Special values of generalized multiple Hurwitz zeta function at non-positive integers

TL;DR

This paper introduces a new method using Raabe's formula and Bernoulli numbers to evaluate the generalized multiple Hurwitz zeta function at non-positive integers.

Contribution

It presents an alternative approach for calculating these special values, expanding the computational techniques for zeta functions.

Findings

Derived explicit formulas for the zeta function at non-positive integers.

Demonstrated the effectiveness of Raabe's formula in this context.

Connected Bernoulli numbers to the evaluation process.

Abstract

In this paper, we provide an alternative method to calculate the values of generalized multiple Hurwitz zeta function at non-positive integers by means of \emph{Raabe}'s formula and the \textit{Bernoulli} numbers.

Equations

$${\zeta }_n(\underline{\alpha };s_1,\dots ,s_n)=\sum _{\underline{m}=(m_1,\dots ,m_n)\in {\mathbb{N}}^n}\frac{1}{(m_1+\alpha {)}^{s_1}\dots (m_1+\cdots +m_n+\alpha {)}^{s_n}}$$

$${\zeta }_n(\underline{\alpha };\underline{s})=\sum _{\underline{m}=(m_1,\dots ,m_n)\in {\mathbb{N}}^n}\prod _{i=1}^n\frac{1}{(m_1+\cdots +m_i+{\alpha }_i{)}^{s_i}}$$

$${\alpha }_i\ne 0,-1,-2,...,$$

$${\zeta }_n(\underline{\alpha };\underline{s})$$

$$=$$

$$Y_n(\underline{\alpha };\underline{s})={\int }_{[0,+\infty {[}^n}\prod _{i=1}^n\frac{1}{(x_1+\cdots +x_i+{\alpha }_i{)}^{s_i}}d\underline{x}.$$

$$(s_n-1)(s_n+s_{n-1}-2+k_n)...\left(\sum _{i=1}^n{s}_i-n+\sum _{i=2}^n{k}_i\right)=\prod _{j=1}^n\left(\sum _{i=j}^n{s}_i-n+j-1+\sum _{i=j+1}^n{k}_i\right)=0.$$

$$Y_n(\underline{\alpha };\underline{\text{s}})=(-1{)}^n$$

$$\sum _{\underline{k}=(k_2,...,k_n)\in {\mathbb{N}}^{n-1}}\frac{\left(\genfrac{}{}{0ex}{}{N_n+1}{k_n}\right)\left(\genfrac{}{}{0ex}{}{N_n+N_{n-1}+2-k_n}{k_{n-1}}\right)...\left(\genfrac{}{}{0ex}{}{{\sum }_{i=2}^n{N}_i+n-{\sum }_{i=3}^n{k}_i}{k_2}\right){\alpha }_1^{\left(-{\sum }_{i=1}^n{s}_i+n-{\sum }_{i=2}^n{k}_i\right)}}{{\prod }_{j=1}^n\left({\sum }_{i=j}^n{N}_i+n-j+1-{\sum }_{i=j+1}^n{k}_i\right)}\prod _{j=2}^n{\alpha }_j^{k_j}$$

$$T(\underline{\text{N}}):=\left\{\underline{k}=(k_2,...,k_n)\in {\mathbb{N}}^{n-1}:0\le k_j\le \sum _{i=j}^n{N}_i+n-j+1-\sum _{i=j+1}^n{k}_i,\forall 2\le j\le n\right\}.$$

$$\begin{array}[]{ccc}&\zeta_{n}(\underline{\alpha};-\underline{\textbf{N}})=(-1)^{n}&\\ &\sum_{\underline{k}=(k_{2},...,k_{n})\in\mathbb{N}^{n-1}}{\sum_{\underline{v}=(v_{1},...,v_{n})\in\mathbb{N}^{n}\atop v_{j}\leq k_{j}\;\forall\;2\leq j\leq n;v_{1}\leq\left(\sum_{i=1}^{n}{N_{i}}+n-\sum_{i=2}^{n}{k_{i}}\right)}A(-\underline{\textbf{N}})\;B_{\underline{v}}\prod_{j=1}^{n}{\frac{1}{\left(\sum_{i=j}^{n}{N_{i}}+n-j+1-\sum_{i=j+1}^{n}{k_{i}}\right)}}}\end{array}$$

