}_{k,\lambda }(1-e_{\lambda }(-2t))}{t(e_{\lambda }(t)+1)}t\\ & =\sum _{n=0}^{\mathrm{\infty }}t{E}_{n,\lambda }^{(k)}(1)\frac{t^n}{n!}+\sum _{n=0}^{\mathrm{\infty }}t{E}_{n,\lambda }^{(k)}\frac{t^n}{n!}\\ & =\sum _{n=0}^{\mathrm{\infty }}(E_{n,\lambda }^{(k)}(1)+E_{n,\lambda }^{(k)})\frac{t^{n+1}}{n!}\\ & =\sum _{n=1}^{\mathrm{\infty }}(n{E}_{n-1,\lambda }^{(k)}(1)+n{E}_{n-1,\lambda }^{(k)})\frac{t^n}{n!}.\end{array}$(28) On the other hand (29) $\begin{array}{rl}L{i}_{k,\lambda }(1-e_{\lambda }(-2t))& =\sum _{m=1}^{\mathrm{\infty }}\frac{(-\lambda {)}^{m-1}(1{)}_{m,1/\lambda }}{(m-1)!m^k}(1-e_{\lambda }(-2t){)}^m\\ & =\sum _{m=1}^{\mathrm{\infty }}\frac{(-\lambda {)}^{m-1}(1{)}_{m,1/\lambda }}{m^{k-1}}\frac{1}{m!}(e_{\lambda }(-2t)-1{)}^m(-1{)}^m\\ & =\sum _{m=1}^{\mathrm{\infty }}\frac{(-\lambda {)}^{m-1}(1{)}_{m,1/\lambda }}{m^{k-1}}\sum _{n=m}^{\mathrm{\infty }}{S}_{2,\lambda }(n,m)\frac{(-2t{)}^n}{n!}(-1{)}^m\\ & =\sum _{m=1}^{\mathrm{\infty }}\frac{{\lambda }^{m-1}(1{)}_{m,1/\lambda }}{m^{k-1}}\sum _{n=m}^{\mathrm{\infty }}(-1{)}^{n-1}{2}^n{S}_{2,\lambda }(n,m)\frac{t^n}{n!}\\ & =\sum _{n=1}^{\mathrm{\infty }}\left(\sum _{m=1}^n\frac{(-1{)}^{n-1}{2}^n{\lambda }^{m-1}(1{)}_{m,1/\lambda }}{m^{k-1}}{S}_{2,\lambda }(n,m)\right)\frac{t^n}{n!}.\end{array}$(29) Thus, comparing the coefficients of (28(28) $\begin{array}{rl}L{i}_{k,\lambda }(1-e_{\lambda }(-2t))& =\frac{L{i}_{k,\lambda }(1-e_{\lambda }(-2t))}{t(e_{\lambda }(t)+1)}t(e_{\lambda }(t)+1)\\ & =\frac{L{i}_{k,\lambda }(1-e_{\lambda }(-2t))}{t(e_{\lambda }(t)+1)}t{e}_{\lambda }(t)+\frac{L{i}_{k,\lambda }(1-e_{\lambda }(-2t))}{t(e_{\lambda }(t)+1)}t\\ & =\sum _{n=0}^{\mathrm{\infty }}t{E}_{n,\lambda }^{(k)}(1)\frac{t^n}{n!}+\sum _{n=0}^{\mathrm{\infty }}t{E}_{n,\lambda }^{(k)}\frac{t^n}{n!}\\ & =\sum _{n=0}^{\mathrm{\infty }}(E_{n,\lambda }^{(k)}(1)+E_{n,\lambda }^{(k)})\frac{t^{n+1}}{n!}\\ & =\sum _{n=1}^{\mathrm{\infty }}(n{E}_{n-1,\lambda }^{(k)}(1)+n{E}_{n-1,\lambda }^{(k)})\frac{t^n}{n!}.\end{array}$(28) ) and (29(29) $\begin{array}{rl}L{i}_{k,\lambda }(1-e_{\lambda }(-2t))& =\sum _{m=1}^{\mathrm{\infty }}\frac{(-\lambda {)}^{m-1}(1{)}_{m,1/\lambda }}{(m-1)!m^k}(1-e_{\lambda }(-2t){)}^m\\ & =\sum _{m=1}^{\mathrm{\infty }}\frac{(-\lambda {)}^{m-1}(1{)}_{m,1/\lambda }}{m^{k-1}}\frac{1}{m!}(e_{\lambda }(-2t)-1{)}^m(-1{)}^m\\ & =\sum _{m=1}^{\mathrm{\infty }}\frac{(-\lambda {)}^{m-1}(1{)}_{m,1/\lambda }}{m^{k-1}}\sum _{n=m}^{\mathrm{\infty }}{S}_{2,\lambda }(n,m)\frac{(-2t{)}^n}{n!}(-1{)}^m\\ & =\sum _{m=1}^{\mathrm{\infty }}\frac{{\lambda }^{m-1}(1{)}_{m,1/\lambda }}{m^{k-1}}\sum _{n=m}^{\mathrm{\infty }}(-1{)}^{n-1}{2}^n{S}_{2,\lambda }(n,m)\frac{t^n}{n!}\\ & =\sum _{n=1}^{\mathrm{\infty }}\left(\sum _{m=1}^n\frac{(-1{)}^{n-1}{2}^n{\lambda }^{m-1}(1{)}_{m,1/\lambda }}{m^{k-1}}{S}_{2,\lambda }(n,m)\right)\frac{t^n}{n!}.\end{array}$(29) ), we obtain identity (26(26) $n{E}_{n-1,\lambda }^{(k)}(1)+n{E}_{n-1,\lambda }^{(k)}=\sum _{m=1}^n\frac{(-1{)}^{n-1}{2}^n{\lambda }^{m-1}(1{)}_{m,1/\lambda }}{m^{k-1}}{S}_{2,\lambda }(n,m).$(26) ).

