A Note on Kawashima Functions

TL;DR

This paper surveys Kawashima functions, highlighting their generalization of the digamma function and exploring their applications to multiple zeta values and related formulas.

Contribution

It provides a comprehensive overview of Kawashima functions and connects their properties to multiple zeta values and recent research developments.

Findings

Kawashima functions generalize the digamma function.

Various formulas for the digamma function are extended to Kawashima functions.

Connections between Kawashima functions and multiple zeta value relations are discussed.

Abstract

This note is a survey of results on the function $F_{\mathbf{k}}(z)$ introduced by G. Kawashima, and its applications to the study of multiple zeta values. We stress the viewpoint that the Kawashima function is a generalization of the digamma function $\psi(z)$, and explain how various formulas for $\psi(z)$ are generalized. We also discuss briefly the relationship of the results on the Kawashima functions with a recent work on Kawashima's MZV relation by M. Kaneko and the author.

Equations

$$\psi (z+1)=-\gamma +\sum _{n=1}^{\infty }\frac{(-1{)}^{n-1}}{n}\left(\genfrac{}{}{0ex}{}{z}{n}\right).$$

$$\psi (N+1)=-\gamma +\sum _{n=1}^N\frac{1}{n}.$$

$$\psi (z+1)=-\gamma +{\int }_0^1\frac{1-t^z}{1-t}dt.$$

$$\psi(z+1)=-\gamma+\sum_{n=1}^{\infty}\biggl{(}\frac{1}{n}-\frac{1}{n+z}\biggr{)}.$$

$$\psi (z+1)=-\gamma +\sum _{m=1}^{\infty }(-1{)}^{m-1}\zeta (m+1)z^m.$$

$$s(\mathbf{k},N)$$

$$s^{\star }(\mathbf{k},N)$$

$$S(\mathbf{k},N)$$

$$S^{\star }(\mathbf{k},N)$$

$$A(\mathbf{k})$$

$$\Delta (\mathbf{k})$$

$$s^{\star }(\mathbf{k},N)$$

$$S^{\star }(\mathbf{k},N)$$

$$\delta (j)=\{\begin{array}{ll}0& \text{if j∈/A(k)},\\ 1& \text{if j∈A(k)}.\end{array}$$

$$|\mathbf{k}|=|{\mathbf{k}}^{\vee }|,A(\mathbf{k})⨿A({\mathbf{k}}^{\vee })=\{1,2,\dots ,|\mathbf{k}|-1\}.$$

$$s^{\star }(\mathbf{k},N)$$

$$S^{\star }(\mathbf{k},N)$$

$$(1-t_k{)}^{N-1}$$

$$\frac{1-(1-t_k{)}^N}{1-(1-t_k)}$$

$$F_{\mathbf{k}}(z)=\sum _{n=1}^{\infty }(-1{)}^{n-1}{s}^{\star }({\mathbf{k}}^{\vee },n)(\genfrac{}{}{0ex}{}{z}{n}).$$

$$\rho =\{\begin{array}{ll}l+1& \text{if q≥1},\\ l& \text{if q=0}.\end{array}$$

$$F_{\mathbf{k}}(N)=S^{\star }(\mathbf{k},N).$$

$$f(N)=\sum _{n=0}^N{a}_n\left(\genfrac{}{}{0ex}{}{N}{n}\right)$$

$$F_{\mathbf{k}}(z)={\int }_{\Delta (\mathbf{k})}{\omega }_{\delta (1)}(t_1)\cdots {\omega }_{\delta (k-1)}(t_{k-1})\frac{1-t_k^z}{1-t_k}d{t}_k.$$

$$\frac{1-(1-t_k{)}^z}{1-(1-t_k)}=\sum _{n=1}^{\infty }(-t_k{)}^{n-1}(\genfrac{}{}{0ex}{}{z}{n}).\text{qed}$$

$$\psi^{(m)}(z+1)=\biggl{(}\frac{d}{dz}\biggr{)}^{m}\int_{0}^{1}\frac{1-t^{z}}{1-t}dt=-\int_{0}^{1}(\log t)^{m}\frac{t^{z}}{1-t}dt.$$