$$A(-\underline{\text{N}})=\left(\genfrac{}{}{0ex}{}{{\sum }_{i=1}^n{N}_i+n-{\sum }_{i=2}^n{k}_i}{v_1}\right){\alpha }^{\left({\sum }_{i=1}^n{N}_i+n-{\sum }_{i=2}^n{v}_i\right)}\prod _{j=2}^n\left(\genfrac{}{}{0ex}{}{{\sum }_{i=j}^n{N}_i+n-j+1-{\sum }_{i=j+1}^n{k}_i}{k_j}\right)\left(\genfrac{}{}{0ex}{}{k_j}{v_j}\right).$$

$$T(\underline{\text{N}}):=\left\{\underline{k}=(k_2,...,k_n)\in {\mathbb{N}}^{n-1}:0\le k_j\le \sum _{i=j}^n{N}_i+n-j+1-\sum _{i=j+1}^n{k}_i,\forall 2\le j\le n\right\}.$$

$$B_{\underline{v}}=\prod _{j=1}^n{B}_{v_j}$$

$$Y_n(\underline{\alpha };\underline{\text{s}})={\int }_{[0,+\infty {[}^n}\prod _{i=1}^n(x_1+\cdots +x_i+{\alpha }_i{)}^{-s_i}d\underline{x}.$$

$$y_i=x_1+\cdots +x_i+{\alpha }_i$$

$$Y_n(\underline{\alpha };\underline{\text{s}})={\int }_{{\prod }_{i=1}^n[{\alpha }_i,+\infty {[}^n}\prod _{i=1}^n(y_1+\cdots +y_i{)}^{-s_i}d\underline{x}.$$

$$z_i=y_1+\cdots +y_i-\sum _{j=2}^i{\alpha }_j$$

$$\left\{\begin{array}[]{ccc}y_{1}=z_{1}&\\ y_{i}=z_{i}-z_{i-1}+\alpha_{i},&\forall\;2\leq i\leq n\end{array}\right.$$

$$\underline{z}\in V_n=\left\{\underline{z}\in {\mathbb{R}}^n:{\alpha }_1\le z_1\le z_2\le \cdots \le z_n\right\}$$

$$Y_n(\underline{\alpha };\underline{\text{s}})={\int }_{V_n}\prod _{i=1}^n(z_i+\sum _{j=2}^i{\alpha }_j{)}^{-s_i}d\underline{z}.$$

$$Y_n(\underline{\alpha };\underline{\text{s}})={\int }_{V_{n-1}}\prod _{i=1}^{n-1}(z_i+\sum _{j=2}^i{\alpha }_j{)}^{-s_i}\left({\int }_{z_{n-1}}^{+\infty }(z_n+\sum _{j=2}^n{\alpha }_j{)}^{-s_n}d{z}_n\right)d{z}_1...d{z}_{n-1}$$

$$\begin{array}[]{ccc}\int_{z_{n-1}}^{+\infty}{(z_{n}+\sum_{j=2}^{n}\alpha_{j})^{-s_{n}}\;dz_{n}}&=&\frac{(z_{n-1}+\sum_{j=2}^{n-1}\alpha_{j})^{-s_{n}+1}}{s_{n}-1}\left(1+\frac{\alpha_{n}}{z_{n-1}+\sum_{j=2}^{n-1}\alpha_{j}}\right)^{-s_{n}+1}\\ &=&\sum_{k_{n}\in\mathbb{N}}{\left(-s_{n}+1\atop k_{n}\right)\frac{(z_{n-1}+\sum_{j=2}^{n-1}\alpha_{j})^{-s_{n}+1-k_{n}}}{s_{n}-1}\;\alpha_{n}^{k_{n}}}\end{array}$$

