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Generating function of multiple polylog of Hurwitz type

Published online by Cambridge University Press:  21 November 2022

Kentaro Ihara*
Affiliation:
Faculty of Science and Engineering, Kindai University, 3-4-1 Kowakae, Higashi-Osaka, Japan
Yusuke Kusunoki
Affiliation:
EX Corporation, 3-19-3 Toyosaki, Osaka, Japan e-mail: [email protected]
Yayoi Nakamura
Affiliation:
Faculty of Science and Engineering, Kindai University, 3-4-1 Kowakae, Higashi-Osaka, Japan e-mail: [email protected]
Hitomi Saeki
Affiliation:
Uenomiya Taishi Senior High School, Minamikawachi, Osaka, Japan e-mail: [email protected]
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Abstract

We introduce interpolated multiple Hurwitz polylogs and interpolated multiple Hurwitz zeta values. In addition, we discuss the generating functions for the sum of the polylogs/zeta values of fixed weight, depth, and all heights. The functions are expressed in terms of generalized hypergeometric functions. Compared with the pioneering results of Ohno and Zagier on the generating function, our setup generalizes the results in three directions, namely, at general heights, with a t-interpolation, and as a Hurwitz type. As an application, by fixing the Hurwitz parameter to rational numbers, the generating functions for multiple zeta values with level are given.

Type
Article
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

1 Introduction

Let ${\mathbb {N}}$ be a set of positive integers. For an index ${\boldsymbol k}=(k_1, \ldots , k_d)\in {\mathbb {N}}^d$ with $d\ge 1$ , the multiple polylog (MP) is a holomorphic function defined as

$$ \begin{align*} \mathrm{Li}_{{\boldsymbol k}}(z)=\sum_{m_{1}>\cdots>m_d>0} \frac{z^{m_1}}{{m_{1}}^{k_{1}}\cdots {m_d}^{k_d}} \qquad (|z|<1). \end{align*} $$

The series converges even at $|z|=1$ for an index ${\boldsymbol k}$ satisfying $k_1>1$ . Such index is called admissible. In particular, the special value of $\mathrm {Li}_{{\boldsymbol k}}(z)$ at $z=1$ for an admissible index ${\boldsymbol k}$ coincides with the multiple zeta value (MZV) defined by

$$ \begin{align*} \zeta({\boldsymbol k})=\sum_{m_{1}>\cdots > m_d>0} \frac{1}{{m_{1}}^{k_{1}}\cdots {m_d}^{k_d}}. \end{align*} $$

The weight, depth, and height of ${{\boldsymbol k}}$ are defined by $\mathrm {wt}({\boldsymbol k})=\sum _{i=1}^d k_{i}$ , $\mathrm {dep}({\boldsymbol k})=d$ , and $\mathrm {ht}({\boldsymbol k})=\sharp \{i\mid k_i\geq 2\}$ , respectively. In 2001, Ohno and Zagier [Reference Ohno and Zagier17] defined the generating function of the sum of MPs (or the sum of MZVs) of a fixed weight, depth, and height, and then proved that the generating function is expressed as a Gaussian hypergeometric function, ${}_2F_1$ . In particular, the Ohno–Zagier formula implies that the sum of the MZVs of a fixed weight, depth, and height is a polynomial in terms of the Riemann zeta values.

In 2008, Li [Reference Li11] generalized the result due to Ohno–Zagier by extending the concept height to i-heights. The author described the generating function of the sum of MPs (or the sum of MZVs) of fixed weight, depth, and all i-heights in terms of the generalized hypergeometric function, ${}_{r+1}F_r$ .

Several studies have extended the formulas developed by Ohno–Zagier and/or by Li for variants of the MZVs (or MPs), such as for multiple zeta star-values, for the q-analog of multiple zeta (star) values, as well as for interpolated MZVs. See [Reference Aoki, Kombu and Ohno2Reference Bradley5, Reference Li12Reference Li and Wakabayashi14, Reference Ohno and Okuda16, Reference Okuda and Takeyama18Reference Wakabayashi20, Reference Zhao22] for further details. The latest study by Li and Wakabayashi [Reference Li and Wakabayashi14] achieved the most extended result, providing a formula for the interpolated multiple q-zeta values for general i-heights.

In our paper, we extend the formula in a new direction. The Hurwitz zeta function

$$\begin{align*}\zeta(s; a)=\sum_{m\ge 0}\frac{1}{(m+a)^s},\qquad \Re(s)>1, \end{align*}$$

generalizes the Riemann zeta function and plays an important role in the number theory. Several authors have introduced multiple versions of Hurwitz zeta functions/values in slightly different ways and have studied various properties. See [Reference Kamano8, Reference Matsumoto15] for examples. In this paper, we introduce interpolated multiple Hurwitz polylogs (MHPs) and interpolated multiple zeta values (MHZs) and demonstrate a Ohno–Zagier type formula for general i-heights as the main result (Theorems 3.5 and 3.6). Consequently, the generating function of the Hurwitz zeta values of height one is expressed explicitly in terms of the generalized hypergeometric function, ${}_{3}F_2$ (Proposition 4.1). Furthermore, the generating function of the Hurwitz zeta values with indices in the form of $ {\boldsymbol k} = (n, n,\ldots , n)$ is expressed as ${}_{n+1}F_n$ (Theorem 4.3). Both of these results directly generalize the cases of typical MZVs. As one of the benefits describing the generating functions in terms of hypergeometric functions, a symmetry in the generating functions might be obtained through transformation formulas of the hypergeometric functions.

As an application, in Section 4, we discuss the MZVs with levels. Some authors have studied the properties of MZVs of level two in several different contexts (e.g., [Reference Hoffman7, Reference Kaneko and Tsumura9, Reference Kaneko and Tsumura10]). By specifying the Hurwitz parameter, we obtain the generating functions of MZVs with all levels.

In Section 2, several basic analytic properties, such as the convergence and differential formulas of (interpolated) MHPs/MHZs, are discussed. In Section 3, the generating function of the sum of (interpolated) MHPs is introduced, and the differential equation of the function is discussed. In Section 4, we present examples of the main theorem and its applications for MZVs with levels.

The results in this paper were obtained through discussions conducted at a master’s seminar held at Kindai University in 2018, for which all authors participated. An extension to the interpolated MHZs was subsequently obtained in 2020.

2 Multiple Hurwitz polylogs

Several authors have introduced multiple Hurwitz zeta values (MHZs) in specific contexts (e.g., [Reference Kamano8, Reference Matsumoto15]). In this section, we define a MHP, which generalizes both MPs (in one variable) and MHZs. Furthermore, we introduce an interpolated variant. Analytical properties, such as convergence and differential equations satisfied by interpolated MHPs, are then discussed.

Let ${\boldsymbol k}=(k_1,\dots ,k_d)\in {\mathbb {N}}^d$ be an index for $d\ge 1$ and $a\in {{\mathbb {C}}}$ with $a\notin {{\mathbb {Z}}}_{\le 0}$ . The multiple Hurwitz zeta (MHZ) value $\zeta ({\boldsymbol k};a)$ for the admissible ${\boldsymbol k}$ discussed in this paper is defined by the series

(2.1) $$ \begin{align} \zeta({\boldsymbol k};a)=\sum_{m_{1}>\cdots>m_d\ge 0} \dfrac{1}{(m_{1}+a)^{k_{1}}\cdots (m_d+a)^{k_d}}, \end{align} $$

and the power series $\widetilde {\mathrm {Li}}_{{\boldsymbol k}}(z ;a)$ is defined as

(2.2) $$ \begin{align} \widetilde{\mathrm{Li}}_{{\boldsymbol k}}(z ;a)=\sum_{m_{1}>\cdots>m_d\ge0} \dfrac{z^{m_{1}}}{(m_{1}+a)^{k_{1}}\cdots (m_d+a)^{k_d}}. \end{align} $$

Because $a\notin {\mathbb {Z}}_{\le 0}$ , the denominators in the RHSs of equations (2.1) and (2.2) never vanish.

