The $q$-multiple gamma functions of Barnes-Milnor type

TL;DR

This paper introduces a new generalization called the $q$-BM multiple gamma function, unifying and extending properties of existing Barnes-Milnor and $q$-multiple gamma functions.

Contribution

It defines the $q$-BM multiple gamma function and proves properties that connect it to existing gamma functions, advancing the theoretical understanding.

Findings

Defined the $q$-BM multiple gamma function.

Proved properties linking it to Barnes-Milnor functions.

Unified existing gamma functions under a new framework.

Abstract

The multiple gamma functions of BM (Barnes-Milnor) type and the $q$-multiple gamma functions have been studied independently. In this paper, we introduce a new generalization of the multiple gamma functions called the $q$-BM multiple gamma function including those functions and prove some properties the BM multiple gamma functions satisfy for them.

Equations

$${\Gamma }_r(w;\mathbf{\omega })=\exp \left({\frac{\partial }{\partial s}{\zeta }_r(s,w;\mathbf{\omega })|}_{s=0}\right),$$

$${\zeta }_r(s,w;\mathbf{\omega })=\sum _{\mathbf{n}\ge 0}(\mathbf{n}\cdot \mathbf{\omega }+w{)}^{-s}(\mathrm{Re}(\mathrm{s})>\mathrm{r}),$$

$${\Gamma }_r^q(w;\mathbf{\omega })=\frac{{\Gamma }_{r+1}(w;(\mathbf{\omega },{\tau }^{\prime })){\Gamma }_{r+1}(w;(\mathbf{\omega },-{\tau }^{\prime }))}{{\Gamma }_r(w;\mathbf{\omega })},$$

$${\Gamma }_{r,k}(w;\mathbf{\omega })=\exp \left({\frac{\partial }{\partial s}{\zeta }_r(s,w;\mathbf{\omega })|}_{s=-k}\right).$$

$${\int }_0^w\left(\log {\Gamma }_{r,k-1}(t;\mathbf{\omega })-\frac{1}{k}{\zeta }_r(1-k,t;\mathbf{\omega })\right)dt=\frac{1}{k}\log \frac{{\Gamma }_{r,k}(w;\mathbf{\omega })}{{\Gamma }_{r,k}(0;\mathbf{\omega })}.$$

$$\underset{\genfrac{}{}{0ex}{}{{\mu }_j\to \infty }{1\le j\le l}}{\lim }\frac{|\mathbf{\alpha }{|}_{\times }}{|\mathbf{\mu }{|}_{\times }}\log {\Gamma }_{r+l,k}\left(w;\left(\mathbf{\omega },\frac{\mathbf{\alpha }}{\mathbf{\mu }}\right)\right)$$

$$\displaystyle=\frac{(-1)^{l}k!}{(l+k)!}\biggl{(}\log\Gamma_{r,l+k}(w;{\boldsymbol{\omega}})\biggr{.}\qquad\qquad\qquad\qquad\qquad$$

$$+\left(\frac{1}{k+1}+\cdots +\frac{1}{k+l}\right){\zeta }_r(-k-l,w;\mathbf{\omega })),$$

$${\Gamma }_{r,k}^q(w;\mathbf{\omega })=\frac{{\Gamma }_{r+1,k}(w;(\mathbf{\omega },{\tau }^{\prime })){\Gamma }_{r+1,k}(w;(\mathbf{\omega },-{\tau }^{\prime }))}{{\Gamma }_{r,k}(w;\mathbf{\omega })}.$$

$${\Gamma }_{r,k}^q(w;\mathbf{\omega })=\exp \left(\frac{k!}{(\log q{)}^k}\sum _{\mathbf{n}\ge 0}{\mathrm{Li}}_{k+1}(q^{\mathbf{n}\cdot \mathbf{\omega }+w})\right).$$

$$\underset{\genfrac{}{}{0ex}{}{{\mu }_j\to \infty }{1\le j\le l}}{\lim }{\Gamma }_{r+l,k}^q{\left(w;\left(\mathbf{\omega },\frac{\mathbf{\alpha }}{\mathbf{\mu }}\right)\right)}^{|\mathbf{\alpha }{|}_{\times }/|\mathbf{\mu }{|}_{\times }}={\Gamma }_{r,l+k}^q(w;\mathbf{\omega }{)}^{\frac{(-1{)}^{l}k!}{(l+k)!}},$$

$${\zeta }_r^q(s,w;\mathbf{\omega })={\zeta }_{r+1}(s,w;(\mathbf{\omega },{\tau }^{\prime }))+{\zeta }_{r+1}(s,w;(\mathbf{\omega },-{\tau }^{\prime }))$$

$$-{\zeta }_r(s,w;\mathbf{\omega }).$$

$${\zeta }_r^q(s,w;\mathbf{\omega })$$

$$\sum _{n\in \mathbb{Z}}(n+z{)}^{-s}=\frac{(-2\pi i{)}^s}{\Gamma (s)}\sum _{n=1}^{\infty }{n}^{s-1}{e}^{2\pi inz}.$$