$$Y_n(\underline{\alpha };\underline{\text{s}})$$

$$\Re \left(\sum _{i=1}^n{s}_i\right)-n+j-1+\sum _{i=2}^n{k}_i>0$$

$$\Re (s_n)-1>0.$$

$$(s_n-1)(s_n+s_{n-1}-2+k_n)...\left(\sum _{i=1}^n{s}_i-n+\sum _{i=2}^n{k}_i\right)=\prod _{j=1}^n\left(\sum _{i=j}^n{s}_i-n+j-1+\sum _{i=j+1}^n{k}_i\right)=0.$$

$$\left(\genfrac{}{}{0ex}{}{N_n+1}{k_n}\right)...\left(\genfrac{}{}{0ex}{}{{\sum }_{i=2}^n{N}_i+n-{\sum }_{i=3}^n{k}_i}{k_2}\right)=\prod _{j=2}^n\left(\genfrac{}{}{0ex}{}{{\sum }_{i=j}^n{N}_i+n-j+1-{\sum }_{i=j+1}^n{k}_i}{k_j}\right)=0$$

$$k_j>\sum _{i=j}^n{N}_i+n-j+1-\sum _{i=j+1}^n{k}_i.$$

$$T(\underline{\text{N}}):=\left\{\underline{k}=(k_2,...,k_n)\in {\mathbb{N}}^{n-1}:0\le k_j\le \sum _{i=j}^n{N}_i+n-j+1-\sum _{i=j+1}^n{k}_i,\forall 2\le j\le n\right\}$$

$$Y_n(\underline{\alpha };\underline{\text{s}})=(-1{)}^n$$

Full text

Mathematics Subject Classifications: 11M32; 11M41.

Key words: Generalized multiple Hurwitz zeta function; integral representation; special values; Bernoulli numbers; Raabe’s formula.

Introduction and notations

The multiple Hurwitz zeta function is defined by

[TABLE]

where $\alpha\neq 0,-1,-2,...,$ and $(s_{1},\dots,s_{2})s\in\mathbb{C}^{n}$, which introduced by Akiyama and Ishikawa and proved by Akiyama and Ishikawa [3]. Matsumoto and Tanigawa proved the analytic continuation of wide class of multiple Dirichlet series and multiple Hurwitz zeta functions in [12] and [13], and the analytic continuation of the series (0.1) is a special case of [12, Theorem 1].

Our main result in this work is the values at non positive integers of the following series

[TABLE]

where, $\underline{\alpha}=(\alpha_{1},\dots,\alpha_{n})\in\mathbb{R}^{n}$ verified some conditions, this series is called the generalized multiple Hurwitz zeta function.

The key of this study is the use of the Raabe formula [7] which expresses the integral in terms of the sum.

In what follows, for any elements $\underline{\textbf{x}}=(x_{1},\dots,x_{n})$ and $\underline{\textbf{y}}=(y_{1},\dots,y_{n})$ of $\mathbb{C}^{n}$ and $\underline{\textbf{s}}=(s_{1},...,s_{n})$ denote a vector in $\mathbb{C}^{n}$ .

1 Main results

For real numbers $\underline{\alpha}=(\alpha_{1},\dots,\alpha_{n})\in\mathbb{R}^{n}$, such that, for all $1\leq i\leq n$:

[TABLE]

So, for a complex $n-$tuples $\underline{s}=(s_{1},\dots,s_{n})\in\mathbb{C}^{n}$, we define the generalized multiple Hurwitz zeta function by

[TABLE]

and the corresponding integral function associated to the generalized multiple Hurwitz zeta function by

[TABLE]

Remark 1.1**.**

We remark that:

• •

If $\underline{\alpha}=(\alpha,\alpha,\dots,\alpha)$, then the series (1.1) corresponding to the classical multiple Hurwitz zeta function.