$$(\log t)^{m}=\biggl{(}-\int_{t}^{1}\frac{du}{u}\biggr{)}^{m}=(-1)^{m}m!\int_{1>u_{1}>\cdots>u_{m}>t}\frac{du_{1}}{u_{1}}\cdots\frac{du_{m}}{u_{m}},$$

$${\psi }^{(m)}(z+1)$$

$$\displaystyle=(-1)^{m}m!\bigl{(}F_{m+1}(z)-\zeta(m+1)\bigr{)}.$$

$$\zeta (m+1)={\int }_{1>u_1>\cdots >u_m>t>0}\frac{d{u}_1}{u_1}\cdots \frac{d{u}_m}{u_m}\frac{1}{1-t}dt.$$

$$F_{m+1}(z)=\frac{(-1{)}^m}{m!}{\psi }^{(m)}(z+1)+\zeta (m+1)$$

$$F_{\mathbf{k}}(z)=\sum_{n=1}^{\infty}\biggl{(}s^{\star}(\mathbf{k},n)-\frac{F_{\mathbf{k}_{-}}(n+z)}{(n+z)^{k_{r}}}\biggr{)}.$$

$$\frac{d{t}_{k^{\prime }}}{1-t_{k^{\prime }}}{\int }_{t_{k^{\prime }}}^1\frac{d{t}_{k^{\prime }+1}}{t_{k^{\prime }+1}}{\int }_0^{t_{k^{\prime }+1}}\frac{d{t}_{k^{\prime }+2}}{t_{k^{\prime }+2}}$$

$$\cdots {\int }_0^{t_{k-2}}\frac{d{t}_{k-1}}{t_{k-1}}{\int }_0^{t_{k-1}}\frac{1-t_k^z}{1-t_k}d{t}_k$$

$$=\sum _{n=1}^{\infty }\frac{d{t}_{k^{\prime }}}{1-t_{k^{\prime }}}{\int }_{t_{k^{\prime }}}^1\frac{d{t}_{k^{\prime }+1}}{t_{k^{\prime }+1}}{\int }_0^{t_{k^{\prime }+1}}\frac{d{t}_{k^{\prime }+2}}{t_{k^{\prime }+2}}$$

Full text

1. Introduction

In [2], G. Kawashima introduced a family of special functions $F_{\mathbf{k}}(z)$, where $\mathbf{k}=(k_{1},\ldots,k_{r})$ is a sequence of positive integers, and proved some remarkable properties of them. As an application, he obtained a large class of algebraic relations among the multiple zeta values (MZVs), called Kawashima’s relation. Kawashima’s relation can be used to derive some of other classes of relations (duality, Ohno’s relation, quasi-derivation relation and cyclic sum formula; see [2, 6, 7]), and is expected to imply all algebraic relations.

In this note, we survey results on these functions $F_{\mathbf{k}}(z)$, which we call the Kawashima functions, and their connections with MZVs. We stress the viewpoint that the Kawashima function is a multiple version of the digamma function. Recall that the digamma function $\psi(z)$ is defined as the logarithmic derivative of the gamma function: $\psi(z)=\frac{d}{dz}\log\Gamma(z)$. This is one of the well-studied functions in classical analysis. Here we list some formulas on $\psi(z)$ ($\gamma$ denotes the Euler-Mascheroni constant):

• •

Newton series:

[TABLE]

• •

Interpolation property: For an integer $N\geq 0$,

[TABLE]

• •

Integral representation:

[TABLE]

• •

Partial fraction series:

[TABLE]

• •

Taylor series:

[TABLE]

In §2.2, we define the Kawashima function $F_{\mathbf{k}}(z)$ by a Newton series generalizing (1.1). Then we explain how the formulas (1.2), (1.3) and (1.4) are extended to $F_{\mathbf{k}}(z)$, in §2.3, §2.4 and §2.5 respectively.

The Taylor expansion of $F_{\mathbf{k}}(z)$ at $z=0$, which generalizes (1.5), is descirbed in §3.2. In fact, there are three methods to compute the Taylor coefficients, each of which expresses the coefficients in terms of MZVs (Proposition 3.1, Proposition 3.2 and Corollary 3.5). In §3.3, we treat another important property of Kawashima functions, the harmonic relation (Theorem 3.7). Then by combining it with the Taylor series (3.3), we deduce Kawashima’s algebraic relation for MZVs (Corollary 3.8).

