Analytic Continuation for Multiple Zeta Values using Symbolic Representations

TL;DR

This paper develops a symbolic method to analytically continue multiple zeta values at negative indices, extending known results and connecting to Bernoulli polynomials and Faulhaber's formula.

Contribution

It introduces a novel symbolic representation for harmonic sums at negative indices, enabling new recurrence relations, generating functions, and reinterpretations of analytic continuation.

Findings

Recovered and extended recurrence relations for harmonic sums

Connected analytic continuation to Bernoulli polynomials and Faulhaber's formula

Provided a natural renormalization perspective for multiple zeta functions

Abstract

We introduce a symbolic representation of $r$-fold harmonic sums at negative indices. This representation allows us to recover and extend some recent results by Duchamp et al., such as recurrence relations and generating functions for these sums. This approach is also applied to the study of the family of extended Bernoulli polynomials, which appear in the computation of harmonic sums at negative indices. It also allows us to reinterpret the Raabe analytic continuation of the multiple zeta function as both a constant term extension of Faulhaber's formula, and as the result of a natural renormalization procedure for Faulhaber's formula.

Equations

$$\sum _{k=1}^N{k}^n=\frac{B_{n+1}(N+1)-B_{n+1}}{n+1},n\ge 0,N\ge 1,$$

$$H_{-n_1,\dots ,-n_r}(N):=\sum _{N>i_1>\cdots >i_r>0}{i}_1^{n_1}\cdots i_r^{n_r},$$

$${\mathrm{Li}}_{-n_1,\dots ,-n_r}(z;N):=\sum _{N>i_1>\cdots >i_r>0}{i}_1^{n_1}\cdots i_r^{n_r}{z}^{i_1}.$$

$$\zeta (s_1,s_2,\dots ,s_r)=\sum _{n_1>n_2>\cdots >n_r>0}\frac{1}{n_1^{s_1}{n}_2^{s_2}\cdots n_r^{s_r}},$$

$$H_{-n_1,\dots ,-n_r}(N)=\prod _{k=1}^r{\mathcal{H}}_{1,\dots ,k}^{n_k}$$

$${\mathcal{F}}_r(w_1,\dots ,w_r;N)=\frac{{\mathcal{F}}_{r-1}(w_1,\dots ,w_{r-1}+w_r;N)-{\mathcal{F}}_{r-1}(w_1,\dots ,w_{r-1};N)}{e^{w_r}-1}$$

$${\mathcal{F}}_r(w_1,\dots ,w_r;N):=\sum _{n_1,\dots ,n_r=0}^{\infty }\frac{w_1^{n_1}\cdots w_r^{n_r}}{n_1!\cdots n_r!}{H}_{-n_1,\dots ,-n_r}(N)$$

$${\mathcal{B}}^n=B_n,$$

$$\sum _{n=0}^{\infty }\frac{B_n}{n!}{z}^n=\frac{z}{e^z-1}.$$

$$e^{z\mathcal{B}}=\frac{z}{e^z-1},$$

$$e^{z\mathcal{B}}=\sum _{n=0}^{\infty }\frac{{\mathcal{B}}^n}{n!}{z}^n=\sum _{n=0}^{\infty }\frac{B_n}{n!}{z}^n.$$

$$\sum _{n=0}^{\infty }\frac{B_n(x)}{n!}{z}^n=\frac{z}{e^z-1}{e}^{zx}$$

$$B_n(x)=(\mathcal{B}+x{)}^n,$$

$$B_n(x)=\sum _{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)B_{n-k}{x}^k=\sum _{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right){\mathcal{B}}^{n-k}{x}^k=(\mathcal{B}+x{)}^n.$$

$${\mathcal{B}}_1^n{\mathcal{B}}_2^m=B_n{B}_m.$$

$${\mathcal{U}}^n=\frac{1}{n+1}.$$

$$f(x+\mathcal{U})={\int }_0^{1}f(x+u)\mathrm{d}u.$$

$$e^{z\mathcal{U}}=\sum _{n=0}^{\infty }\frac{z^n{\mathcal{U}}^n}{n!}=\sum _{n=0}^{\infty }\frac{z^n}{(n+1)!}=\frac{e^z-1}{z}.$$

$$e^{z(\mathcal{U}+\mathcal{B})}=e^{z\mathcal{U}}{e}^{z\mathcal{B}}=\frac{e^z-1}{z}\cdot \frac{z}{e^z-1}=1,$$