Lemma 2.1 Let $ a \in {{\mathbb {C}}}$ with $ a \notin {{\mathbb {Z}}}_{\le 0}$ .

  1. (i) The series $\widetilde {\mathrm {Li}}_{{\boldsymbol k}} (z ;a)$ converges absolutely in $|z|<1$ for any index ${\boldsymbol k}$ , and in $|z|\le 1$ for an admissible index ${\boldsymbol k}$ . Consequently, $\widetilde {\mathrm {Li}}_{{\boldsymbol k}} (z ;a)$ is a holomorphic function in $|z|< 1$ for any index ${\boldsymbol k}$ .

  2. (ii) $\zeta ({\boldsymbol k};a)$ converges absolutely for an admissible index ${\boldsymbol k}$ .

Proof (i) Let $|z|\le 1$ . Applying $M=\min \{m\in {\mathbb {Z}}_{\ge 0}\mid {\Re }(m+a)\geq 1\},$ we have

(2.3) $$ \begin{align} &\left|\widetilde{\mathrm{Li}}_{{\boldsymbol k}}(z ;a)\right| \leq \sum_{m_{1}>\cdots>m_d\ge 0}\dfrac{|z|^{m_1}}{|m_{1}+a|^{k_{1}}\cdots|m_d+a|^{k_d}}\notag\\ &=\sum_{i=0}^d \sum_{m_{1}>\cdots>m_i\ge M>m_{i+1}>\cdots>m_d\ge 0}\dfrac{|z|^{m_1}}{|m_{1}+a|^{k_{1}}\cdots|m_d+a|^{k_d}}\notag\\ &=\sum_{i=0}^d \Big(\sum_{m_{1}>\cdots>m_i\ge M}\dfrac{|z|^{m_1}}{|m_{1}+a|^{k_{1}}\cdots|m_i+a|^{k_i}}\notag \\ &\qquad\quad \sum_{M>m_{i+1}>\cdots>m_d\ge 0}\dfrac{1}{|m_{i+1}+a|^{k_{i+1}}\cdots|m_d+a|^{k_d}}\Big)\notag\\ &\le \sum_{i=0}^d \Big(\sum_{m_{1}>\cdots>m_i\ge M}\dfrac{|z|^{m_1}}{|m_{1}+a|^{k_{1}}\cdots|m_i+a|^{k_i}}\cdot C_i \Big) \end{align} $$

for certain positive constants $C_i \ (0\le i\le d)$ (where $C_d$ can be 1). Because

$$ \begin{align*} |m+a|\ge {\Re }(m+a)\ge m-M+1, \end{align*} $$

for $m\ge M$ based on the definition of M, the following holds:

(2.4) $$ \begin{align} (3) &\le \sum_{i=0}^d \Big(\sum_{m_{1}>\cdots>m_i\ge M}\dfrac{|z|^{m_1}}{(m_1-M+1)^{k_{1}}\cdots (m_i-M+1)^{k_i}}\cdot C_i \Big)\notag\\ &\le \sum_{i=0}^d \Big(\sum_{m_{1}>\cdots>m_i>0}\dfrac{|z|^{m_1+M-1}}{m_{1}^{k_{1}}\cdots m_i^{k_i}}\cdot C_i\Big)\notag\\ &=|z|^{M-1}\sum_{i=0}^d C_i {\mathrm{Li}}_{k_1, \ldots, k_i}(|z|). \end{align} $$

It is well known that the usual MP ${\mathrm {Li}}_{{\boldsymbol k}}(z)$ , defined as

$$ \begin{align*} \mathrm{Li}_{{\boldsymbol k}}(z)=\sum_{m_{1}>\cdots>m_d>0} \dfrac{z^{m_{1}}}{{m_{1}}^{k_{1}}\cdots {m_d}^{k_d}}, \end{align*} $$

converges absolutely at $|z|<1$ for any ${\boldsymbol k}$ , and even at $|z|=1$ for an admissible ${\boldsymbol k}$ . Indeed, ${\mathrm {Li}}_{{\boldsymbol k}}(1)$ agrees with MZVs $\zeta ({{\boldsymbol k}})$ . Hence, equation (2.4) is bounded from above. This concludes (i).

  1. (ii) Because $\widetilde {\mathrm {Li}}_{{\boldsymbol k}} (1 ;a)=\zeta ({\boldsymbol k};a)$ for an admissible ${\boldsymbol k}$ , the assertion directly follows from (i).

The multiple Hurwitz polylog (MHP) is defined as

$$ \begin{align*} \mathrm{Li}_{{\boldsymbol k}}(z ;a):=z^{a}\, \widetilde{\mathrm{Li}}_{{\boldsymbol k}}(z ;a) =\sum_{m_{1}>\cdots>m_d\ge0}\dfrac{z^{m_{1}+a}}{(m_{1}+a)^{k_{1}}\cdots (m_d+a)^{k_d}}. \end{align*} $$

This yields a multivalued function in z on $|z|\le 1$ caused by the complex power $z^a$ . To make this a single-valued function, let R (respectively, $\overline {R}$ ) be a unit disc (respectively, a closed unit disc) with a slit

$$ \begin{align*} R=\{z\in {{\mathbb{C}}}\mid |z|<1\}- {\mathbb{R}}_{\le 0},\qquad \overline{R}=\{z\in {{\mathbb{C}}}\mid |z|\le1\}- {\mathbb{R}}_{\le 0}. \end{align*} $$

In addition, we choose the principal branch $\pi <\arg z\le \pi $ for $z\in R$ (respectively, $z\in \overline {R}$ ). Lemma 2.1(i) implies that $\mathrm {Li}_{{\boldsymbol k}}(z ;a)$ is single-valued and holomorphic in R for any index ${\boldsymbol k}$ and converges in $\overline {R}$ for an admissible ${\boldsymbol k}$ .

When $a = 1$ , $\mathrm {Li}_{{\boldsymbol k}}(z ;a)$ reduces to the usual MP

$$\begin{align*}\mathrm{Li}_{{\boldsymbol k}}(z ;1)=\sum_{m_{1}>\cdots>m_d\ge0}\dfrac{z^{m_{1}+1}}{(m_{1}+1)^{k_{1}}\cdots (m_d+1)^{k_d}} =\sum_{m_{1}>\cdots>m_d>0}\dfrac{z^{m_{1}}}{{m_{1}}^{k_{1}}\cdots {m_d}^{k_d}}=\mathrm{Li}_{{\boldsymbol k}}(z), \end{align*}$$

and to the MHZs when $z = 1$ for an admissible ${\boldsymbol k}$ , that is, $\mathrm {Li}_{{\boldsymbol k}}(1 ;a)=\zeta ({\boldsymbol k};a)$ . The differential formula behaves better for $\mathrm {Li}_{{\boldsymbol k}}(z ;a)$ than $\widetilde {\mathrm {Li}}_{{\boldsymbol k}}(z ;a)$ . The lemma below is generalized in Proposition 2.3.