Let k = 1, from Theorem 2.1, we get the following corollary.

Corollary 2.2

For $n\in \mathbb{N}$, we have $\sum _{m=1}^n(-1{)}^{n-1}{2}^n{\lambda }^{m-1}(1{)}_{m,1/\lambda }{S}_{2,\lambda }(n,m)=2n{\delta }_{0,n-1},$where ${\delta }_{n,k}$ is the Kronecker's symbol.

Proof.

From (26(26) $n{E}_{n-1,\lambda }^{(k)}(1)+n{E}_{n-1,\lambda }^{(k)}=\sum _{m=1}^n\frac{(-1{)}^{n-1}{2}^n{\lambda }^{m-1}(1{)}_{m,1/\lambda }}{m^{k-1}}{S}_{2,\lambda }(n,m).$(26) ), we take k = 1, then $\begin{array}{rl}\sum _{m=1}^n(-1{)}^{n-1}{2}^n{\lambda }^{m-1}(1{)}_{m,1/\lambda }{S}_{2,\lambda }(n,m)& =n{E}_{n-1,\lambda }(1)+n{E}_{n-1,\lambda }\\ & =n(E_{n-1,\lambda }(1)+E_{n-1,\lambda })\\ & =2n{\delta }_{0,n-1}.\end{array}$Hence, we get the desired result.

Theorem 2.3

For $n\in \mathbb{N},k\in \mathbb{Z}$ and $d\equiv 1(\mathrm{mod}2),n\ge 0$ $E_{n,\lambda }^{(k)}(x)=\sum _{j=0}^n\sum _{i=0}^{d-1}\sum _{l=1}^{n-j+1}(\genfrac{}{}{0ex}{}{n}{j})d^j{E}_{j,\lambda /d}\left(\frac{i+x}{d}\right)\frac{{\lambda }^{l-1}(-1{)}^{n-j+i}{2}^{n-j}(1{)}_{l,1/\lambda }}{(n-j+1)l^{k-1}}{S}_{2,\lambda }(n-j+1,l).$

Proof.