$$(x+\mathcal{B}+\mathcal{U}{)}^n=x^n$$

$$P(x+\mathcal{B}+\mathcal{U})=P(x).$$

$$H_{n_1,\dots ,n_r}(N)=\sum _{N>i_1>\cdots >i_r>0}\frac{1}{i_1^{n_1}\cdots i_r^{n_r}},$$

$${\left(\mathcal{H}(N)\right)}^n=H_{-n}(N)=1^n+2^n+\cdots +(N-1{)}^n.$$

$$B_{n+1}(x+1)-B_{n+1}(x)=(n+1)x^n,$$

$${\left(\mathcal{H}(N)\right)}^n$$

$$H_{-n,-m}(N)=\sum _{k=0}^m\sum _{l=0}^{p-1-k}\sum _{q=0}^{p-k-l}\frac{B_k{B}_l}{(m+1)(p-k)}\left(\genfrac{}{}{0ex}{}{m+1}{k}\right)\left(\genfrac{}{}{0ex}{}{p-k}{l}\right)\left(\genfrac{}{}{0ex}{}{p-k-l}{q}\right)(N-1{)}^q.$$

$$H_{-n,-m}(N)$$

$$H_{-n,-m}(N)=\frac{1}{m+1}\sum _{k=0}^m\left(\genfrac{}{}{0ex}{}{m+1}{k}\right){\mathcal{B}}_2^k{\left(\mathcal{H}(N)\right)}^{p-k-1}.$$

$${\mathcal{H}}_1=\mathcal{H}(N),{\mathcal{H}}_{1,2}=\mathcal{H}\left({\mathcal{H}}_1\right),$$

$$H_{-n,-m}(N)={\mathcal{H}}_1^n(N)\cdot \frac{{\left({\mathcal{B}}_2+{\mathcal{H}}_1\right)}^{m+1}-{\mathcal{B}}_2^{m+1}}{m+1}={\mathcal{H}}_1^n\cdot {\mathcal{H}}_{1,2}^m.$$

Full text

1 Introduction

Faulhaber’s classical formula

[TABLE]

exemplifies the importance of Bernoulli polynomials $B_{n}(x)$ in various summation problems. In a recent article (Duchamp et al., 2017), Duchamp et al. proposed an extension of this formula to different types of sums such as the multiple nested sums

[TABLE]

and their polynomial version, the truncated polylogarithmic function at negative indices

[TABLE]

Their results highlight interesting combinatorial aspects of these nested sums, together with a natural generalization of the usual Bernoulli polynomials, called extended Bernoulli polynomials.

Obviously, both sums (1.1) and (1.2) are related to the multiple zeta values (MZVs)

[TABLE]

the study of which is also the ultimate aim for this work. In a previous work (Jiu et al., 2016), we exhibited a natural symbolic approach to the analytic continuation of multiple zeta values at negative integers. Since $\zeta(s_{1},s_{2},\ldots,s_{r})$ involves multiple variables, Hartog’s phenomenon indicates that there may be multiple analytic coninuations, in constrast with the one dimensional case. However, we showed that a simple symbolic computation rule allows us to express in a simple way the analytic continuation of the multiple zeta values as obtained by Sadaoui (Sadaoui, 2014, Thm. 1) using Raabe’s identity. Furthermore, the symbolic approach allowed us to discover that, surprisingly, Raabe’s analytic continuation method generates the same values for the multiple zeta function at negative integers as those obtained from the Euler-Maclaurin summation formula (Akiyama et al., 2001, eq. 6). Though these may or may not be identical analytic continuations, the fact that they agree at non-positive integer points is surprising.

In this paper, we demonstrate two more appearances of the same values for the analytic continuation of the multiple zeta function; one is as the constant term of an extension of Faulhaber’s formula to nested sums of the form (1.1), and the other is as the result of a natural renormalization procedure applied to (1.1) in the $N\to\infty$ limit. Although Hartog’s phenomenon suggests multiple possible analytic continuations, we wish to highlight the natural appearance of the Raabe–type analytic continuation (again!) as a motivation for its further study.