• •

If $\underline{\alpha}=(1,1,\dots,1)$, then the series (1.1) corresponding to the multiple zeta function.

For the meromorphic continuation of the integral (1.4) and the series (1.1), we refer the reader to the work [12].

We first give well-known elementary result for the integral function.

Lemma 1.1**.**

*Let $\underline{\textbf{N}}=(N_{1},\dots,N_{n})$ be a point of $\mathbb{N}^{n}$,

1. (1)

The point $(\underline{\textbf{s}}=-\underline{\textbf{N}})$ is a polar divisor for the function $Y_{n}(\underline{\alpha};\underline{\textbf{s}})$ if and only if there exists a $\underline{k}=(k_{2},...,k_{n})\in\mathbb{N}^{n-1}$ such that

[TABLE] 2. (2)

If $(\underline{\textbf{s}}=-\underline{\textbf{N}})$ is not a polar divisor for the integral function, then the value of this function at this point exists and is given by

[TABLE]

with

[TABLE]

We give now a similar result for the generalized multiple Hurwitz zeta function.

Theorem 1**.**

Let $\underline{\textbf{N}}=(N_{1},\dots,N_{n})$ a point of $\mathbb{N}^{n}$, if the point $(\underline{\textbf{s}}=-\underline{\textbf{N}})$ is not a polar divisor for the integral function $Y_{n}(\underline{\alpha};\underline{\textbf{s}})$, then the value of the generalized multiple Hurwitz zeta function $\zeta_{n}(\underline{\alpha};\underline{\textbf{s}})$ at the point $(\underline{\textbf{s}}=-\underline{\textbf{N}})$ exists and is given by

[TABLE]

with

[TABLE]

and

[TABLE]

and

[TABLE]

where $B_{v_{j}}$ is the ${v_{j}}^{-th}$ Bernoulli number.

2 Proof of lemma 1.1

Let the integral function

[TABLE]

If we use the following change of variables:

[TABLE]

for all $1\leq i\leq n$, we find

[TABLE]

Now, using the following change of variables:

[TABLE]

for all $1\leq i\leq n$. This change gives

[TABLE]

Since $\underline{y}=(y_{1},...,y_{n})\in\prod_{i=1}^{n}[\alpha_{i},+\infty[$, this gives

[TABLE]

and, we find

[TABLE]

This integral can be rewritten as follows.

[TABLE]

with

[TABLE]

if and only if $\Re(s_{n})-1>0$.

Inductively on $n$, we find

[TABLE]

if and only if for all $1\leq i\leq n-1$

[TABLE]

and

[TABLE]

Therefore, for any point $\underline{\textbf{N}}=(N_{1},\dots,N_{n})\in\mathbb{N}^{n}$

The point $(\underline{\textbf{s}}=-\underline{\textbf{N}})$ is a polar divisor for the function $Y_{n}(\underline{\alpha};\underline{\textbf{s}})$ if there exists a $\underline{k}=(k_{2},...,k_{n})\in\mathbb{N}^{n-1}$ such that

[TABLE] 2. 2)

If $(\underline{\textbf{s}}=-\underline{\textbf{N}})$ is not a polar divisor we get

[TABLE]

if and only if there exists an $\underline{k}=(k_{2},...,k_{n})\in\mathbb{N}^{n-1}$ and $2\leq j\leq n$, such that

[TABLE]

Let

[TABLE]

which is finite, then

[TABLE]

3 An intermediate approximation

For $\underline{a}=(a_{1},...,a_{n})\in\mathbb{R}_{+}^{n}$ and $\underline{\textbf{s}}=(s_{1},...,s_{n})\in\mathbb{C}^{n}$, we define the function

[TABLE]

We prove the following useful result.