For $d\in \mathbb{N}$, and $d\equiv 1(\mathrm{mod}2)$, we have (30) $\begin{array}{rl}& \sum _{n=0}^{\mathrm{\infty }}{E}_{n,\lambda }^{(k)}(x)\frac{t^n}{n!}=\frac{L{i}_{k,\lambda }(1-e_{\lambda }(-2t))}{t(e_{\lambda }(t)+1)}{e}_{\lambda }^x(t)=\frac{L{i}_{k,\lambda }(1-e_{\lambda }(-2t))}{t(e_{\lambda /d}(dt)+1)}\sum _{i=0}^{d-1}(-1{)}^i{e}_{\lambda }^{i+x}(t)\\ & =\frac{1}{2t}\sum _{i=0}^{d-1}(-1{)}^i{e}_{\lambda /d}^{\frac{i+x}{d}}(dt)\frac{2}{e_{\lambda /d}(dt)+1}L{i}_{k,\lambda }(1-e_{\lambda }(-2t))\\ & =\frac{1}{2t}\sum _{i=0}^{d-1}(-1{)}^i\sum _{j=0}^{\mathrm{\infty }}{E}_{j,\lambda /d}\left(\frac{i+x}{d}\right)\frac{(dt{)}^j}{j!}L{i}_{k,\lambda }(1-e_{\lambda }(-2t))\\ & =\frac{1}{2t}\sum _{i=0}^{d-1}(-1{)}^i\sum _{j=0}^{\mathrm{\infty }}{d}^j{E}_{j,\lambda /d}\left(\frac{i+x}{d}\right)\frac{t^j}{j!}\sum _{l=1}^{\mathrm{\infty }}\frac{(-\lambda {)}^{l-1}(1{)}_{l,1/\lambda }}{(l-1)!l^k}(1-e_{\lambda }(-2t){)}^l\\ & =\frac{1}{2t}\sum _{j=0}^{\mathrm{\infty }}{d}^j\sum _{i=0}^{d-1}(-1{)}^i{E}_{j,\lambda /d}\left(\frac{i+x}{d}\right)\frac{t^j}{j!}\sum _{l=1}^{\mathrm{\infty }}\frac{(-\lambda {)}^{l-1}(1{)}_{l,1/\lambda }}{l^{k-1}}\frac{1}{l!}(-1{)}^l(e_{\lambda }(-2t)-1{)}^l\\ & =\frac{1}{2t}\sum _{j=0}^{\mathrm{\infty }}{d}^j\sum _{i=0}^{d-1}(-1{)}^i{E}_{j,\lambda /d}\left(\frac{i+x}{d}\right)\frac{t^j}{j!}\sum _{l=1}^{\mathrm{\infty }}\frac{(-\lambda {)}^{l-1}(1{)}_{l,1/\lambda }}{l^{k-1}}(-1{)}^l\sum _{m=l}^{\mathrm{\infty }}{S}_{2,\lambda }(m,l)\frac{(-2t{)}^m}{m!}\\ & =\frac{1}{2t}\sum _{j=0}^{\mathrm{\infty }}{d}^j\sum _{i=0}^{d-1}(-1{)}^i{E}_{j,\lambda /d}\left(\frac{i+x}{d}\right)\frac{t^j}{j!}\sum _{m=l}^{\mathrm{\infty }}(\sum _{l=1}^m\frac{{\lambda }^{l-1}(-1{)}^{m-1}(1{)}_{l,1/\lambda }{2}^m}{l^{k-1}}{S}_{2,\lambda }(m,l))\frac{t^m}{m!}\\ & =\sum _{j=0}^{\mathrm{\infty }}{d}^j\sum _{i=0}^{d-1}{E}_{j,\lambda /d}\left(\frac{i+x}{d}\right)\frac{t^j}{j!}\sum _{m=1}^{\mathrm{\infty }}(\sum _{l=1}^m\frac{{\lambda }^{l-1}(-1{)}^{m+i-1}(1{)}_{l,1/\lambda }{2}^{m-1}}{l^{k-1}}{S}_{2,\lambda }(m,l))\frac{t^{m-1}}{m!}\\ & =\sum _{j=0}^{\mathrm{\infty }}{d}^j\sum _{i=0}^{d-1}{E}_{j,\lambda /d}\left(\frac{i+x}{d}\right)\frac{t^j}{j!}\sum _{m=0}^{\mathrm{\infty }}(\sum _{l=1}^{m+1}\frac{{\lambda }^{l-1}(-1{)}^{m+i}(1{)}_{l,1/\lambda }{2}^m}{l^{k-1}}{S}_{2,\lambda }(m+1,l))\frac{t^m}{(m+1)!}\\ & =\sum _{n=0}^{\mathrm{\infty }}(\sum _{j=0}^n{d}^j\sum _{i=0}^{d-1}{E}_{j,\lambda /d}\left(\frac{i+x}{d}\right)\sum _{l=1}^{n-j+1}\frac{{\lambda }^{l-1}(-1{)}^{n-j+i}(1{)}_{l,1/\lambda }{2}^{n-j}}{l^{k-1}}{S}_{2,\lambda }(n-j+1,l))\frac{t^n}{j!(n-j+1)!}\\ & =\sum _{n=0}^{\mathrm{\infty }}(\sum _{j=0}^n\sum _{i=0}^{d-1}\sum _{l=1}^{n-j+1}(\genfrac{}{}{0ex}{}{n}{j})d^j{E}_{j,\lambda /d}\left(\frac{i+x}{d}\right)\frac{{\lambda }^{l-1}(-1{)}^{n-j+i}(1{)}_{l,1/\lambda }{2}^{n-j}}{(n-j+1)l^{k-1}}{S}_{2,\lambda }(n-j+1,l))\frac{t^n}{n!}.