Lemma 2.2 Let ${\boldsymbol k}=(k_1, \ldots , k_d)\in {\mathbb {N}}^d$ . The MHP satisfies the following:

$$\begin{align*}\dfrac{d}{dz}\mathrm{Li}_{{\boldsymbol k}}(z ;a) =\left\{\!\!\begin{array}{ll} \dfrac{1}{z}\ \mathrm{Li}_{(k_{1}-1,k_{2},\ldots,k_d)}(z; a), & (k_{1}\geq2, \ d\ge 1),\\[6pt] \dfrac{1}{1-z}\,\mathrm{Li}_{(k_{2},k_{3},\ldots,k_d)}(z; a), & (k_{1}=1, \ d\geq2), \end{array} \right. \end{align*}$$

and $\frac {d}{dz}\mathrm {Li}_{(1)}(z; a)=z^a/(1-z)$ for $z\in R$ .

Proof It is easy to confirm this by a termwise differentiation of $\mathrm {Li}_{{\boldsymbol k}}(z ;a)$ .

Except for formula of $\mathrm {Li}_{1}(z; a)$ , the differential formula of the usual MP $\mathrm {Li}_{{\boldsymbol k}}(z)$ is the same in appearance as that of Lemma 2.2. Therefore, we expect the generating function of the Ohno–Zagier type for the MHPs to behave similar to that of the MPs.

For generalization, using Yamamoto’s method in [Reference Yamamoto21], let us define the t-multiple Hurwitz polylog (t-MHP), and t-multiple Hurwitz zeta value (t-MHZ) as

(2.5) $$ \begin{align} \mathrm{Li}^t_{{\boldsymbol k}}(z;a)=\sum_{{\boldsymbol p}}t^{\sigma({{\boldsymbol p}})}\mathrm{Li}_{{{\boldsymbol p}}}(z;a), \qquad {\zeta}^t({\boldsymbol k};a)=\sum_{{\boldsymbol p}}t^{\sigma({{\boldsymbol p}})}{\zeta}({\boldsymbol p};a), \end{align} $$

where, in each summation, ${\boldsymbol p}$ runs over all indices of the form ${\boldsymbol p}=(k_1\Box k_2\Box \cdots \Box k_d)$ obtained by filling in boxes $\Box $ with symbols “ ” or “ $+$ ,” and $\sigma ({{\boldsymbol p}}):=d-\mathrm {dep}\,{\boldsymbol p}$ , or the number of boxes in which “ $+$ ” is filled in.

The star-type MHP (MHSP) and the star-type MHZ (MHSZ) are, respectively, defined for $a\notin {\mathbb {Z}}_{\le 0}$ as

$$ \begin{align*} \mathrm{Li}^{\star}_{{\boldsymbol k}}(z ;a) &=\sum_{m_{1}\ge \cdots\ge m_d\ge0}\dfrac{z^{m_{1}+a}}{(m_{1}+a)^{k_{1}}\cdots (m_d+a)^{k_d}}, \\{\zeta}^{\star}({\boldsymbol k} ;a) &=\sum_{m_{1}\ge \cdots\ge m_d\ge0}\dfrac{1}{(m_{1}+a)^{k_{1}}\cdots (m_d+a)^{k_d}}. \end{align*} $$

The t-MHP (respectively, t-MHZ) is a polynomial in t and interpolates the MHP (respectively, MHZ) at $t=0$ and MHSP (respectively, MHSZ) at $t=1$ , i.e., $\mathrm {Li}^0_{{\boldsymbol k}}(z;a)=\mathrm {Li}_{{\boldsymbol k}}(z ;a)$ and $\mathrm {Li}^1_{{\boldsymbol k}}(z;a)=\mathrm {Li}^{\star }_{{\boldsymbol k}}(z ;a)$ (respectively, ${\zeta }^0({\boldsymbol k} ;a)={\zeta }({\boldsymbol k} ;a)$ and ${\zeta }^1({\boldsymbol k} ;a)={\zeta }^{\star }({\boldsymbol k} ;a)$ ). See the summary in Table 1.

Table 1

When ${\boldsymbol k}$ is an index of depth $1$ , $\mathrm {Li}^t_{{\boldsymbol k}}(z;a)=\mathrm {Li}_{{\boldsymbol k}}(z;a)$ clearly holds by definition.

The t-MHP satisfies the following differential relation.

Proposition 2.3 Let ${\boldsymbol k}=(k_1, \ldots , k_d)\in {\mathbb {N}}^d$ . The t-MHP satisfies

$$\begin{align*}\dfrac{d}{dz}\mathrm{Li}^{t}_{{\boldsymbol k}}(z ;a) =\left\{\!\!\begin{array}{ll} \dfrac{1}{z}\ \mathrm{Li}^{t}_{(k_{1}-1,k_{2},\ldots,k_d)}(z; a), & (k_{1}\geq2, \ d\ge 1),\\[6pt] \left(\dfrac{t}{z}+\dfrac{1}{1-z}\right)\,\mathrm{Li}^{t}_{(k_{2},k_{3},\ldots,k_d)}(z; a), & (k_{1}=1, \ d\geq2), \end{array} \right. \end{align*}$$

and $\frac {d}{dz}\mathrm {Li}^{t}_{(1)}(z; a)=z^a/(1-z)$ for $z\in R$ .

Proof The former case follows easily from the former formula provided in Lemma 2.2 and equation (2.5). For the latter case, we have

$$ \begin{align*} \frac{d}{dz}\mathrm{Li}^t_{(1,k_2,\ldots,k_d)}(z; a) &=\sum_{{\boldsymbol p}=(1\Box k_2\Box\cdots\Box k_d)}\frac{d}{dz}\mathrm{Li}_{{\boldsymbol p}}(z; a)t^{\sigma({\boldsymbol p})}\\ &=\sum_{{\boldsymbol p}=(1+k_2\Box\cdots\Box k_d)}\frac{d}{dz}\mathrm{Li}_{{\boldsymbol p}}(z; a)t^{\sigma({\boldsymbol p})} +\sum_{{\boldsymbol p}=(1,k_2\Box\cdots\Box k_d)}\frac{d}{dz}\mathrm{Li}_{{\boldsymbol p}}(z; a)t^{\sigma({\boldsymbol p})}\\ &=\sum_{{\boldsymbol p}=(k_2\Box\cdots\Box k_d)}\frac{1}{z}\,\mathrm{Li}_{{\boldsymbol p}}(z; a)t^{\sigma({\boldsymbol p})+1} +\sum_{{\boldsymbol p}=(k_2\Box\cdots\Box k_d)}\frac{1}{1-z}\mathrm{Li}_{{\boldsymbol p}}(z; a)t^{\sigma({\boldsymbol p})}\\ &=\left(\frac{t}{z}+\dfrac{1}{1-z}\right)\mathrm{Li}^t_{(k_2,\dots,k_d)}(z; a). \end{align*} $$

Because $\mathrm {Li}^{t}_{(1)}(z; a)=\mathrm {Li}_{(1)}(z; a)$ , the last assertion follows from Lemma 2.2.