Encouraged by the simplification that symbolic computation may bring to the manipulation of complicated sums, in this article we first revisit the approach by Duchamp et al. in order to gain insight on the significance of the generalized Faulhaber formula and the extended Bernoulli polynomials that naturally emerge from it. Minh (Minh, 2003) has obtained results which are similar to ours, though non-symbolic. One of the major results of the present study is the simple product representation of the multiple nested sums

[TABLE]

from Theorem 1, where the symbols $\mathcal{H}_{1,\ldots,k}^{n_{k}}$ are defined below. This simple product representation allows us in turn to prove several consequences, such as that in Theorem 4 the recurrence

[TABLE]

satisfied by the generating function for the harmonic sums

[TABLE]

and other relations and recurrences for nested harmonic sums. Moreover, these symbolic representations allow us to handle complicated multiple sums related to the analytic continuation of the multiple zeta function.

2 Symbols

2.1 The $\mathcal{B}$ and $\mathcal{U}$ symbols

In what follows, we will frequently use the Bernoulli symbol $\mathcal{B}$ with the evaluation rule

[TABLE]

where $B_{n}$ is the $n$-th Bernoulli number, defined by the generating function

[TABLE]

Given this definition, we have

[TABLE]

since

[TABLE]

Moreover, the Bernoulli polynomials $B_{n}(x)$ with generating function

[TABLE]

then have the simple symbolic expression

[TABLE]

since

[TABLE]

Additionally, two symbols $\mathcal{B}_{1}$ and $\mathcal{B}_{2}$ are called *independent *if they satisfy, $\forall n,m\in\mathbb{N}$,

[TABLE]

Another useful symbol is the uniform symbol $\mathcal{U}$, with the evaluation rule

[TABLE]

This is equivalent to the operational action

[TABLE]

We deduce, for example,

[TABLE]

Finally, from the identity

[TABLE]

we deduce that both symbols $\mathcal{B}$ and $\mathcal{U}$ annihilate each other, in the sense

[TABLE]

for an arbitrary positive integer $n$, an identity that extends by linearity to any polynomial $P$:

[TABLE]

Our main objects of study, the multiple harmonic numbers, are defined as the truncated multiple zeta value

[TABLE]

and we focus here on their values at negative indices, i.e., the multiple power sums $H_{-n_{1},\ldots,-n_{r}}(N)$, defined in (1.1). In particular, we shall show that these multiple power sums can be expressed in two equivalent ways using two new symbols which are described next.

2.2 The $\mathcal{H}$ symbol

For the single harmonic sum, define, for $N\in\mathbb{N}$, the symbol $\mathcal{H}$ by

[TABLE]

Since the Bernoulli polynomials satisfy the identity

[TABLE]

we have

[TABLE]

which extends naturally to the case $\left(\mathcal{H}(x)\right)^{n}$ for $x\not\in\mathbb{N}$, as the famous Faulhaber formula.

An extension to bivariate power sums can be found in (Duchamp et al., 2017, Thm. 1): with $p=n+m+2$,

[TABLE]

This complicated triple-sum formula suggests the introduction of symbolic computation as follows: considering two independent Bernoulli symbols $\mathcal{B}_{1}$ and $\mathcal{B}_{2}$ and summing over $q$, this identity can be simplified as

[TABLE]

Therefore, using Faulhaber’s formula (2.2), we obtain

[TABLE]

We further deduce, by denoting

[TABLE]

that

[TABLE]

The general case is given in Theorem 1 below.

2.3 The $\mathcal{V}$ symbol

In order to obtain another symbolic expression, we first consider the double sum case case, i.e.,

[TABLE]

Introduce the “uniform over $[0,z]$” symbol $\mathcal{V}(z)$ as follows: for any polynomial $P$,

[TABLE]

so that

[TABLE]

where $Q$ is an antiderivative of $P$. By Faulhaber’s formula (2.2), the inner sum (to which we have added the null term corresponding to $i=0$) in $H_{-n,-m}(N)$ reads

[TABLE]

Thus,

[TABLE]

Using the binomial formula yields

[TABLE]

Choosing $P(x)=x^{k}$, $z=j$ in (2.6) and then evaluating at $x=0$, we obtain

[TABLE]

so that

[TABLE]

Therefore, substitution yields

[TABLE]

where we have used the identity

[TABLE]

Reindexing the sum ($l=k+1$) gives

[TABLE]

so that, defining the nested symbols

[TABLE]

we obtain

[TABLE]

The general case is provided in Theorem 1 below.