\end{array}$(30) Then, from (30(30) $\begin{array}{rl}& \sum _{n=0}^{\mathrm{\infty }}{E}_{n,\lambda }^{(k)}(x)\frac{t^n}{n!}=\frac{L{i}_{k,\lambda }(1-e_{\lambda }(-2t))}{t(e_{\lambda }(t)+1)}{e}_{\lambda }^x(t)=\frac{L{i}_{k,\lambda }(1-e_{\lambda }(-2t))}{t(e_{\lambda /d}(dt)+1)}\sum _{i=0}^{d-1}(-1{)}^i{e}_{\lambda }^{i+x}(t)\\ & =\frac{1}{2t}\sum _{i=0}^{d-1}(-1{)}^i{e}_{\lambda /d}^{\frac{i+x}{d}}(dt)\frac{2}{e_{\lambda /d}(dt)+1}L{i}_{k,\lambda }(1-e_{\lambda }(-2t))\\ & =\frac{1}{2t}\sum _{i=0}^{d-1}(-1{)}^i\sum _{j=0}^{\mathrm{\infty }}{E}_{j,\lambda /d}\left(\frac{i+x}{d}\right)\frac{(dt{)}^j}{j!}L{i}_{k,\lambda }(1-e_{\lambda }(-2t))\\ & =\frac{1}{2t}\sum _{i=0}^{d-1}(-1{)}^i\sum _{j=0}^{\mathrm{\infty }}{d}^j{E}_{j,\lambda /d}\left(\frac{i+x}{d}\right)\frac{t^j}{j!}\sum _{l=1}^{\mathrm{\infty }}\frac{(-\lambda {)}^{l-1}(1{)}_{l,1/\lambda }}{(l-1)!l^k}(1-e_{\lambda }(-2t){)}^l\\ & =\frac{1}{2t}\sum _{j=0}^{\mathrm{\infty }}{d}^j\sum _{i=0}^{d-1}(-1{)}^i{E}_{j,\lambda /d}\left(\frac{i+x}{d}\right)\frac{t^j}{j!}\sum _{l=1}^{\mathrm{\infty }}\frac{(-\lambda {)}^{l-1}(1{)}_{l,1/\lambda }}{l^{k-1}}\frac{1}{l!}(-1{)}^l(e_{\lambda }(-2t)-1{)}^l\\ & =\frac{1}{2t}\sum _{j=0}^{\mathrm{\infty }}{d}^j\sum _{i=0}^{d-1}(-1{)}^i{E}_{j,\lambda /d}\left(\frac{i+x}{d}\right)\frac{t^j}{j!}\sum _{l=1}^{\mathrm{\infty }}\frac{(-\lambda {)}^{l-1}(1{)}_{l,1/\lambda }}{l^{k-1}}(-1{)}^l\sum _{m=l}^{\mathrm{\infty }}{S}_{2,\lambda }(m,l)\frac{(-2t{)}^m}{m!}\\ & =\frac{1}{2t}\sum _{j=0}^{\mathrm{\infty }}{d}^j\sum _{i=0}^{d-1}(-1{)}^i{E}_{j,\lambda /d}\left(\frac{i+x}{d}\right)\frac{t^j}{j!}\sum _{m=l}^{\mathrm{\infty }}(\sum _{l=1}^m\frac{{\lambda }^{l-1}(-1{)}^{m-1}(1{)}_{l,1/\lambda }{2}^m}{l^{k-1}}{S}_{2,\lambda }(m,l))\frac{t^m}{m!}\\ & =\sum _{j=0}^{\mathrm{\infty }}{d}^j\sum _{i=0}^{d-1}{E}_{j,\lambda /d}\left(\frac{i+x}{d}\right)\frac{t^j}{j!}\sum _{m=1}^{\mathrm{\infty }}(\sum _{l=1}^m\frac{{\lambda }^{l-1}(-1{)}^{m+i-1}(1{)}_{l,1/\lambda }{2}^{m-1}}{l^{k-1}}{S}_{2,\lambda }(m,l))\frac{t^{m-1}}{m!}\\ & =\sum _{j=0}^{\mathrm{\infty }}{d}^j\sum _{i=0}^{d-1}{E}_{j,\lambda /d}\left(\frac{i+x}{d}\right)\frac{t^j}{j!}\sum _{m=0}^{\mathrm{\infty }}(\sum _{l=1}^{m+1}\frac{{\lambda }^{l-1}(-1{)}^{m+i}(1{)}_{l,1/\lambda }{2}^m}{l^{k-1}}{S}_{2,\lambda }(m+1,l))\frac{t^m}{(m+1)!}\\ & =\sum _{n=0}^{\mathrm{\infty }}(\sum _{j=0}^n{d}^j\sum _{i=0}^{d-1}{E}_{j,\lambda /d}\left(\frac{i+x}{d}\right)\sum _{l=1}^{n-j+1}\frac{{\lambda }^{l-1}(-1{)}^{n-j+i}(1{)}_{l,1/\lambda }{2}^{n-j}}{l^{k-1}}{S}_{2,\lambda }(n-j+1,l))\frac{t^n}{j!(n-j+1)!}\\ & =\sum _{n=0}^{\mathrm{\infty }}(\sum _{j=0}^n\sum _{i=0}^{d-1}\sum _{l=1}^{n-j+1}(\genfrac{}{}{0ex}{}{n}{j})d^j{E}_{j,\lambda /d}\left(\frac{i+x}{d}\right)\frac{{\lambda }^{l-1}(-1{)}^{n-j+i}(1{)}_{l,1/\lambda }{2}^{n-j}}{(n-j+1)l^{k-1}}{S}_{2,\lambda }(n-j+1,l))\frac{t^n}{n!}.\end{array}$(30) ), we arrive at the desired result.