3 Generating function of MHPs

Following previous studies [Reference Li11, Reference Li and Wakabayashi14, Reference Ohno and Zagier17], we introduce the generating function of the MHPs as follows:

For $i\in {\mathbb N}$ , we define the i-th height of ${{\boldsymbol k}}=(k_{1},\ldots ,k_d)\in {\mathbb {N}}^d$ as i-ht $({{\boldsymbol k}})= \sharp \{l\, |\, k_{l}> i\}$ . The usual height coincides with the $1$ -height, that is, $1$ -ht $({{\boldsymbol k}})=$ ht $({\boldsymbol k})$ . For a fixed $r\in {\mathbb N}$ , and any integers $k,d,h_{1},\ldots ,h_{r}\geq 0$ , and $0\le j\le r-1$ , we define the set of indices as

$$ \begin{align*} I(k,d,h_{1},h_{2},\ldots,h_{r})&=\{{{\boldsymbol k}}\,|\,\mathrm{wt}({{\boldsymbol k}})=k,\mathrm{dep}({{\boldsymbol k}})=d, i\mbox{-ht}({{\boldsymbol k}})=h_{i}\quad (i=1,\ldots, r)\}, \\ I_{j}(k,d,h_{1},h_{2},\ldots,h_{r})&=\{{{\boldsymbol k}}=(k_1,\ldots, k_d)\,|\,{{\boldsymbol k}}\in I(k,d,h_{1},\ldots,h_{r}),k_{1}\geq j+2\}. \end{align*} $$

Clearly, $\{I_j\}$ gives a decreasing filtration of I:

$$\begin{align*}I(k,d,h_{1},\ldots,h_{r})\supset I_{0}(k,d,h_{1},\ldots,h_{r})\supset I_{1}(k,d,h_{1},\ldots,h_{r})\supset\cdots \supset I_{r-1}(k,d,h_{1},\ldots,h_{r}).\end{align*}$$

Note that

$$ \begin{align*} I(k,d,h_{1},\ldots,h_{r})\neq\emptyset &\Longrightarrow k\geq d+\textstyle \sum_{j=1}^{r}h_{j}, \quad d\geq h_{1}\geq\cdots\geq h_{r}\geq 0, \\ I_{j}(k,d,h_{1},\ldots,h_{r})\neq\emptyset &\Longrightarrow k\geq d+\textstyle \sum_{j=1}^{r}h_{j}, \quad d\geq h_{1}\geq\cdots\geq h_{r}\geq 0, \quad h_{j+1}\geq 1. \end{align*} $$

For $k,d,h_{1},\ldots ,h_{r}\geq 0$ , the sums of the MHPs of a fixed weight, depth, and ith heights for all $1\le i \le r$ , are defined as

$$ \begin{align*} G^t(k,d,h_1,\ldots,h_r; z ;a)&=\sum_{{{\boldsymbol k}}\in I(k,d,h_{1},\ldots,h_{r})}\mathrm{Li}_{{\boldsymbol k}}^t(z ;a), \\ G_j^t(k,d,h_{1},\ldots,h_{r}; z ;a)&=\sum_{{{\boldsymbol k}}\in I_{j}(k,d,h_{1},\ldots,h_{r})}\mathrm{Li}_{{\boldsymbol k}}^t(z ;a)\quad (0\le j\le r-1). \end{align*} $$

Each sum is understood to be zero if $I(k,d,h_{1},\ldots ,h_{r})$ or if $I_j(k,d,h_{1},\ldots ,h_{r})$ is empty except when $(k,d,h_{1},\ldots ,h_{r})=(0,0,0, \ldots , 0)$ for $G^t$ . In this case, we conventionally apply

(3.1) $$ \begin{align} G^t(0,0,\ldots,0; z ;a)=c^t(z) &:= \left. \frac{d}{dz}\mathrm{Li}^t_{(1)}(z ;a)\right/ \left(\frac{t}{z}+\frac{1}{1-z}\right) \notag \\ &=\left. \frac{z^{a}}{1-z}\right/ \left(\frac{t}{z}+\frac{1}{1-z}\right) =\frac{z^{a+1}}{z+t(1-z)}. \end{align} $$

It should be noted that $c^0(z)=z^{a}$ and $c^1(z)=z^{a+1}$ .

Lemma 3.1 Put $h=\sum _{j=1}^rh_j$ .

  1. (1) For $k\geq d+h,\ d\geq h_{1}\geq \cdots \geq h_{r}\geq 1$ , we have

    $$ \begin{align*} &\dfrac{d}{dz}G^t_{r-1}(k,d,h_{1},\ldots,h_{r}; z ;a)\\& \quad =\dfrac{1}{z}\{G^t_{r-1}(k-1,d,h_{1},\ldots,h_{r}; z ;a) +G^t_{r-2}(k-1,d,h_{1},\ldots,h_{r-1},h_{r}-1; z ;a)\\& \qquad\quad -G^t_{r-1}(k-1,d,h_{1},\ldots,h_{r-1},h_{r}-1, z ;a)\}.\end{align*} $$
  2. (2) For $k\geq d+h,\ d\geq h_{1}\geq \cdots \geq h_{r}\geq 0$ , and $h_{j+1}\geq 1$ ( $1\leq j\leq r-2$ ), we have

    $$ \begin{align*} &\dfrac{d}{dz}\left\{G^t_{j}(k,d,h_{1},\ldots,h_{r}; z ;a)-G^t_{j+1} (k,d,h_{1},\ldots,h_{r}; z ;a)\right\}\\& \quad = \dfrac{1}{z}\{G^t_{j-1}(k-1,d,h_{1}, \ldots,h_{j},h_{j+1}-1,h_{j+2},\ldots,h_{r}; z ;a)\\& \qquad \quad -G^t_{j}(k-1,d,h_{1},\ldots,h_{j},h_{j+1}-1,h_{j+2},\ldots,h_{r}; z ;a)\}. \end{align*} $$
  3. (3) For $k\geq d+h,\ d\geq h_{1}\geq \cdots \geq h_{r}\geq 0$ , and $h_{1}\geq 1$ , we have

    $$ \begin{align*} & \kern-6pt\dfrac{d}{dz}\left\{G^t_{0}(k,d,h_{1},\ldots,h_{r}; z ;a)-G^t_{1}(k,d,h_{1},\ldots,h_{r}; z ;a)\right\}\\& \kern-6pt\quad =\dfrac{1}{z}\{G^t(k-1,d,h_{1}-1, h_2, \ldots, h_{r}; z ;a) -G^t_{0}(k-1,d,h_{1}-1,h_2, \ldots,h_{r}; z ;a)\}. \end{align*} $$
  4. (4) For $k\geq d+h,\ d\geq h_{1}\geq \cdots \geq h_{r}\geq 0,\ d\geq 2$ , we have

    $$ \begin{align*} &\dfrac{d}{dz}\left\{G^t(k,d,h_{1},\ldots,h_{r}; z ;a)-G^t_{0}(k,d,h_{1},\ldots,h_{r}; z ;a)\right\}\\& \quad =\left(\dfrac{t}{z}+\dfrac{1}{1-z}\right)G^t(k-1,d-1,h_{1},\ldots,h_{r}; z ;a). \end{align*} $$

Proof Each of these cases can be shown in the same way as in Lemma 2.2 in [Reference Li11] (or in Lemma 6 in [Reference Li and Wakabayashi14]) by replacing their function $G_j$ or $G^t_j$ with Hurwitz-type $G^t_{j}(k,d,h_{1},\ldots ,h_{r}; z; a)$ .