2.4 General polynomial version

The former results extend naturally to polynomials as follows: given $P$ and $Q$ two polynomials without constant terms

[TABLE]

a polynomial version of identity (2.8) reads

[TABLE]

or equivalently

[TABLE]

3 Multiple power sums

3.1 Symbolic expression of multiple power sums

Theorem 1**.**

The $r$-fold multiple power sums (1.1) can be expressed as

[TABLE]

where $\mathcal{H}_{1}=\mathcal{H}(N)$ and recursively $\mathcal{H}_{1,\ldots k}=\mathcal{H}\left(\mathcal{H}_{1,\ldots,k-1}\right)$, for $k=2,3,\ldots,r$. Equivalently, defining

[TABLE]

produces

[TABLE]

These identities extend to the case of $r$ polynomials $P_{1},\dots,P_{r}$ without constant terms as follows

[TABLE]

Remark*.*

Both identities (3.1) and (3.3) still hold when $N$ is a real number. 2. 2.

Comparing (3.1) and (3.3) shows that the symbols $\mathcal{H}_{1,\ldots,k}$ and $\mathcal{V}_{1,\ldots,k}$ are related as

[TABLE] 3. 3.

The computation rules for both symbols $\mathcal{H}_{1,\ldots,k}$ and $\mathcal{V}_{1,\ldots,k}$ are reminiscent of the chain rule for differentiation

[TABLE] 4. 4.

This method also applies to the sums

[TABLE]

and yields

[TABLE]

where the symbol $\bar{\mathcal{H}}$ is now defined by $\bar{\mathcal{H}}_{1}=\mathcal{H}(N+1)$ and recursively $\bar{\mathcal{H}}_{1,\ldots k}=\mathcal{H}\left(\bar{\mathcal{H}}_{1,\ldots,k-1}+1\right)$, for $k=2,3,\ldots,r$.

Proof.

It suffices to show (3.3) by induction, since (3.1) is similar. From the definition (1.1), the recurrence

[TABLE]

holds. Note that when $N<r$, $H_{-n_{1},\dots,-n_{r}}(N)=0$, since the indices cannot satisfy the relation

[TABLE]

so that the sum is empty. Therefore, we can further extend the summation range of $i_{1}$ to $1\leq i_{1}\leq N-1$. Also, the identity (3.3) can be expressed, by shifting the indices, as

[TABLE]

where

[TABLE]

In order to evaluate $H_{-n_{1},\dots,-n_{r}}(N)$, we expand the nested $\mathcal{V}$ symbols to obtain, using (2.7), a polynomial in the variable $\mathcal{B}_{2}+\mathcal{V}(i_{1})$ with coefficients $a_{k}$ that depend on the remaining symbols $\mathcal{B}_{3},\ldots,\mathcal{B}_{r}$, as

[TABLE]

Thus,

[TABLE]

as desired. ∎

3.2 Polylogarithmic Function

Duchamp et al. (Duchamp et al., 2017) also considered the truncated polylogarithmic function

[TABLE]

and its values at negative indices as expressed by (1.2). An analog of our previous theorem is now derived for such truncated multiple polylogarithms. We first recall the definition of the Apostol-Bernoulli polynomials $B_{n}\left(x|\lambda\right)$ (Apostol, 1951), with generating function

[TABLE]

These polynomials satisfy an extended version of Faulhaber’s identity, namely

[TABLE]

Notice that in the case $\lambda=1$, this is the usual Faulhaber identity, and the Apostol Bernoulli polynomials reduce to the Bernoulli polynomials. Let us introduce the symbol $\mathcal{A}_{\lambda}$ defined by

[TABLE]

Choosing $x=0,$ we deduce

[TABLE]

while in the general case

[TABLE]

The main justification for introducing this symbol is the following identity

[TABLE]

This identity produces the following symbolic expression for the truncated multiple polylogarithms.

Theorem 2**.**

Let $\bar{\mathcal{V}}_{1}:=\mathcal{V}(N)$, $\bar{\mathcal{V}}_{1,2}=\mathcal{V}\left(\mathcal{A}_{z}+\bar{\mathcal{V}}_{1}\right)$, and $\mathcal{\bar{V}}_{1,\ldots,k}=\mathcal{V}\left(\mathcal{B}_{k}+\mathcal{\bar{V}}_{1,\ldots,k-1}\right)$ for $k=3,4,\ldots,r$. Then, we have

[TABLE]

Note that this expression coincides with (3.3) (3.3), except for the replacement of $\mathcal{B}_{1}$ by $\mathcal{A}_{z}$.