Theorem 2.4

For $m,h,p\in \mathbb{N}$, $k\in \mathbb{Z}$, and $m\equiv 1(\mathrm{mod}2),h\equiv 1(\mathrm{mod}2)$, we have $\begin{array}{rl}& m^p{T}_{p,\lambda }^{(k)}(h,m)+h^p{T}_{p,\lambda }^{(k)}(m,h)\\ & =\sum _{\mu =0}^{m-1}\sum _{j=0}^p\sum _{\nu =0}^{h-1}\sum _{l=1}^{p-j+1}(\genfrac{}{}{0ex}{}{p}{j})(mh{)}^{j-1}\frac{{\lambda }^{l-1}(-1{)}^{p-j}{2}^{p-j+1}(1{)}_{l,1/\lambda }}{(p-j+1)l^{k-1}}{S}_{2,\lambda }(p-j+1,l)(-1{)}^{\mu +\nu }\\ & \times ((\mu h)m^{p-j}{\overline{E}}_{j,\lambda /h}(\frac{\mu }{m}+\frac{\nu }{h})+(m\nu )h^{p-j}{\overline{E}}_{j,\lambda /m}(\frac{\mu }{m}+\frac{\nu }{h})).\end{array}$

Proof.

For $m,h\in \mathbb{N}$, and $m\equiv 1(\mathrm{mod}2),h\equiv 1(\mathrm{mod}2)$, by (25(25) $T_{p,\lambda }^{(k)}(h,m)=2\sum _{\mu =1}^{m-1}(-1{)}^{\mu }\frac{\mu }{m}{\overline{E}}_{p,\lambda }^{(k)}\left(\frac{h\mu }{m}\right),$(25) ) and Theorem 2.3, we get (31) $\begin{array}{rl}& m^p{T}_{p,\lambda }^{(k)}(h,m)+h^p{T}_{p,\lambda }^{(k)}(m,h)\\ & =2m^p\sum _{\mu =0}^{m-1}(-1{)}^{\mu }\frac{\mu }{m}{\overline{E}}_{p,\lambda }^{(k)}\left(\frac{h\mu }{m}\right)+2h^p\sum _{\nu =0}^{h-1}(-1{)}^{\nu }\frac{\nu }{h}{\overline{E}}_{p,\lambda }^{(k)}\left(\frac{m\nu }{h}\right)\\ & =2m^p\sum _{\mu =0}^{m-1}(-1{)}^{\mu }\frac{\mu }{m}\sum _{j=0}^p\sum _{\nu =0}^{h-1}\sum _{l=1}^{p-j+1}(\genfrac{}{}{0ex}{}{p}{j})\frac{{\lambda }^{l-1}(-1{)}^{p-j+\nu }{2}^{p-j}(1{)}_{l,1/\lambda }}{(p-j+1)l^{k-1}}{S}_{2,\lambda }(p-j+1,l)\\ & \times h^j{\overline{E}}_{j,\lambda /h}(\frac{\mu }{m}+\frac{\nu }{h})\\ & +2h^p\sum _{\nu =0}^{h-1}(-1{)}^{\nu }\frac{\nu }{h}\sum _{j=0}^p\sum _{\mu =0}^{m-1}\sum _{l=1}^{p-j+1}(\genfrac{}{}{0ex}{}{p}{j})\frac{{\lambda }^{l-1}(-1{)}^{p-j+\mu }{2}^{p-j}(1{)}_{l,1/\lambda }}{(p-j+1)l^{k-1}}{S}_{2,\lambda }(p-j+1,l)\\ & \times m^j{\overline{E}}_{j,\lambda /m}(\frac{\mu }{m}+\frac{\nu }{h})\\ & =\sum _{\mu =0}^{m-1}\frac{\mu }{m}\sum _{j=0}^p\sum _{\nu =0}^{h-1}\sum _{l=1}^{p-j+1}(\genfrac{}{}{0ex}{}{p}{j})\frac{{\lambda }^{l-1}(-1{)}^{p-j+\nu +\mu }{2}^{p-j+1}(1{)}_{l,1/\lambda }}{(p-j+1)l^{k-1}}{S}_{2,\lambda }(p-j+1,l)\\ & \times