$${\Gamma }_{r,k}^q(w;\mathbf{\omega })$$

$$\underset{s\to -k}{\lim }\frac{d}{ds}\frac{a^s}{\Gamma (s)}=\frac{k!}{(-a{)}^k}.$$

$${\Gamma }_r^q(w;\mathbf{\omega })=\prod _{\mathbf{n}\ge 0}(1-q^{\mathbf{n}\cdot \mathbf{\omega }+w}{)}^{-1}.$$

$$\underset{l}{\underbrace{{\int }_0^1\cdots {\int }_0^1}}\log {\Gamma }_{r+l,k}^q(w+\mathbf{t}\cdot \mathbf{\alpha };(\mathbf{\omega },\mathbf{\alpha }))d\mathbf{t}$$

$$=\underset{\genfrac{}{}{0ex}{}{{\mu }_j\to \infty }{1\le j\le l}}{\lim }\frac{1}{|\mathbf{\mu }{|}_{\times }}\log {\Gamma }_{r+l,k}\left(w;\left(\mathbf{\omega },\frac{\mathbf{\alpha }}{\mathbf{\mu }}\right)\right).$$

$${\int }_0^1\cdots {\int }_0^1{\zeta }_{r+l,k}^q(w+\mathbf{t}\cdot \mathbf{\alpha };(\mathbf{\omega },\mathbf{\alpha }))d\mathbf{t}$$

$$=$$

$$\biggl{.}-n\tau^{\prime}+w+\frac{{\bf k}\cdot{\boldsymbol{\alpha}}}{{\boldsymbol{\mu}}}\biggr{)}^{-s}.$$

$${\int }_0^1\cdots {\int }_0^1{\zeta }_{r+l,k}^q(w+\mathbf{t}\cdot \mathbf{\alpha };(\mathbf{\omega },\mathbf{\alpha }))d\mathbf{t}$$

$$=$$

$$\underset{l}{\underbrace{{\int }_0^1\cdots {\int }_0^1}}\log {\Gamma }_{r+l,k}^q(w+\mathbf{t}\cdot \mathbf{\alpha };(\mathbf{\omega },\mathbf{\alpha }))d\mathbf{t}$$

$$\displaystyle=\frac{(-1)^{l}k!}{|{\boldsymbol{\alpha}}|_{\times}(l+k)!}\biggl{(}\log\Gamma^{q}_{r,l+k}(w;{\boldsymbol{\omega}})\biggr{.}\qquad\qquad\qquad$$

$$+\left(\frac{1}{k+1}+\cdots +\frac{1}{k+l}\right){\zeta }_r^q(-k-l,w;\mathbf{\omega })).$$

$${\int }_0^{{\alpha }_1}\cdots {\int }_0^{{\alpha }_l}{\zeta }_{r+l}^q(s,w+|\mathbf{t}|;(\mathbf{\omega },\mathbf{\alpha }))d\mathbf{t}=\frac{{\zeta }_r^q(s-l,w;\mathbf{\omega })}{{\prod }_{i=1}^l(s-i)}$$

$${\int }_0^{{\alpha }_l}{\zeta }_{r+l}^q(s,w+|\mathbf{t}|;(\mathbf{\omega },\mathbf{\alpha }))d{t}_l=\frac{{\zeta }_{r+l-1}^q(s-1,w+|\mathbf{t}⟨l⟩|;(\mathbf{\omega };\mathbf{\alpha }⟨l⟩))}{s-1},$$

$${\int }_0^{{\alpha }_l}{\zeta }_{r+l}^q(s,w+|\mathbf{t}|;(\mathbf{\omega },\mathbf{\alpha }))d{t}_l$$

$$=$$

Full text

1. Introduction

The $q$-gamma function has been studied since Barnes [2]. Jackson defined it as the reciprocal of the $q$-Pochhammer symbol, which satisfies the periodicity and some transformation properties like the usual gamma functions. Almost 90 years later, Kurokawa [3] introduced the $q$-multiple gamma and sine function. This generalization is also derived from the multiple gamma functions defined by Barnes [1]. In [3], some properties, such as the multiplication formula and the periodicity, were proved. A bit later, Tanaka [8] pointed out that the $q$-multiple gamma functions can be regarded as Appell’s O-function.

Let $w,\omega_{1},\omega_{2},\cdots,\omega_{r}$ be complex numbers with positive real parts. Barnes’ multiple gamma functions are defined by

[TABLE]

where $\zeta_{r}$ are the multiple Hurwitz’ zeta functions

[TABLE]

${\bf 0}=(0,\cdots,0)$, ${\bf n}=(n_{1},\cdots,n_{r})$, ${\boldsymbol{\omega}}=(\omega_{1},\cdots,\omega_{r})$, $\boldsymbol{n}\geq\boldsymbol{0}\overset{\mathrm{def}}{\Leftrightarrow}n_{i}\geq{0}\,(i=1,\cdots,r)$ and ${\bf n}\cdot{\boldsymbol{\omega}}=n_{1}\omega_{1}+\cdots+n_{r}\omega_{r}$. We use similar notation for other vectors. Kurokawa’s definition of the $q$-multiple gamma functions is

[TABLE]

where $q=\exp(-2\pi i/\tau^{\prime})$.