Proof.

By (3.3),

[TABLE]

Using (3.6), we have

[TABLE]

The proof is completed by further simplification, in the same way as in the proof of Theorem 1. ∎

Remark*.*

For the multivariate version of these sums, a similar calculation produces

[TABLE]

where $\tilde{\mathcal{V}}_{1}=\mathcal{V}(N)$ and recursively $\tilde{\mathcal{V}}_{1,\ldots,k+1}=\mathcal{V}\left(\mathcal{A}_{z_{k}}+\tilde{\mathcal{V}}_{1,\ldots,k}\right)$, for $k=2,3,\ldots,r$. Notice that when $z_{i}=1$, $\mathcal{A}_{z_{i}}=\mathcal{B}_{k}$ and identities (3.7) and (3.3) coincide.

3.3 Recurrence

The symbolic representation (3.1) is now exploited to deduce the following recurrence identity on harmonic sums.

Theorem 3**.**

The $r$-fold multiple power sums satisfy the recurrence

[TABLE]

Remark*.*

This identity can be seen as a multivariate generalization of identity (2.5), and is different from identity (3.4).

Proof.

A straightforward computation produces

[TABLE]

as desired. ∎

3.4 Generating function

Theorem 4**.**

The generating function of the $r$-fold harmonic sums, defined as

[TABLE]

satisfies, for $n\geq 2,$ the recurrence

[TABLE]

with the initial value

[TABLE]

Proof.

Starting from

[TABLE]

we expand the last factor, using the integration rule (2.6) and the recurrence (3.2), to obtain

[TABLE]

where, from (2.1),

[TABLE]

Further simplification completes the proof, and computation of the initial value $\mathcal{F}_{1}$ is elementary. ∎

4 Extended Bernoulli polynomials

4.1 Definitions

Duchamp et al. (Duchamp et al., 2017) expressed the multiple power sums in terms of extended Bernoulli polynomials with multiple indices $B_{n_{1},\dots,n_{r}}(z)$ as follows

[TABLE]

where $b^{\prime}_{n_{1},\dots,n_{r}}$ is a sequence of numbers defined recursively (see (Duchamp et al., 2017, Definition 1)) and the extended Bernoulli polynomials $B_{n_{1},\ldots,n_{r}}\left(z\right)$ are defined recursively as follows.

Definition 5**.**

For $z\in\mathbb{C}$,

[TABLE]

where the simple index polynomial $B_{n}\left(z\right)$ coincides with the usual Bernoulli polynomial of degree $n$. It appears that the recursive rule (4.1) does not allow to determine the value $B_{n_{1},\ldots,n_{r}}(0);$ however, this value is not needed anywhere in the forthcoming results. Hence we define the shifted extended Bernoulli polynomials (without constant term) as

[TABLE]

and notice that they satisfy the same recurrence as the extended Bernoulli polynomials, namely

[TABLE]

In the next section, an explicit symbolic expression for these polynomials is derived.

4.2 Symbolic expression

We use the following result that can be found as Lemma 3 in (Duchamp et al., 2017).

Lemma 6**.**

Consider the difference equation

[TABLE]

where $P$ is a polynomial and $f:\mathbb{N}\rightarrow\mathbb{R}$ is an unknown function. Let $P\left(x\right)=\sum_{j=0}^{d}a_{j}{x\choose j}$ be the decomposition of $P$ in the polynomial basis $\left\{{x\choose j}\right\}_{j\geq 0}$. Then the unique solution $f$ without constant term of equation (4.3) is $f_{0}\left(x\right)=\sum_{j=0}^{d}a_{j}{x\choose j+1}$.

Before stating the general case, we provide the explicit symbolic computation of the single-index and double-index shifted extended Bernoulli polynomials.

Example 7**.**

For a single index, since $B_{n}(z)=\left(\mathcal{B}+z\right)^{n}$ is the usual Bernoulli polynomial, identity (2.7) shows that

[TABLE]

For the double-index case, from

[TABLE]

we express the polynomial on the right-hand side as

[TABLE]

The Stirling numbers of the second kind $\genfrac{\{}{\}}{0.0pt}{}{n}{k}$ satisfy

[TABLE]

which implies

[TABLE]

and therefore, using Lemma 6,

[TABLE]

Now since

[TABLE]

we use the identity (Hansen, 1975, Entry 52.2.33)

[TABLE]

to obtain

[TABLE]

and

[TABLE]

Then, we deduce

[TABLE]

The general case is given next.