m^{p-j}(mh{)}^j{\overline{E}}_{j,\lambda /h}(\frac{\mu }{m}+\frac{\nu }{h})\\ & +\sum _{\nu =0}^{h-1}\frac{\nu }{h}\sum _{j=0}^p\sum _{\mu =0}^{m-1}\sum _{l=1}^{p-j+1}(\genfrac{}{}{0ex}{}{p}{j})\frac{{\lambda }^{l-1}(-1{)}^{p-j+\mu +\nu }{2}^{p-j+1}(1{)}_{l,1/\lambda }}{(p-j+1)l^{k-1}}{S}_{2,\lambda }(p-j+1,l)\\ & \times h^{p-j}(mh{)}^j{\overline{E}}_{j,\lambda /m}(\frac{\mu }{m}+\frac{\nu }{h})\\ & =\sum _{\mu =0}^{m-1}\sum _{j=0}^p\sum _{\nu =0}^{h-1}\sum _{l=1}^{p-j+1}(\genfrac{}{}{0ex}{}{p}{j})(mh{)}^{j-1}\frac{{\lambda }^{l-1}(-1{)}^{p-j+\nu +\mu }{2}^{p-j+1}(1{)}_{l,1/\lambda }}{(p-j+1)l^{k-1}}{S}_{2,\lambda }(p-j+1,l)\\ & \times m^{p-j}(\mu h){\overline{E}}_{j,\lambda /h}(\frac{\mu }{m}+\frac{\nu }{h})\\ & +\sum _{\mu =0}^{m-1}\sum _{j=0}^p\sum _{\nu =0}^{h-1}\sum _{l=1}^{p-j+1}(\genfrac{}{}{0ex}{}{p}{j})(mh{)}^{j-1}\frac{{\lambda }^{l-1}(-1{)}^{p-j+\mu +\nu }{2}^{p-j+1}(1{)}_{l,1/\lambda }}{(p-j+1)l^{k-1}}{S}_{2,\lambda }(p-j+1,l)\\ & \times h^{p-j}(m\nu ){\overline{E}}_{j,\lambda /m}(\frac{\mu }{m}+\frac{\nu }{h})\\ & =\sum _{\mu =0}^{m-1}\sum _{j=0}^p\sum _{\nu =0}^{h-1}\sum _{l=1}^{p-j+1}(\genfrac{}{}{0ex}{}{p}{j})(mh{)}^{j-1}\frac{{\lambda }^{l-1}(-1{)}^{p-j}{2}^{p-j+1}(1{)}_{l,1/\lambda }}{(p-j+1)l^{k-1}}{S}_{2,\lambda }(p-j+1,l)(-1{)}^{\mu +\nu }\\ & \times ((\mu h)m^{p-j}{\overline{E}}_{j,\lambda /h}(\frac{\mu }{m}+\frac{\nu }{h})+(m\nu )h^{p-j}{\overline{E}}_{j,\lambda /m}(\frac{\mu }{m}+\frac{\nu }{h})).\end{array}$(31) Therefore, by (31(31) $\begin{array}{rl}& m^p{T}_{p,\lambda }^{(k)}(h,m)+h^p{T}_{p,\lambda }^{(k)}(m,h)\\ & =2m^p\sum _{\mu =0}^{m-1}(-1{)}^{\mu }\frac{\mu }{m}{\overline{E}}_{p,\lambda }^{(k)}\left(\frac{h\mu }{m}\right)+2h^p\sum _{\nu =0}^{h-1}(-1{)}^{\nu }\frac{\nu }{h}{\overline{E}}_{p,\lambda }^{(k)}\left(\frac{m\nu }{h}\right)\\ & =2m^p\sum _{\mu =0}^{m-1}(-1{)}^{\mu }\frac{\mu }{m}\sum _{j=0}^p\sum _{\nu =0}^{h-1}\sum _{l=1}^{p-j+1}(\genfrac{}{}{0ex}{}{p}{j})\frac{{\lambda }^{l-1}(-1{)}^{p-j+\nu }{2}^{p-j}(1{)}_{l,1/\lambda }}{(p-j+1)l^{k-1}}{S}_{2,\lambda }(p-j+1,l)\\ & \times h^j{\overline{E}}_{j,\lambda /h}(\frac{\mu }{m}+\frac{\nu }{h})\\ & +2h^p\sum _{\nu =0}^{h-1}(-1{)}^{\nu }\frac{\nu }{h}\sum _{j=0}^p\sum _{\mu =0}^{m-1}\sum _{l=1}^{p-j+1}(\genfrac{}{}{0ex}{}{p}{j})\frac{{\lambda }^{l-1}(-1{)}^{p-j+\mu }{2}^{p-j}(1{)}_{l,1/\lambda }}{(p-j+1)l^{k-1}}{S}_{2,\lambda }(p-j+1,l)\\ & \times m^j{\overline{E}}_{j,\lambda /m}(\frac{\mu }{m}+\frac{\nu }{h})\\ & =\sum _{\mu =0}^{m-1}\frac{\mu }{m}\sum _{j=0}^p\sum _{\nu =0}^{h-1}\sum _{l=1}^{p-j+1}(\genfrac{}{}{0ex}{}{p}{j})\frac{{\lambda }^{l-1}(-1{)}^{p-j+\nu +\mu }{2}^{p-j+1}(1{)}_{l,1/\lambda }}{(p-j+1)l^{k-1}}{S}_{2,\lambda }(p-j+1,l)\\ & \times m^{p-j}(mh{)}^j{\overline{E}}_{j,\lambda /h}(\frac{\mu }{m}+\frac{\nu }{h})\\ & +\sum _{\nu =0}^{h-1}\frac{\nu }{h}\sum _{j=0}^p\sum _{\mu =0}^{m-1}\sum _{l=1}^{p-j+1}(\genfrac{}{}{0ex}{}{p}{j})\frac{{\lambda }^{l-1}(-1{)}^{p-j+\mu +\nu }{2}^{p-j+1}(1{)}_{l,1/\lambda }}{(p-j+1)l^{k-1}}{S}_{2,\lambda }(p-j+1,l)\\ & \times h^{p-j}(mh{)}^j{\overline{E}}_{j,\lambda /m}(\frac{\mu }{m}+\frac{\nu }{h})\\ & =\sum _{\mu =0}^{m-1}\sum _{j=0}^p\sum _{\nu =0}^{h-1}\sum _{l=1}^{p-j+1}(\genfrac{}{}{0ex}{}{p}{j})(mh{)}^{j-1}\frac{{\lambda }^{l-1}(-1{)}^{p-j+\nu +\mu }{2}^{p-j+1}(1{)}_{l,1/\lambda }}{(p-j+1)l^{k-1}}{S}_{2,\lambda }(p-j+1,l)\\ & \times m^{p-j}(\mu h){\overline{E}}_{j,\lambda /h}(\frac{\mu }{m}+\frac{\nu }{h})\\ & +\sum _{\mu =0}^{m-1}\sum _{j=0}^p\sum _{\nu =0}^{h-1}\sum _{l=1}^{p-j+1}(\genfrac{}{}{0ex}{}{p}{j})(mh{)}^{j-1}\frac{{\lambda }^{l-1}(-1{)}^{p-j+\mu +\nu }{2}^{p-j+1}(1{)}_{l,1/\lambda }}{(p-j+1)l^{k-1}}{S}_{2,\lambda }(p-j+1,l)\\ & \times h^{p-j}(m\nu ){\overline{E}}_{j,\lambda /m}(\frac{\mu }{m}+\frac{\nu }{h})\\ & =\sum _{\mu =0}^{m-1}\sum _{j=0}^p\sum _{\nu =0}^{h-1}\sum _{l=1}^{p-j+1}(\genfrac{}{}{0ex}{}{p}{j})(mh{)}^{j-1}\frac{{\lambda }^{l-1}(-1{)}^{p-j}{2}^{p-j+1}(1{)}_{l,1/\lambda }}{(p-j+1)l^{k-1}}{S}_{2,\lambda }(p-j+1,l)(-1{)}^{\mu +\nu }\\ & \times ((\mu h)m^{p-j}{\overline{E}}_{j,\lambda /h}(\frac{\mu }{m}+\frac{\nu }{h})+(m\nu )h^{p-j}{\overline{E}}_{j,\lambda /m}(\frac{\mu }{m}+\frac{\nu }{h})).\end{array}$(31) ), we obtain the reciprocity relations for the degenerate poly-Dedekind-type DC sums.