Remark

  1. (i) Lemma 3.1(2) gives Lemma 3.1(3) when $j=0$ by interpreting $G^t_{-1}$ as $G^t$ .

  2. (ii) By convention (3.1), Lemma 3.1(4) is valid even when $d=1$ .

For a fixed $r\in {\mathbb {N}}$ , we define the generating functions

$$\begin{align*}\Phi^t=\Phi^t(x_1,\dots,x_{r+2}; z; a), \quad \Phi^t_j=\Phi^t_j(x_1,\dots,x_{r+2}; z; a)\quad (1\leq j\leq r-1)\end{align*}$$

as a formal power series in $x_1, \ldots , x_{r+2}$ by

$$ \begin{align*} \Phi^t&=\sum_{k\geq d+h \atop d\geq h_{1}\geq\cdots\geq h_{r}\geq0}G^t(k,d,h_{1},\ldots,h_{r}; z; a)x_{1}^{k-d-h}x_{2}^{d-h_{1}}x_{3}^{h_{1}-h_{2}}\cdots x_{r+1}^{h_{r-1}-h_{r}}x_{r+2}^{h_{r}},\\ \Phi^t_{j}&=\sum_{k\geq d+h \atop d\geq h_{1}\geq\cdots\geq h_{r}\geq0}G^t_{j}(k,d,h_{1},\ldots,h_{r}; z; a)x_{1}^{k-d-h}x_{2}^{d-h_{1}}x_{3}^{h_{1}-h_{2}}\cdots x_{r+1}^{h_{r-1}-h_{r}}x_{r+2}^{h_{r}}, \end{align*} $$

where $\sum _{j=1}^r h_j$ is abbreviated as h to shorten the equation. The notations $\Phi ^{t, (r)}:=\Phi ^t$ and $\Phi ^{t, (r)}_{j}:=\Phi ^t_{j}$ are used to emphasize the dependence on r. Note that for $r\in {\mathbb {N}}$ ,

$$ \begin{align*} \Phi^{t, (r)}=\Phi^{t, (r+1)}\big|_{x_{r+3}=x_1x_{r+2}},\qquad \Phi^{t, (r)}_j=\Phi^{t, (r+1)}_j\big|_{x_{r+3}=x_1x_{r+2}} \quad (1\leq j\leq r-1). \end{align*} $$

These generating functions satisfy the following differential equations. Let $D=z\frac {d}{dz}$ be the Euler operator.

Proposition 3.2 $\Phi ^t$ and $\Phi ^t_j$ satisfy:

  1. (1) $D\Phi ^t_{r-1}=x_{1}\Phi ^t_{r-1}+\dfrac {x_{r+2}}{x_{r+1}}(\Phi ^t_{r-2}-\Phi ^t_{r-1})$ ,

  2. (2) $D(\Phi ^t_{j}-\Phi ^t_{j+1})=\dfrac {x_{j+3}}{x_{j+2}}(\Phi ^t_{j-1}-\Phi ^t_{j})\qquad (1\le j\le r-2),$

  3. (3) $D(\Phi ^t_{0}-\Phi ^t_{1})=\dfrac {x_{3}}{x_{2}}(\Phi ^t-\Phi ^t_{0}-c^t(z)),$

  4. (4) $D(\Phi ^t-\Phi ^t_{0}-c^t(z))=\left (t+\dfrac {z}{1-z}\right )x_2 \Phi ^t$ ,

where $c^t(z)={z^{a}}/(z+t (1-z))$ is a constant term of $\Phi ^t$ defined in equation (3.1).

Proof (1) and (2) can be shown in the same way as Lemma 2.2 in [Reference Li11] (or Lemma 6 in [Reference Li and Wakabayashi14]) by replacing their function $\Phi ^t$ (respectively, $\Phi ^t_j$ ) with our Hurwitz-type $\Phi ^t(x_1,\dots ,x_{r+2}; z; a)$ (respectively, $\Phi ^t_j(x_1,\dots ,x_{r+2}; z; a)$ ). We will next demonstrate (3) and (4). In the following, $h=\sum _{j=1}^r h_j$ . For (3), because

$$ \begin{align*} \Phi^t_{0}&-\Phi^t_{1}\\ =&\hspace{-2ex} \sum_{k\geq d+h, h_1\ge 1\atop d\geq h_{1}\geq\cdots\geq h_{r}\geq0} \hspace{-3ex}\{G^t_{0}(k,d,h_{1},\ldots,h_{r}; z; a)-G^t_{1}(k,d,h_{1},\ldots,h_{r}; z; a)\} x_{1}^{k-d-h}x_{2}^{d-h_{1}}x_{3}^{h_{1}-h_{2}}\cdots x_{r+2}^{h_{r}}, \end{align*} $$

by applying Lemma 3.1(3), we have

$$ \begin{align*} D&(\Phi^t_{0}-\Phi^t_{1})\\ =&\hspace{-2ex}\sum_{k\geq d+h, h_1\ge 1\atop d\geq h_{1}\geq\cdots\geq h_{r}\geq0} \hspace{-3ex} \{G^t(k-1,d,h_{1}-1,h_{2},\ldots,h_{r}; z; a)-G^t_{0}(k-1,d,h_{1}-1,h_{2},\ldots,h_{r}; z; a)\} \\ &\times x_{1}^{k-d-h}x_{2}^{d-h_{1}}x_{3}^{h_{1}-h_{2}}\cdots x_{r+2}^{h_{r}}\\ =&\hspace{-2ex}\sum_{k, d, h_{1}, \ldots, h_{r}\geq0} \hspace{-3ex} \{G^t(k-1,d,h_{1}-1,h_{2},\ldots,h_{r}; z; a)-G^t_{0}(k-1,d,h_{1}-1,h_{2},\ldots,h_{r}; z; a)\} \\ &\times x_{1}^{k-d-h}x_{2}^{d-h_{1}}x_{3}^{h_{1}-h_{2}}\cdots x_{r+2}^{h_{r}} -G^t(0,0,0,0,\ldots,0;a;z)\dfrac{x_3}{x_2}. \end{align*} $$

By shifting the indices $k-1$ to k, and $h_1-1$ to $h_1$ in the sum above,

$$ \begin{align*} &=\sum_{k, d, h_{1}, \ldots, h_{r}\geq0} \hspace{-3ex} \{G^t(k,d,h_{1},h_{2},\ldots,h_{r}; z; a)-G^t_{0}(k,d,h_{1},h_{2},\ldots,h_{r}; z; a)\} \\ &\quad \times x_{1}^{k-d-h}x_{2}^{d-h_{1}}x_{3}^{h_{1}-h_{2}}\cdots x_{r+2}^{h_{r}}\dfrac{x_3}{x_2} -c^t(z)\dfrac{x_3}{x_2}\\ &=\dfrac{x_{3}}{x_{2}}(\Phi^t-\Phi^t_{0}-c^t(z)). \end{align*} $$