In 2007, Kurokawa and Ochiai [4] constructed the theory of the multiple gamma functions of BM (Barnes-Milnor) type:

[TABLE]

The case of $k=0$ is the Barnes’ multiple gamma functions $\Gamma_{r}$. The parameters $r$ and $k$ are usually called “order” and “depth” respectively. Their generalization of Kinkelin’s formula enables the function $\Gamma_{r,k}$ to bring down its depth by integration:

Theorem 1.1** ([4], Theorem 4).**

For $k\geq{1}$, we have

[TABLE]

Moreover, Kurokawa and Wakayama [5] investigated period deformations and a generalization of Raabe’s integral formula for the BM multiple gamma functions. These results can be written as

[TABLE]

[TABLE]

where $|{\boldsymbol{\omega}}|_{\times}=\prod_{i=1}^{r}\omega_{i}$. This theorem allows adjusting the order and the depth of $\Gamma^{q}_{r,k}$ by removing some parts of periods. Our purpose in this paper is finding the $q$-analogue of the above theorems and its applications. To prove them, we define the $q$-BM multiple gamma functions as follows:

[TABLE]

At the first onset, we generalize Shibukawa’s expression ([7], Corollary 4.9):

Theorem 1.2**.**

Let $\mathrm{Li}_{s}(x)=\sum_{n\geq{1}}x^{n}/n^{s}$ be the usual polylogarithm function. For $k\geq{0}$, we have

[TABLE]

Additionally, we show that the $q$-multiple gamma functions satisfies the result of Kurokawa and Ochiai more concisely than the multiple gamma functions.

Theorem 1.3**.**

We have

[TABLE]

where $\displaystyle\frac{{\boldsymbol{\alpha}}}{{\boldsymbol{\mu}}}=\left(\frac{\alpha_{1}}{\mu_{1}},\cdots,\frac{\alpha_{l}}{\mu_{l}}\right)$.

2. Product expression of $\Gamma_{r,k}^{q}$

In this section, we prove Theorem 1.2.

Proof.

Let

[TABLE]

Then we show that

[TABLE]

where we used Lipschitz’ formula

[TABLE]

Hence we get

[TABLE]

and we can easily get

[TABLE]

∎

Corollary 2.1** ([8], Theorem 1).**

We have

[TABLE]

3. Period deformation and Raabe’s formula

We prove period deformation and Raabe’s formula of $\Gamma^{q}_{r,k}$ in the similar form of [5].

Proposition 3.1** (Period deformation).**

We have

[TABLE]

[TABLE]

Proof.

We obtain

[TABLE]

[TABLE]

[TABLE]

Here, by substituting $\mu_{j}m+k_{j}=\nu_{j}$, we have

[TABLE]

[TABLE]

Thus the statement is given by the differentiation at $s=-k$. ∎

Theorem 3.2** (Raabe’s formula).**

We have

[TABLE]

Proof.

We plan to prove is similar to that of [5], Theorem 4. We only have to show

[TABLE]

when $\mathrm{Re}(s)>-r-l$ because both sides are meromorphically extendable on the whole plane $\mathbb{C}$. Moreover, it is sufficient that we get

[TABLE]

where ${\boldsymbol{\alpha}}\langle{l}\rangle$ means $(\alpha_{1},\cdots,\alpha_{l-1})$. Thus we show this. By substitution $w+|{\bf t}|\mapsto{z}$, we get

[TABLE]

[TABLE]

where we used the ladder structure of the $q$-multiple Hurwitz’ zeta functions

[TABLE]

which was proved in [8], Proposition 2.1. Therefore we obtain

[TABLE]

[TABLE]

From the above, we get the desired result inductively. ∎

4. Proof of Theorem 1.3 and corollaries

In this section, we prove Theorem 1.3 and show its applications.

Proof.

It follows that

[TABLE]

[TABLE]

since

[TABLE]

Therefore we obtain the statement. ∎

This theorem has some applications. In the following corollaries, we see two of them. The first is a proof of the transformation property (or the modular property) of Dedekind’s eta function:

[TABLE]

This can be derived from the following corollary which is found in [8].

Corollary 4.1** ([8], Theorem 2).**

We have

[TABLE]

This is the generalization of Shintani’s result [6], Proposition 2 (1).

The second is the vanishing property of $\zeta^{q}_{r}(s,w;{\boldsymbol{\omega}})$. By comparing Theorem 1.3 and Theorem 3.2, we get the following:

Corollary 4.2** ([7], Corollary 4.8 (2)).**

For $n\geq{0}$, we have

[TABLE]

Proof.

It is easy to see that

[TABLE]

∎