At the Lyon Conference, the author talked on a new proof of Kawashima’s MZV relation based on the double shuffle relation and the regularization theorem, which is a part of the work with M. Kaneko [4]. In §3.4, we briefly discuss the relationship between this proof and the results on Kawashima functions presented in §3.2 and §3.3.

Though this is basically an expository article on known results (largely due to Kawashima), it includes some results which appear in print for the first time; Proposition 2.7, Proposition 2.9 and Corollary 2.10. On the other hand, we should also note that we leave out some important works related with Kawashima functions and Kawashima’s MZV relation; particularly, their $q$-analogue studied by Takeyama [5], and the generalization of Kawashima’s relation to ‘interpolated’ MZVs by Tanaka and Wakabayashi [8]. For details, we refer the reader to their original articles.

2. Definition and Formulas of the Kawashima function

In this section, we define the Kawashima function by generalizing the Newton series in (1.1), and present generalizations of (1.2), (1.3) and (1.4).

2.1. Multiple harmonic sums

Let $\mathbf{k}=(k_{1},\ldots,k_{r})$ be an index, i.e., a sequence of positive integers of finite length $r$. We call $\lvert{\mathbf{k}}\rvert:=k_{1}+\cdots+k_{r}$ the weight of $\mathbf{k}$. We regard the sequence of length [math] as an index, the empty index denoted by $\varnothing$, though we mainly consider nonempty indices.

For a nonempty index $\mathbf{k}=(k_{1},\ldots,k_{r})$ and an integer $N\geq 0$, we put

[TABLE]

In [9], integral representations of $s^{\star}(\mathbf{k},N)$ and $S^{\star}(\mathbf{k},N)$ are given:

Theorem 2.1**.**

For a nonempty index $\mathbf{k}=(k_{1},\ldots,k_{r})$, put $k=\lvert{\mathbf{k}}\rvert$ and

[TABLE]

Then we have

[TABLE]

where $\omega_{0}(t)=\frac{dt}{t}$, $\omega_{1}(t)=\frac{dt}{1-t}$ and

[TABLE]

Proof.

The first formula (2.1) is [9, Theorem 1.2], stated in different symbols (in [9], the inverse order is adopted for the index). The second (2.2) is an immediate consequence of the first, since $\sum_{n=1}^{N}t_{k}^{n-1}=\frac{1-t_{k}^{N}}{1-t_{k}}$. ∎

As noted in [9], the integral representations (2.1) and (2.2) imply the following identities, known as Hoffman’s duality:

Theorem 2.2** ([1, 2]).**

Let $\mathbf{k}^{\vee}$ be the Hoffman dual of $\mathbf{k}$, i.e., the index characterized by

[TABLE]

Then we have

[TABLE]

Proof.

Under the change of variables $t_{i}\mapsto 1-t_{i}$, $\omega_{0}(t_{i})$ and $\omega_{1}(t_{i})$ are interchanged and $\Delta(\mathbf{k})$ maps onto $\Delta(\mathbf{k}^{\vee})$. Hence the identities follow from

[TABLE]

2.2. Newton series (definition)

Following Kawashima [2], we define the Kawashima function by a Newton series:

Definition 2.3**.**

For a nonempty index $\mathbf{k}$, we define the Kawashima function $F_{\mathbf{k}}(z)$ as

[TABLE]

As a convention, we put $F_{\varnothing}(z)=1$.

From the Newton series formula for the digamma function (1.1), we see that $F_{1}(z)=\psi(z+1)+\gamma$. Hence the Kawashima function may be viewed as a generalization of (a slight modification of) the digamma function.

With regard to the convergence of the series (2.5), Kawashima proved:

Proposition 2.4** ([2, Proposition 5.1]).**

Let $\mathbf{k}$ be a nonempty index and $\rho$ the last component of the Hoffman dual of $\mathbf{k}$. Then the Newton series $F_{\mathbf{k}}(z)$ has the abscissa of convergence $-\rho$, i.e., converges uniformly on compact sets in the half plane $\operatorname{Re}(z)>-\rho$, and diverges on $\operatorname{Re}(z)<-\rho$.