Proposition 3.1**.**

Let $\underline{\textbf{N}}=(N_{1},...,N_{n})$ a point of $\mathbb{N}^{n}$, then we have for $\underline{a}\in\mathbb{R}^{+}$

[TABLE]

with

[TABLE]

and

[TABLE]

Proof.

Let $\underline{a}\in\mathbb{R}^{n}_{+}$, such that for all $\underline{x}=(x_{1},...,x_{n})\in[\alpha,+\infty[^{n}$ and for all $1\leq i\leq n$

[TABLE]

we have

[TABLE]

This integral can be written as follows

[TABLE]

Since for $\Re(s_{n})>1$ we have

[TABLE]

condition (3.5) yields

[TABLE]

If for $1\leq j\leq n-1$

[TABLE]

then inductively we find

[TABLE]

But, for all $2\leq j\leq n$ we have

[TABLE]

and

[TABLE]

which yields

[TABLE]

with

[TABLE]

Setting $\underline{\textbf{s}}=-\underline{\textbf{N}}=-(N_{1},...,N_{n})\in\mathbb{N}^{n}$ yields (3.2) and ends the proof of Proposition 3.1. ∎

4 Proof of Theorem 1

The proof relies on the Raabe formula [7], which expresses the integral in terms of the sum.

Proposition 4.1**.**

1. (1)

Raabe formula:

for all $\underline{\textbf{s}}\in\mathbb{C}^{n}$, outside the possible polar divisors of $Y_{n}(\underline{\alpha};\underline{\textbf{s}})$, we have:

[TABLE]

where:

[TABLE]

and $d{\underline{\textbf{t}}}$ is the Lebesgue measure on $\mathbb{R}^{n}$. 2. (2)

For a fixed point $\underline{\textbf{N}}=(N_{1},...,N_{n})$ in $\mathbb{N}^{n}$ the maps $\displaystyle{\underline{\textbf{a}}\mapsto Y_{n,\underline{a}}(\underline{\alpha};-\underline{\textbf{N}})}$ and $\displaystyle{\underline{\textbf{a}}\mapsto\zeta_{n,\underline{\textbf{t}}}(\underline{\alpha};-\underline{\textbf{N}})}$ are polynomials in $\underline{\textbf{a}}=(a_{1},...,a_{n})\in\mathbb{R}_{+}^{n}$.

Proof.

1. (1)

Let $\underline{\textbf{s}}\in\mathbb{C}^{n}$ be chosen in such a way that the integral function and the generalized multiple Hurwitz zeta function are absolutely convergent.

Thus, for $\underline{\textbf{t}}\in\mathbb{R}_{+}^{n}$, we have:

[TABLE]

This last equality which is verified for all $\underline{\textbf{s}}\in\mathbb{C}^{n}$ follows by analytic continuation outside the polar divisors. 2. (2)

follows from (3.2) combined with the Raabe formula.

∎

Lemma 4.1** ([6]).**

Let $P$ and $Q$ to be two polynomials in $n$ variables linked by

[TABLE]

Write out

[TABLE]

where $h_{\underline{\textbf{L}}}\in\mathbb{C}$ and $\underline{\textbf{L}}=(L_{1},...,L_{n})\in\mathbb{N}^{n}$ ranges over a finite set of multi-index. Then

[TABLE]

*where the $B_{L_{i}}(a_{i})$ are the Bernoulli polynomials [4].

Conversely, if $Q$ is given by (4.4), then the relations (4.2) and (4.3) yield equivalent formulas for the polynomial $P$.*

Proposition 4.2**.**

If we write out the polynomial $Y_{\underline{\textbf{a}}}(\underline{\alpha};-\underline{\textbf{N}})$ as a sum of monomials,

[TABLE]

*with $\displaystyle{\underline{\textbf{a}}^{\underline{\textbf{L}}}=\prod_{i=1}^{n}{a_{i}^{L_{i}}}}$ and $C_{\underline{\textbf{L}}}=C_{\underline{\textbf{L}}}(\underline{\textbf{N}})\in\mathbb{C}$.