Hence, (3) can be proven. For (4), because

$$ \begin{align*} \Phi&^t-\Phi^t_{0}\\ =&\hspace{-2ex}\sum_{k, d, h_1,\ldots, h_{r}\geq0} \hspace{-3ex}\{G^t(k,d,h_{1},\ldots,h_{r}; z; a)-G^t_{0}(k,d,h_{1},\ldots,h_{r}; z; a)\} x_{1}^{k-d-h}x_{2}^{d-h_{1}}x_{3}^{h_{1}-h_{2}}\cdots x_{r+2}^{h_{r}}, \end{align*} $$

by applying Lemma 3.1(4) and based on the fact mentioned in Remark (ii) above, we have

$$ \begin{align*} &D(\Phi^t-\Phi^t_{0})\\&= \sum_{k, d, h_1,\ldots, h_{r}\geq0} \hspace{-4ex}D \{G^t(k,d,h_{1},\ldots,h_{r}; z; a)-G^t_{0}(k,d,h_{1},\ldots,h_{r}; z; a)\} x_{1}^{k-d-h}x_{2}^{d-h_{1}}x_{3}^{h_{1}-h_{2}}\cdots x_{r+2}^{h_{r}} \\&=Dc^t(z)+\hspace{-3ex}\sum_{ (k,d,\ldots,h_r)\atop \not=(0,0,\ldots,0)} \hspace{-1ex}\left(t+\frac{z}{1-z}\right)G^t(k-1,d-1,h_{1},\ldots,h_{r}; z; a) x_{1}^{k-d-h}x_{2}^{d-h_{1}}x_{3}^{h_{1}-h_{2}}\cdots x_{r+2}^{h_{r}}. \end{align*} $$

By shifting the indices $k-1$ to k, and $d-1$ to d in the sum above,

$$ \begin{align*} &=Dc^t(z)+\hspace{-3ex}\sum_{k, d, h_1,\ldots, h_{r}\geq0} \hspace{-1ex}\left(t+\frac{z}{1-z}\right)G^t(k,d,h_{1},\ldots,h_{r}; z; a) x_{1}^{k-d-h}x_{2}^{d-h_{1}}x_{3}^{h_{1}-h_{2}}\cdots x_{r+2}^{h_{r}}x_2\\ &=Dc^t(z)+\left(t+\frac{z}{1-z}\right)x_2 \Phi^t. \end{align*} $$

This shows that (4) holds and thus completes the proof.

Let us define

$$\begin{align*}y_{0}=\Phi^t_{r-1}, \quad y_{j}=\Phi^t_{r-1-j}-\Phi^t_{r-j}\quad(1\le j\le r-1), \quad y_{r}=\Phi^t-\Phi^t_{0}.\end{align*}$$

Proposition 3.3 We have the following:

  1. (1) $Dy_{0}=x_{1}y_{0}+\dfrac {x_{r+2}}{x_{r+1}}y_{1}$ ,

  2. (2) $Dy_{j}=\dfrac {x_{r+2-j}}{x_{r+1-j}}y_{j+1}\qquad (1\le j\le r-2)$ ,

  3. (3) $Dy_{r-1}=\dfrac {x_{3}}{x_{2}}(y_{r}-c^t(z))$ ,

  4. (4) $D(y_{r}-c^t(z))=\left (t+\dfrac {z}{1-z}\right )x_{2} \displaystyle \sum _{j=0}^r y_j$ ,

  5. (5) $y_{j}=\dfrac {x_{r+2-j}}{x_{r+2}}D^{j}y_{0}-\dfrac {x_{1}x_{r+2-j}}{x_{r+2}}D^{j-1}y_{0}\qquad (1\le j\le r-1)$ ,

  6. (6) $y_{r}-c^t(z)=\dfrac {x_{2}}{x_{r+2}}D^{r}y_{0}-\dfrac {x_{1}x_{2}}{x_{r+2}}D^{r-1}y_{0}$ .

Proof Because $\Phi ^t=\sum _{j=0}^ry_j$ and $\Phi ^t_0=\sum _{j=0}^{r-1}y_j$ , (1)–(4) are rewriting of Proposition 3.2. In addition, (5) is proven through the induction of j and (2). To prove (6), using (5) and (3), we have

$$ \begin{align*} y_r-c^t(z)&=\frac{x_2}{x_3}Dy_{r-1}=\frac{x_2}{x_3}\left( \frac{x_3}{x_{r+2}}D^ry_0- \frac{x_1x_3}{x_{r+2}}D^{r-1}y_0\right)\\[3pt] &=\frac{x_2}{x_{r+2}}D^ry_0-\frac{x_1x_2}{x_{r+2}}D^{r-1}y_0. \\[-40pt] \end{align*} $$

Proposition 3.4 Here, $y_{0}=\Phi ^t_{r-1}$ satisfies the following condition:

$$ \begin{align*} &\Big[ D^{r+1}-(x_1+tx_2)D^r-t\sum_{j=0}^{r-1}(x_{r+2-j}-x_1x_{r+1-j})D^j\\[-1ex] &-z\big\{ D^{r+1}-(x_1+(t-1)x_{2})D^{r}-(t-1)\sum_{j=0}^{r-1}(x_{r+2-j}-x_{1}x_{r+1-j})D^{j} \big\} \Big] y_{0}=x_{r+2}z^{a}. \end{align*} $$

Proof Proposition 3.3(6) implies that

(3.2) $$ \begin{align} D(y_{r}-c^t(z))=\dfrac{x_{2}}{x_{r+2}}D^{r+1}y_{0}-\dfrac{x_{1}x_{2}}{x_{r+2}}D^{r}y_{0}. \end{align} $$

Furthermore, Proposition 3.3(5) and (6) give

(3.3) $$ \begin{align} \sum_{j=0}^r y_j-c^t(z) &=y_{0}+\sum_{j=1}^{r}\left(\dfrac{x_{r+2-j}}{x_{r+2}}D^{j}y_{0}-\dfrac{x_{1}x_{r+2-j}}{x_{r+2}}D^{j-1}y_{0}\right) \notag\\ &=\sum_{j=0}^{r}\dfrac{x_{r+2-j}}{x_{r+2}}D^{j}y_{0} -\sum_{j=0}^{r-1}\dfrac{x_{1}x_{r+1-j}}{x_{r+2}}D^{j}y_{0}\notag \\ &=\sum_{j=0}^{r-1}\dfrac{x_{r+2-j}-x_{1}x_{r+1-j}}{x_{r+2}}D^{j}y_{0}+\dfrac{x_{2}}{x_{r+2}}D^{r}y_{0}. \end{align} $$

Substituting equations (3.2) and (3.3) into the equation

$$ \begin{align*} D(y_{r}-c^t(z))=\left(t+\dfrac{z}{1-z}\right)x_{2} \sum_{j=0}^r y_j, \end{align*} $$

in Proposition 3.3(4), we have

$$ \begin{align*} \frac{x_{2}}{x_{r+2}}&D^{r+1}y_{0}-\dfrac{x_{1}x_{2}}{x_{r+2}}D^{r}y_{0}\\ &=\left(t+\dfrac{z}{1-z}\right)x_{2} \left(\sum_{j=0}^{r-1}\dfrac{x_{r+2-j}-x_{1}x_{r+1-j}}{x_{r+2}}D^{j}y_{0}+\dfrac{x_{2}}{x_{r+2}}D^{r}y_{0}+c^t(z)\right). \end{align*} $$

By multiplying both sides by $(1-z)x_2^{-1}x_{r+2}$ ,

$$ \begin{align*} (1-z)&(D^{r+1}y_{0}-x_{1}D^{r}y_{0})\\ &=\left(t-(t-1)z\right)\left(\sum_{j=0}^{r-1}(x_{r+2-j}-x_{1}x_{r+1-j})D^{j}y_{0}+x_{2}D^{r}y_{0}+x_{r+2}c^t(z)\right)\\ &=\left(t-(t-1)z\right)\left(\sum_{j=0}^{r-1}(x_{r+2-j}-x_{1}x_{r+1-j})D^{j}y_{0}+x_{2}D^{r}y_{0}\right)+x_{r+2}z^{a}. \end{align*} $$