In particular, all Kawashima functions are defined and holomorphic on $\operatorname{Re}(z)>-1$. Hence, at least, it makes sense to consider the Taylor expansion at $z=0$. We present explicit results in §3.2.

Remark 2.5**.**

If we write $\mathbf{k}=(k_{1},\ldots,k_{q},\underbrace{1,\ldots,1}_{l})$, where $k_{q}>1$ or $q=0$, then $\rho$ is given by

[TABLE]

In [2, Proposition 5.1], the latter case seems to be missed.

2.3. Interpolation property

Proposition 2.6**.**

For any integer $N\geq 0$, we have

[TABLE]

Conversely, if a Newton series $f(z)=\sum_{n=0}^{\infty}a_{n}\binom{z}{n}$ satisfies $f(N)=S^{\star}(\mathbf{k},N)$ for all $N\geq 0$, then $f(z)$ coincides with $F_{\mathbf{k}}(z)$ coefficientwise (i.e., $a_{n}=(-1)^{n-1}s^{\star}(\mathbf{k}^{\vee},n)$ hold for all $n$).

Proof.

The identity (2.6) follows from (2.4). For the second assertion, note the fact that the identity

[TABLE]

determines inductively the coefficients $a_{n}$ by the values $f(N)$. ∎

This characterization of the Kawashima function by its values at non-negative integers plays an essential role in Kawashima’s proofs of the fraction series expansion (Theorem 2.12) and the harmonic relation (Theorem 3.7).

2.4. Integral representation

Proposition 2.7**.**

With the same notation as in Theorem 2.1, we have

[TABLE]

Proof.

Just as in the proof of (2.4), make the change of variables $t_{i}\mapsto 1-t_{i}$ and use the identity

[TABLE]

Example 2.8**.**

Let us describe the relation between the polygamma function \psi^{(m)}(z)=\bigl{(}\frac{d}{dz}\bigr{)}^{m}\psi(z) and the Kawashima function. For $m=0$, we already know that $F_{1}(z)=\psi^{(0)}(z+1)+\gamma$. For $m>0$, we have

[TABLE]

Since

[TABLE]

we have

[TABLE]

Here we use the integral representation (2.7) for $F_{m+1}(z)$ together with the iterated integral expression

[TABLE]

Hence we get

[TABLE]

for integers $m>0$. Note that this also holds for $m=0$ if we interpret $\zeta(1)$ as $\gamma$.

2.5. Fraction series

Here we give two generalizations of (1.4). The first is an inductive formula:

Proposition 2.9**.**

Let $\mathbf{k}=(k_{1},\ldots,k_{r})$ be a nonempty index and write $\mathbf{k}_{-}=(k_{1},\ldots,k_{r-1})$ (when $r=1$, $\mathbf{k}_{-}$ is the empty index $\varnothing$). Then we have

[TABLE]

Proof.

Put $k=\lvert{\mathbf{k}}\rvert$ and $k^{\prime}=\lvert{\mathbf{k}_{-}}\rvert$. Then the tail of the multiple integral (2.7) is written as

[TABLE]

Hence the whole integral is equal to

[TABLE]

For $\mathbf{k}=(1)$, the above formula (2.10) is the same as the formula (1.4) for the digamma function. See Example 2.13 below.

Corollary 2.10**.**

With the same notation as in Proposition 2.9, Kawashima functions satisfy the difference equation

[TABLE]

Proof.

Since both sides are analytic, we may assume that $z$ is real. From Proposition 2.9, we obtain

[TABLE]

hence the proposition follows from that

[TABLE]

Moreover, from Proposition 2.7, we see that $F_{\mathbf{k}}(z)$ is monotone increasing for $z\geq 0$. Therefore, it suffices to show that

[TABLE]

Now we have an estimate

[TABLE]

and the statement is proven by induction on $r$. ∎

Note that, from (2.11), it follows that $F_{\mathbf{k}}(z)$ is meromorphically continued to the whole complex plane.

The second generalization of (1.4), which seems more nontrivial than the first, was given by Kawashima [3]. To present it, we make some definitions.

For a nonempty index $\mathbf{k}=(k_{1},\ldots,k_{r})$, write $\overleftarrow{\mathbf{k}}=(k_{r},\ldots,k_{1})$.