Theorem 8**.**

For $n_{1},\dots,n_{r}>0,$ the shifted extended Bernoulli polynomials are expressed symbolically as the product

[TABLE]

Comparing with (3.3), we deduce the link

[TABLE]

between shifted extended Bernoulli polynomials and $r$-fold multiple power sums.

Proof.

It suffices to show that the right hand side of (4.4) satisfies the recurrence (4.2). We start with the straightforward identity for the derivative of the $\mathcal{V}$ symbol

[TABLE]

which can be described as the “differentiation replaces $\mathcal{V}(z)$ by $z$” rule.

In order to compute

[TABLE]

similarly to the case of multiple power sums, we expand

[TABLE]

a polynomial in $\mathcal{B}_{1}+\mathcal{V}(z)$ with coefficients $a_{k}$ that depend on the symbols $\mathcal{B}_{2},\ldots,\mathcal{B}_{r}$. Thus, using (4.5), we obtain the derivative

[TABLE]

from which, replacing $z$ by $z+\mathcal{U}$ and applying the cancellation property (2.3), we deduce, by using (4.6) and (3.5),

[TABLE]

which produces the desired recurrence

[TABLE]

Remark*.*

Recurrence (4.1) does not determine the constant term for all extended Bernoulli polynomials except for the single-indexed ones, which coincide with the usual Bernoulli polynomials. Therefore, if we alternatively define

[TABLE]

a similar proof produces

[TABLE]

4.3 Generating function

Example 9**.**

Let us first compute the generating function of the extended Bernoulli polynomials with two indices $\beta_{n_{1},n_{2}}(z)$ as follows: with

[TABLE]

define the generating function

[TABLE]

Then

[TABLE]

where $\mathcal{G}_{1}\left(w_{1};z\right)$ is the generating function of the usual shifted Bernoulli polynomials $\beta_{n}(z)=B_{n}(z)-B_{n}(0),$ which is equal to

[TABLE]

The general case is as follows.

Theorem 10**.**

The generating function of the shifted extended Bernoulli polynomials

[TABLE]

satisfies the recurrence

[TABLE]

with initial condition given by (4.7).

Proof.

Using (4.4), we have

[TABLE]

where the last factor is expanded as

[TABLE]

so that

[TABLE]

which is precisely

[TABLE]

This completes the proof. ∎

4.4 Connection among the symbols $\beta$, $\mathcal{B}$ and $\mathcal{H}$

The next result provides a connection formula between the symbols $\beta$, $\mathcal{B}$ and $\mathcal{H}$.

Theorem 11**.**

The symbols $\beta$, $\mathcal{B}$ and $\mathcal{H}$ are related by

[TABLE]

Proof.

From the recurrence (4.1)

[TABLE]

we deduce, summing over $k$ from [math] to $N-1,$

[TABLE]

Using the summation by parts formula

[TABLE]

with

[TABLE]

we deduce

[TABLE]

Assume

[TABLE]

so that

[TABLE]

which completes the proof. ∎

Remark*.*

Since $\beta_{n_{1},\ldots,n_{r}}(z)$ is a polynomial in $z$, identity (4.8) extends to the case of an arbitrary real number $z\in\mathbb{R}$ as

[TABLE]

Example 12**.**

The special case of double-index shifted extended Bernoulli polynomials with $n=2$ reads

[TABLE]

Since

[TABLE]

we deduce

[TABLE]

so that

[TABLE]

5 Analytic Continuation

Having obtained explicit or recurrence expressions for the finite case $N\in\mathbb{N}$, we shall study the limiting behavior of the harmonic sum

[TABLE]

as $N\to\infty$. The series is obviously divergent in this limit, but deducing its value is equivalent to evaluating the analytic continuation of the multiple zeta function at negative integers. We introduce two natural methods to assign this divergent series a value: a natural renormalization procedure, and extension to a constant term, and show that they coincide. In fact, we will then show that they both yield Raabe’s formula (Jiu et al., 2018; Sadaoui, 2014). For what follows, we will extensively use the shorthand

[TABLE]

Then, given the multiple zeta function

[TABLE]

its analytic continuation is evaluated in (Jiu et al., 2018, eq. 13) (with typos in the indices of the $B_{k_{j}}$ corrected, and the prefactor of $1/(n_{r}+1)$ correctly omitted), as the $(r-1)$ fold sum