Hence, we obtain the following desired equation:

$$ \begin{align*} &D^{r+1}y_0-(x_1+tx_2)D^ry_0-t\sum_{j=0}^{r-1}(x_{r+2-j}-x_1x_{r+1-j})D^jy_0\\[-1ex] &-z\big\{D^{r+1}y_0-(x_1+(t-1)x_{2})D^{r}y_0-(t-1)\sum_{j=0}^{r-1}(x_{r+2-j}-x_{1}x_{r+1-j})D^{j}y_0 \big\} =x_{r+2}z^{a}. \\[-22pt] \end{align*} $$

Let $\mathcal {P}_1(X)=\mathcal {P}^{\, t}_1(X;x_1,\ldots , x_{r+2})$ and $\mathcal {P}_2(X)=\mathcal {P}^{\, t}_2(X;x_1,\ldots , x_{r+2})$ be polynomials of degree $r+1$ in X defined by

$$ \begin{align*} \mathcal{P}_1(X)&=X^{r+1}-(x_1+tx_2)X^r-t\sum_{j=0}^{r-1}(x_{r+2-j}-x_1x_{r+1-j})X^j, \\ \mathcal{P}_2(X)&=X^{r+1}-(x_1+(t-1)x_{2})X^{r}-(t-1)\sum_{j=0}^{r-1}(x_{r+2-j}-x_{1}x_{r+1-j})X^{j}. \end{align*} $$

Note that

$$ \begin{align*} \mathcal{P}^{\, t-1}_1(X;x_1,\ldots, x_{r+2})&=\mathcal{P}^{\, t}_2(X;x_1,\ldots, x_{r+2}), \\ \mathcal{P}^{\, st}_1(X;x_1,x_2, \ldots, x_{r+2})&=\mathcal{P}^{\, t}_1(X;x_1,sx_2, \ldots, sx_{r+2}) \end{align*} $$

are valid for any values of parameters s and t. The polynomial

$$ \begin{align*} \mathcal{L}(X):=\mathcal{P}_1(X)-z \mathcal{P}_2(X) \end{align*} $$

satisfies

(3.4) $$ \begin{align} \mathcal{L}(D)\Phi^t_{r-1}=x_{r+2}z^{a} \end{align} $$

by virtue of Proposition 3.4. Let $\{-\alpha _j\}$ and $\{-\beta _j\}$ be the zeros of $\mathcal {P}_2(X)$ and $\mathcal {P}_1(X)$ respectively, that is,

(3.5) $$ \begin{align} \mathcal{P}_1(X)=\prod_{j=1}^{r+1} (X+\beta_j), \qquad \mathcal{P}_2(X)=\prod_{j=1}^{r+1} (X+\alpha_j). \end{align} $$

Because the identity as an operator $ \mathcal {L}(D)z^{a}=z^{a} \mathcal {L}(D+a) $ holds, by applying $\widetilde {\Phi }^t_{r-1}:=z^{-a}\Phi ^t_{r-1}$ and dividing equation (3.4) by $z^{a}$ , we obtain

(3.6) $$ \begin{align} \mathcal{L}(D+a)\widetilde{\Phi}^t_{r-1}=x_{r+2}. \end{align} $$

Applying D to both sides to make this a homogeneous equation, we have

$$ \begin{align*} D\mathcal{L}(D+a)\widetilde{\Phi}^t_{r-1}=0, \end{align*} $$

or equivalently,

(3.7) $$ \begin{align} \{D(D+a+\beta_1) &\cdots (D+a+\beta_{r+1})\notag \\ & -z(D+1)(D+a+\alpha_1)\cdots (D+a+\alpha_{r+1}) \} \widetilde{\Phi}^t_{r-1}=0. \end{align} $$

The generalized hypergeometric function

$$\begin{align*}_{r+2}\,F_{r+1}\left(\!\!\! \begin{array}{c} b_1,\ldots, b_{r+2}\\ c_1,\ldots, c_{r+1} \end{array} \!;\, z\right)\! =\sum_{m=0}^{\infty}\dfrac{[b_1]_m\dots[b_{r+2}]_m}{[c_1]_m\dots [c_{r+1}]_m}\frac{z^m}{m!}, \end{align*}$$

where $[b]_m=\prod _{i=0}^{m-1} (b+i)$ , is a solution to the differential equation

(3.8) $$ \begin{align} \left[ D(D+c_1-1)\cdots (D+c_{r+1}-1)-z(D+b_1)\cdots (D+b_{r+2}) \right] F=0, \end{align} $$

which is holomorphic at $z=0$ . See [Reference Andrews, Askey and Roy1, Reference Gasper and Rahman6] for further details.

Theorem 3.5 Letting $z\in R$ and $a\in {\mathbb {C}}$ with $a\not \in {\mathbb {Z}}_{\leq 0}$ , we have

(3.9) $$ \begin{align} &\Phi^t_{r-1}(x_{1},\ldots,x_{r+2}; z; a)= \frac{x_{r+2}}{\mathcal{P}_1(a)}\,z^{a}\, {}_{r+2}F_{r+1}\left(\!\!\! \begin{array}{lcl} a+\alpha_{1},&\hspace{-2ex}\ldots, &\hspace{-1.5ex} a+\alpha_{r+1},\ 1\\ a+\beta_1+1,&\hspace{-1.5ex} \ldots,&\hspace{-1.5ex} a+\beta_{r+1}+1 \end{array} ;z\right) \end{align} $$
(3.10) $$ \begin{align} &=\frac{x_{r+2}}{\mathcal{P}_2(a-1)}\,z^{a-1}\, \left\{{}_{r+2}F_{r+1}\left(\!\!\! \begin{array}{lcl} a+\alpha_{1}-1,&\hspace{-2ex}\ldots, &\hspace{-1.5ex} a+\alpha_{r+1}-1,\ 1\\ a+\beta_1,&\hspace{-1.5ex} \ldots,&\hspace{-1.5ex} a+\beta_{r+1} \end{array} ;z\right)-1\right\}, \end{align} $$

where $\{\alpha _{j}\}$ and $\{\beta _{j}\}$ are defined as in equation ( 3.5 ).

Proof Applying $\Phi ^{t}_{r-1}=\Phi ^{t}_{r-1}(x_{1},\ldots ,x_{r+2}; z; a)$ and $\widetilde {\Phi }^{t}_{r-1}=\widetilde {\Phi }^t_{r-1}(x_{1},\ldots ,x_{r+2}; z; a)$ , from equations (3.7) and (3.8), we have

$$ \begin{align*} \widetilde{\Phi}^t_{r-1}=C\cdot{_{r+2}F_{r+1}}\left(\!\!\!\begin{array}{lcl} a+\alpha_{1},&\hspace{-2ex}\ldots, &\hspace{-1.5ex} a+\alpha_{r+1},\ 1\\ a+\beta_1+1,&\hspace{-1.5ex} \ldots,&\hspace{-1.5ex} a+\beta_{r+1}+1\end{array} ;z\right),\end{align*} $$

for a constant C, which is independent of z. This shows that C is the constant term of $\widetilde {\Phi }^t_{r-1}$ . Equation (3.6) shows that

(3.11) $$ \begin{align} [\mathcal{P}_1(D+a)-z \mathcal{P}_2(D+a)]\widetilde{\Phi}^t_{r-1}=x_{r+2}. \end{align} $$

The constant terms of both sides of equation (3.11) give the equation $\mathcal {P}_1(a)C=x_{r+2}$ , and thus $C=x_{r+2}/\mathcal {P}_1(a)$ . Because $\Phi ^t_{r-1}=z^{a}\widetilde {\Phi }^t_{r-1}$ , this completes the proof of equation (3.9). Equation (3.10) can be easily obtained by shifting the parameters in the hypergeometric function in equation (3.9).