Definition 2.11**.**

For integers $r>0$ and $n_{1},\ldots,n_{r}>0$, we put

[TABLE]

Then, for a nonempty index $\mathbf{k}=(k_{1},\ldots,k_{r})$ of weight $k=\lvert{\mathbf{k}}\rvert$, we define

[TABLE]

where the sum is taken over all sequences of positive integers $n_{1},\ldots,n_{k}$ satisfying

[TABLE]

(recall that $A(\mathbf{k})$ denotes the set $\{k_{1},k_{1}+k_{2},\ldots,k_{1}+\cdots+k_{r-1}\}$).

For example,

[TABLE]

By the following theorem, this is equal to $F_{1,1,2}(z)$.

Theorem 2.12** ([3, Theorem 4.4]).**

For a nonempty index $\mathbf{k}$, we have

[TABLE]

Example 2.13**.**

Let us consider an index $\mathbf{k}=(k)$ of length $1$. By (2.10) and (2.14), we have

[TABLE]

In particular, when $k=1$, both expressions coincide with the formula (1.4) for the digamma function. For $k>1$, in contrast, it seems not easy to see that these two expressions are equal.

3. Kawashima’s relation of multiple zeta values

In this section, we discuss the connections of Kawashima functions with multiple zeta values.

3.1. Notation related to multiple zeta values

A nonempty index $\mathbf{k}=(k_{1},\ldots,k_{r})$ is said admissible if $k_{r}>1$. For such $\mathbf{k}$, we define the multiple zeta value (MZV) and the multiple zeta-star value (MZSV) by

[TABLE]

We also regard the empty index $\varnothing$ as admissible, and put $\zeta(\varnothing)=\zeta^{\star}(\varnothing)=1$.

Let $\mathfrak{H}^{1}=\bigoplus_{\mathbf{k}}\mathbb{Q}\cdot\mathbf{k}$ be the $\mathbb{Q}$-vector space freely generated by all indices $\mathbf{k}$, and $\mathfrak{H}^{0}$ the subspace generated by the admissible indices. There are two $\mathbb{Q}$-bilinear products $*$ and $\mathbin{\bar{*}}$, called the harmonic products, for which $\varnothing$ is the unit element and which satisfies

[TABLE]

where $\mathbf{k}=(k_{1},\ldots,k_{r})$ and $\mathbf{l}=(l_{1},\ldots,l_{s})$ are any nonempty indices and $\mathbf{k}_{-}=(k_{1},\ldots,k_{r-1})$, $\mathbf{l}_{-}=(l_{1},\ldots,l_{s-1})$. In the following, we also need another product

[TABLE]

defined on the subspace of $\mathfrak{H}^{1}$ generated by all nonempty indices.

We extend the map $\mathbf{k}\mapsto s(\mathbf{k},z)$ to a linear map on $\mathfrak{H}^{1}$. That is, for $v=\sum_{\mathbf{k}}a_{\mathbf{k}}\cdot\mathbf{k}\in\mathfrak{H}^{1}$, we put

[TABLE]

The same rule also applies to $S(\mathbf{k},N)$, $F_{\mathbf{k}}(z)$, $\zeta(\mathbf{k})$ and so on. Then one can see that

[TABLE]

Moreover, we define a linear operator $v\mapsto v^{\star}$ on $\mathfrak{H}^{1}$ by

[TABLE]

so that $s^{\star}(v,N)=s(v^{\star},N)$, $S^{\star}(v,N)=S(v^{\star},N)$ and $\zeta^{\star}(v)=\zeta(v^{\star})$.

3.2. Taylor series

We give three ways to express the Taylor coefficients of $F_{\mathbf{k}}(z)$ at $z=0$ in terms of MZVs. The first is to substitute

[TABLE]

into the definition (2.5) of $F_{\mathbf{k}}(z)$. The result is:

Proposition 3.1** ([2, Proposition 5.2]).**

For any nonempty index $\mathbf{k}$, the Taylor expansion of $F_{\mathbf{k}}(z)$ at $z=0$ is given by

[TABLE]

The second method is to differentiate repeatedly the integral representation (2.7) as in the proof of (2.9). By this method, we obtain the following formula.