Then*

[TABLE]

*where $\displaystyle{B_{\underline{\textbf{L}}}=\prod_{i=1}^{n}}{B_{L_{i}}}$ is a product of Bernoulli numbers.

More generally, for $\underline{\textbf{a}}=(a_{1},...,a_{n})\in\mathbb{R}_{+}^{n}$, we have:*

[TABLE]

where $\displaystyle{B_{\underline{\textbf{L}}}(\underline{\textbf{a}})=\prod_{i=1}^{n}}{B_{L_{i}}}(a_{i})$ is a product of Bernoulli numbers.

Proof.

It follows from the above lemma, with $P(\underline{a})=Y_{n,\underline{a}}(\underline{\alpha};-\underline{\textbf{N}})$ and $\displaystyle{Q(\underline{a})=\zeta_{n,\underline{a}}(\underline{\alpha};-\underline{\textbf{N}})}$. ∎

4.1 Proof of Theorem 1:

Relation (3.2) shows that for all $\underline{a}\in\mathbb{R}_{+}^{n}$

[TABLE]

with

[TABLE]

and

[TABLE]

Setting,

[TABLE]

this gives

[TABLE]

It follows from Proposition 4.2 that

[TABLE]

with

[TABLE]

and $B_{v_{j}}$ is the ${v_{j}}^{-th}$ Bernoulli number, which ends the proof of Theorem 1.

5 Two special cases

In this part, we give two applications of this results:

5.1 Multiple Hurwitz zeta values at non positive integers

In this paragraph, we give the values of the classical multiple zeta values at non positive integers.

So, for $\underline{\alpha}:=(\alpha,\dots,\alpha)$, where $\alpha\neq 0,-1,-2,\dots,$, we find

[TABLE]

which is the multiple Hurwitz zeta function.

Thus, if we apply Theorem 1, we find the same result given in [16].

Corollary 5.1**.**

Let $\underline{\textbf{N}}=(N_{1},\dots,N_{n})$ a point of $\mathbb{N}^{n}$, if the point $(\underline{\textbf{s}}=-\underline{\textbf{N}})$ is not a polar divisor for the integral function $Y_{n}(\alpha;\underline{\textbf{s}})$, then the value of the multiple Hurwitz zeta function $\zeta_{n}(\alpha;\underline{\textbf{s}})$ at the point $(\underline{\textbf{s}}=-\underline{\textbf{N}})$ exists and is given by

[TABLE]

with

[TABLE]

and

[TABLE]

and

[TABLE]

where $B_{v_{j}}$ is the ${v_{j}}^{-th}$ Bernoulli number.

5.2 Multiple zeta values at non positive integers

Now, for $\underline{\alpha}:=(1,\dots,1)$, we find

[TABLE]

which is the multiple zeta function.

Thus, if we apply Theorem 1, we find the same result given in [14].

Corollary 5.2**.**

Let $\underline{\textbf{N}}=(N_{1},\dots,N_{n})$ a point of $\mathbb{N}^{n}$, if the point $(\underline{\textbf{s}}=-\underline{\textbf{N}})$ is not a polar divisor for the integral function $Y(\underline{\textbf{s}})$, then the value of the multiple zeta function $Z(\underline{\textbf{s}})$ at the point $(\underline{\textbf{s}}=-\underline{\textbf{N}})$ exists and is given by 222This corrects a typo in[14, Theorem 1].

[TABLE]

with

[TABLE]

[TABLE]

and

[TABLE]

where $B_{v_{j}}$ is the ${v_{j}}^{-th}$ Bernoulli number.

Acknowledgement

We thank the referees for their numerous and very helpful comments and suggestions which greatly contributed in improving the final presentation.