[TABLE]

5.1 Renormalization

For what follows, notice that, for $k\geq 2,$ the $\mathcal{H}$ symbols satisfy the evaluation rule

[TABLE]

For $k=1$, they are evaluated as

[TABLE]

Theorem 13**.**

Define the following renormalization rules for the symbol $\mathcal{H}_{1,\ldots,k}(\infty)$:

for $k\geq 2$, define recursively

[TABLE]

• 2.

for $k=1$, define

[TABLE]

Then

[TABLE]

which, up to sign, is the value of $\zeta(-n_{1},\ldots,-n_{r})$ given by Raabe’s analytic continuation in (Jiu et al., 2018).

Proof.

Given independent Bernoulli symbols $\{\mathcal{B}_{1},\mathcal{B}_{2},\ldots,\mathcal{B}_{k}\}$, define the $\mathcal{C}$ symbols through the recursive rule

[TABLE]

The main result of (Jiu et al., 2018, Thm. 1) was the symbolic expression

[TABLE]

On the other hand, the renormalization procedure above indicates that the action of the $\mathcal{H}$ and $\mathcal{C}$ symbols exactly coincide; in fact, we have the equality

[TABLE]

which completes the proof. ∎

Heuristically, at depth $k=1$ we discard the divergent contribution from the $N$ term and keep the constant term, while at depth $k\geq 2$ we discard the constant term. This is yet another appearance of Raabe’s analytic continuation, from a natural renormalization procedure.

5.2 Constant Term Interpretation

We begin by studying the depth one case. There, Faulhaber’s formula reads

[TABLE]

Note that there is no constant term, as expected, since $H_{n}^{-}(0)=0$ trivially. However, if such a constant term were to exist, we could evaluate it by putting $k=n+1$ in the summand, giving

[TABLE]

We now explicitly describe the action of the $\mathcal{H}$ symbols, noting that this is a case study in how useful symbols are, since they enable the easy manipulation of multiple sums as those that follow.

Theorem 14**.**

We have the explicit expansion, as a polynomial in $N$,

[TABLE]

Proof.

We prove by induction on the depth $r$. For $r=1$, this is precisely Faulhaber’s formula. Now assume that the formula holds for $r$, and use the inductive hypothesis to write

[TABLE]

We then multiply together both occurrences of $(\mathcal{B}_{1}+\mathcal{V}_{1})$ and expand

[TABLE]

with $n=\sum_{i=1}^{r+1}n_{i}+r-\sum_{i=2}^{r+1}k_{i}$. This provides the innermost $k_{1}$ summation and the $j=1$ term in the product, completing the inductive proof. ∎

Note that there is no constant term in (5.2), since $H_{n_{1},\ldots,n_{r}}(0)=0$.

Theorem 15**.**

If we instead sum over all tuples $(k_{r},k_{r-1},\ldots,k_{1})$ which would give a constant coefficient in (5.2), we obtain Raabe’s analytic continuation $(-1)^{\tilde{n}}\zeta(-n_{1},\ldots,-n_{r})$.

Proof.

This constant term condition over the summation set translates into the equivalent condition $\sum_{i=1}^{r}n_{i}+r=\sum_{i=1}^{r}k_{i}$, giving the sum

[TABLE]

Since $\sum_{i=1}^{r}n_{i}+r=\sum_{i=1}^{r}k_{i}$, the binomial coefficient at $j=1$ reduces to $\binom{k_{1}}{k_{1}}=1$. We can also omit the summation over $k_{1}$ by writing $k_{1}=\sum_{i=1}^{r}n_{i}+r-\sum_{i=2}^{r}k_{i}$, yielding

[TABLE]

By comparing with the explicit expansion (5.1), we see this is exactly Raabe’s continuation, which completes the proof. ∎

Example 16**.**

For depth two, the extended Faulhaber formula reads

[TABLE]

Again we want to extend this to a constant term, so set $n+m+2-k-j=0$ on the summand, giving

[TABLE]

Hence the “constant term” is

[TABLE]

Now Raabe’s analytic continuation for depth $2$ reads

[TABLE]

as desired.

6 Acknowledgment

The first author was partially supported by the Austrian Science Fund (FWF) grant SFB F50 (F5006-N15 and F5009-N15)