Proposition 3.2**.**

With the same notation as in Theorem 2.1, we put

[TABLE]

Then we have

[TABLE]

The third method is based on Theorem 2.12 and a computation of the derivatives of $G_{\mathbf{k}}(z)$ at $z=0$.

Definition 3.3**.**

Let $\mathbf{k}=(k_{1},\ldots,k_{r})$ be a nonempty index of weight $k=\lvert{\mathbf{k}}\rvert$. For an index $\mathbf{l}=(l_{1},\ldots,l_{k})$ of length $k$, define

[TABLE]

where the sum is taken just as in the definition of $G_{\mathbf{k}}(z)$, i.e., over all sequences of positive integers $n_{1},\ldots,n_{k}$ satisfying (2.13).

Proposition 3.4** ([3, Proposition 5.2]).**

For a nonempty index $\mathbf{k}=(k_{1},\ldots,k_{r})$ and an integer $m\geq 1$, put

[TABLE]

Then we have

[TABLE]

Corollary 3.5**.**

For a nonempty index $\mathbf{k}$, we have

[TABLE]

By comparing the above three expressions of the Taylor expansion of $F_{\mathbf{k}}(z)$, we get

[TABLE]

Since each of these expressions can be written as a sum of finitely many MZVs, this identity gives linear relations among MZVs. The relation

[TABLE]

appears in [4] with a different proof (see §3.4 below), while

[TABLE]

($\mathbf{k}^{\vee}$ is replaced by $\mathbf{k}$) is given in [3, Proposition 5.3]. Kawashima also proved the equivalence of (3.10) for $m=1$ and the duality relation.

Example 3.6**.**

Let us consider the case of $\mathbf{k}=(1)$. Then the formula (3.3) says that

[TABLE]

On the other hand, (3.4) and (3.7) give

[TABLE]

which is exactly the classical formula (1.5) in the introduction. Hence we obtain $\zeta(\underbrace{1,\ldots,1}_{m-1},2)=\zeta(m+1)$, which is a special case of the duality.

3.3. Harmonic relation

Theorem 3.7** ([2, Theorem 5.3]).**

For any indices $\mathbf{k}$ and $\mathbf{l}$, we have

[TABLE]

By substituting the Taylor expansion (3.3) into this relation (3.11), we obtain algebraic relations among MZVs.

Corollary 3.8** (Kawashima’s relation).**

For any indices $\mathbf{k}$, $\mathbf{l}$ and any integer $m\geq 1$, we have

[TABLE]

3.4. Remark on the work of Kaneko-Yamamoto

Here we use the notation of [4]. Then the integral $A_{m}(\mathbf{k})$ in Proposition 3.2 is written as

[TABLE]

This means that $A_{m}(\mathbf{k})$ is identical to the value denoted by the same symbol in [4, §6] in the case of $Z=\zeta$.

By the change of variables $t_{i}\mapsto 1-t_{i}$ (in other words, by the duality relation for MZVs), this integral transforms to

[TABLE]

Hence, by combining with (3.9), we obtain

[TABLE]

This is a special case of the “integral-series identity” [4, Theorem 4.1]. In fact, this special case is sufficient to imply the general integral-series identity, under the assumption of the double shuffle relation (see [4, Proposition 5.4]).

In the proof of [4, Theorem 6.7], Kaneko and the author went in reverse, i.e., deduced the equality (3.9) from the integral-series identity (3.14) and the duality (3.13). Then they proved the relation

[TABLE]

corresponding to Kawashima’s relation (3.12) without using Kawashima functions. Indeed, since their aim was to show the statement “the regularized double shuffle relation and the duality imply Kawashima’s relation” in an algebraic setting, transcendental object such as $F_{\mathbf{k}}(z)$ was not available.

Acknowledgments

The author would like to thank Prof. Masanobu Kaneko for valuable discussions. He also wishes to express his gratitude to the organizers of this Lyon Conference 2016 for their invitation and hospitality. This work was supported in part by JSPS KAKENHI JP26247004, JP16H06336 and JP16K13742, as well as JSPS Joint Research Project with CNRS “Zeta functions of several variables and applications,” JSPS Core-to-Core program “Foundation of a Global Research Cooperative Center in Mathematics focused on Number Theory and Geometry” and the KiPAS program 2013–2018 of the Faculty of Science and Technology at Keio University.