Relations for Bernoulli-Barnes numbers and Barnes zeta functions (Q2876607)

The paper under review considers Bernoulli-Barnes numbers, Bernoulli-Barnes polynomials and Barnes zeta functions.NEWLINENEWLINENEWLINEBernoulli-Barnes numbers \(B_k({\mathbf a})\) are generalizations of Bernoulli numbers, which are defined for a fixed vector \({\mathbf a}=(a_1,a_2,\cdots,a_n)\in ({\mathbb R}_{>0})^n\) through NEWLINE\[NEWLINE\frac{z^n}{(e^{a_1z}-1)\cdots(e^{a_n z}-1)}=\sum_{k\geq 0}B_k({\mathbf a}) \frac{z^k}{k!}.NEWLINE\]NEWLINE The first main result of the paper is the following.NEWLINENEWLINENEWLINETheorem 1.1. For \(n\geq 3,m\geq 1\), where \(m\) is odd, and \({\mathbf a}=(a_1,\cdots,a_n) \in ({\mathbb R}_{>0})^n\), NEWLINE\[NEWLINE\sum_{j=n-m}^{n}{{n+j-4} \choose{j-2}}\frac{1}{(m-n+j)!}\sum_{|I|=j} B_{m-n+j}({\mathbf a}_I)=\begin{cases} \frac{1}{2}\;\;&\text{if}\;n=m=3\,,\\ 0&\text{otherwise}\,, \end{cases}NEWLINE\]NEWLINE where the inner sum is over all subsets \(I\subset\{1,2,\dots,n\}\) of cardinality \(j\), and \({\mathbf a}_I=(a_i\,|i\in I)\).NEWLINENEWLINEBernoulli-Barnes polynomials \(B_k(x;{\mathbf a})\) are defined through NEWLINE\[NEWLINE\frac{z^n e^{xz}}{(e^{a_1z}-1)\cdots (e^{a_n z}-1)}=\sum_{k\geq 0} B_k(x;{\mathbf a})\frac{z^k}{k!}\;,NEWLINE\]NEWLINE and these give special values of the Barnes zeta function NEWLINE\[NEWLINE\zeta(s,x;{\mathbf a})=\sum_{m_1,\dots,m_n\in {\mathbb Z}_{\geq 0}}\frac{1} {(x+m_1a_1+\cdots +m_n a_n)^s}NEWLINE\]NEWLINE at negative integers \(-k\). More precisely, we have NEWLINE\[NEWLINE\zeta(-k,x;{\mathbf a})=\frac{(-1)^n k!}{(k+n)!}B_{k+n}(x;{\mathbf a}).NEWLINE\]NEWLINENEWLINENEWLINENEWLINEThen the second main result of the paper expresses \(\zeta(s,x;{\mathbf a})\) in terms of Bernoulli-Barnes polynomials, Hurwitz zeta functions, and Fourier-Dedekind sums defined as NEWLINE\[NEWLINE\sigma_r(a_1,\dots,\hat{a}_j,\dots,a_n;a_j)=\frac{1}{a_j} \sum_{\lambda^{a_j}=1\neq \lambda}\frac{\lambda^r}{\prod_{1\leq k\neq j\leq n} (1-\lambda^{a_k})}.NEWLINE\]NEWLINENEWLINENEWLINENEWLINETheorem 1.3. Let \(a_1,\dots,a_n\) be pairwise coprime positive integers. Then NEWLINE\[NEWLINE\zeta(s,x;{\mathbf a})=\frac{(-1)^{n-1}}{(n-1)!}\sum_{k=0}^{n-1} (-1)^k{{n-1}\choose{k}}B_{n-1-k}(x;{\mathbf a})\zeta(s-k;x)NEWLINE\]NEWLINE NEWLINE\[NEWLINE+\sum_{j=1}^{n}a_j^{-s}\sum_{r=0}^{a_j-1}\sigma_{-r}(a_1,\dots,\hat{a}_j, \dots,a_n;a_j)\zeta\left(s;\frac{x+r}{a_j}\right).NEWLINE\]NEWLINE NEWLINEFrom this, by specializing \(s\) at negative integers, reciprocity laws of the generalization of Apostol-Dedekind sums are obtained.NEWLINENEWLINEIn the last section, various formulas for \(B_m(x;{\mathbf a})\) such as difference, symmetry and recurrence formulas are given.