Corollary 2.5

For $m,h,p\in \mathbb{N}$, and $m\equiv 1(\mathrm{mod}2),h\equiv 1(\mathrm{mod}2)$, then we get $m^p{T}_p(h,m)+h^p{T}_p(m,h)=2\sum _{\mu =0}^{m-1}\sum _{\nu =0}^{h-1}(mh{)}^{p-1}(-1{)}^{\mu +\nu }(\mu h+m\nu ){\overline{E}}_p(\frac{\mu }{m}+\frac{\nu }{h}).$

Proof.

Taking k = 1 and $\lambda \to 0$, by Corollary 2.2 and Theorem 2.4, we have $\begin{array}{rl}& m^p{T}_p(h,m)+h^p{T}_p(m,h)\\ & =\sum _{\mu =0}^{m-1}\sum _{j=0}^p\sum _{\nu =0}^{h-1}\sum _{l=1}^{p-j+1}(\genfrac{}{}{0ex}{}{p}{j})(mh{)}^{j-1}\frac{{\lambda }^{l-1}(-1{)}^{p-j}{2}^{p-j+1}(1{)}_{l,1/\lambda }}{p-j+1}{S}_{2,\lambda }(p-j+1,l)(-1{)}^{\mu +\nu }\\ & \times ((\mu h)m^{p-j}{\overline{E}}_{j,\lambda /h}(\frac{\mu }{m}+\frac{\nu }{h})+(m\nu )h^{p-j}{\overline{E}}_{j,\lambda /m}(\frac{\mu }{m}+\frac{\nu }{h}))\\ & =2\sum _{\mu =0}^{m-1}\sum _{\nu =0}^{h-1}(mh{)}^{p-1}(-1{)}^{\mu +\nu }((\mu h){\overline{E}}_{p,\lambda /h}(\frac{\mu }{m}+\frac{\nu }{h})+(m\nu ){\overline{E}}_{p,\lambda /m}(\frac{\mu }{m}+\frac{\nu }{h}))\\ & =2\sum _{\mu =0}^{m-1}\sum _{\nu =0}^{h-1}(mh{)}^{p-1}(-1{)}^{\mu +\nu }(\mu h+m\nu ){\overline{E}}_p(\frac{\mu }{m}+\frac{\nu }{h}).\end{array}$Consequently, we obtain the result.

3. Conclusion

In this paper, we observe that the poly-Dedekind-type DC sums and their generalizations are defined in terms of poly-Euler functions and their generalizations. So we introduced the degenerate poly-Ddekind-type DC sums $T_{p,\lambda }^{(k)}(h,m)$, which satisfy the reciprocity relations. Meanwhile, we derive several identities and properties for degenerate poly-Euler polynomials and numbers.

As a further generalizations of the poly-Dedekind type DC sums, we would like to further study the reciprocity relations of degenerate poly-Dedekind-type DC sums, some properties of degenerate poly-Dedekind-type DC sums, their generating functions in the exponential form and their applications in physics and mathematics as well as other fields.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research was funded by the National Natural Science Foundation of China (Nos. 12271320, 11871317,11926325, 11926321), Key Research and Development Program of Shaanxi (No. 2021GY-137).