Theorem 3.6 Letting $z\in R$ and $a\in {\mathbb {C}}$ with $a\not \in {\mathbb {Z}}_{\leq 0}$ , we have

$$ \begin{align*} \Phi^t_{0}(x_{1},\ldots,x_{r+2}; z; a) =\frac{1}{\mathcal{P}_1(a)}& \Bigg\{ \sum_{j=0}^{r-1}A_{j} \sum_{l=0}^{j} \begin{pmatrix} a \\ j-l\\ \end{pmatrix}j!z^{a+l} \prod_{i=1}^{r+1}\dfrac{[a+\alpha_{i}]_{l}}{[ a +\beta_{i}+1]_{l}} \\ &\ {}_{r+2}F_{r+1}\left(\!\!\! \begin{array}{lcl} a+\alpha_{1}+l,&\hspace{-3ex}\ldots, &\hspace{-1.5ex} a+\alpha_{r+1}+l,\ l+1\\ a+\beta_1+l+1,&\hspace{-1.5ex} \ldots,&\hspace{-1.5ex} a+\beta_{r+1}+l+1 \end{array} ;z\right) \kern-1pt\!\Bigg\}, \end{align*} $$

where

$$ \begin{align*} A_{j}&=\sum_{i=j}^{r-1}(x_{r+2-i}-x_{1}x_{r+1-i})\displaystyle\genfrac{\{}{\}}{0pt}{}{i}{j}+x_{1}x_{2}\displaystyle\genfrac{\{}{\}}{0pt}{}{r-1}{j} \qquad (0\leq j\leq r-1), \end{align*} $$

$\genfrac {\{}{\}}{0pt}{}{n}{m}$ is the Stirling number of the second kind, and $\{\alpha _j\}$ and $\{\beta _j\}$ are defined in equation ( 3.5 ).

Proof Applying $\Phi ^t_{0}=\Phi ^t_{0}(x_{1},\ldots ,x_{r+2}; z; a)$ , the equation

$$ \begin{align*} \Phi^t_{0} &=\sum_{j=0}^{r-1}\dfrac{x_{r+2-j}-x_1x_{r+1-j}}{x_{r+2}}D^{j}y_{0} +\dfrac{x_1x_2}{x_{r+2}}D^{r-1}y_{0} =\sum_{j=0}^{r-1}\dfrac{A_{j}}{x_{r+2}}z^{j}\left(\dfrac{d}{dz}\right)^{j}y_{0}, \end{align*} $$

obtained in the same way as in Theorem 2.2 in [Reference Li11], and equation (3.9) demonstrate the theorem.

Note that the product on the right-hand side is computed as follows:

$$\begin{align*}\prod_{i=1}^{r+1}\dfrac{[a+\alpha_{i}]_{l}}{[ a +\beta_{i}+1]_{l}} =\frac{{\mathcal P}_2(a)\cdots {\mathcal P}_2(a+l-1)}{{\mathcal P}_1(a+1)\cdots {\mathcal P}_1(a+l)}. \end{align*}$$

4 Special cases and applications

4.1 The case of $r=1$

4.1.1 The case of height one: $h_1=1$

When $r=1$ , Theorems 3.5 and 3.6 reduce to

(4.1) $$ \begin{align} \hspace{-27pt}\Phi^t_{0}(x_{1}, x_2, x_{3}; z; a)&= \frac{x_{3} z^{a}}{\mathcal{P}_1(a)}\, {}_{3}F_{2}\left(\!\!\! \begin{array}{ll} a+\alpha_{1},&a+\alpha_{2},\ 1\\ a+\beta_1+1,&a+\beta_{2}+1 \end{array} ;z\right) \end{align} $$
(4.2) $$ \begin{align} &\qquad\qquad\qquad\qquad\, = \frac{x_{3} z^{a-1}}{\mathcal{P}_2(a-1)}\, \left\{{}_{3}F_{2}\left(\!\!\! \begin{array}{ll} a+\alpha_{1}-1,&a+\alpha_{2}-1,\ 1\\ a+\beta_1,&a+\beta_{2} \end{array} ;z\right)-1\right\} , \end{align} $$

where

$$ \begin{align*} {\mathcal{P}_1(X)}&=X^2-(x_1+tx_2)X-t(x_3-x_1x_2)=(X+\beta_1)(X+\beta_2), \\ {\mathcal{P}_2(X)}&=X^2-(x_1+(t-1)x_2)X-(t-1)(x_3-x_1x_2)=(X+\alpha_1)(X+\alpha_2). \end{align*} $$

To examine the terms of height one in $\Phi ^t_{0}(x_{1}, x_2, x_{3}; z; a)$ , dividing both sides of equation (4.1) (resp. equation (4.2)) by $x_3$ , and applying $x_3=0$ , ${\mathcal {P}_1(X)}=(X-x_1)(X-tx_2)$ and ${\mathcal {P}_2(X)}=(X-x_1)(X-(t-1)x_2)$ yield the following.

Proposition 4.1 We then have

$$ \begin{align*} &\sum_{k>d>0}\mathrm{Li}^t_{(k-d+1, \{1\}_{d-1})}(z; a)x_1^{k-d-1}x_2^{d-1}= \sum_{i, j>0}\mathrm{Li}^t_{(i+1, \{1\}_{j-1})}(z; a)x_1^{i-1}x_2^{j-1} \\&\quad= \frac{z^{a}}{(a-x_1)(a-tx_2)} {}_{3}F_{2}\left(\!\!\!\begin{array}{ll} a-x_1,&a-(t-1)x_2,\ 1\\ a-x_1+1,&a-tx_2+1 \end{array} ;z\right) \\&\quad= \frac{z^{a-1}}{(a-x_1-1)(a-(t-1)x_2-1)} \left\{{}_{3}F_{2}\left(\!\!\! \begin{array}{ll} a-x_1,&a-(t-1)x_2,\ 1\\ a-x_1+1,&a-tx_2+1 \end{array} ;z\right)-1\right\}. \end{align*} $$

In particular, when $z=1$ , we have

(4.3) $$ \begin{align} &\sum_{i, j>0}\zeta^t(i+1, \{1\}_{j-1}; a)x_1^{i-1}x_2^{j-1}\notag \\ &\quad= \frac{1}{(a-x_1)(a-tx_2)} {}_{3}F_{2}\left(\!\!\! \begin{array}{ll} a-x_1,&a-(t-1)x_2,\ 1\\ a-x_1+1,&a-tx_2+1 \end{array} ;1\right) \end{align} $$
(4.4) $$ \begin{align} &= \frac{1}{(a-x_1-1)(a-(t-1)x_2-1)} \left\{{}_{3}F_{2}\left(\!\!\! \begin{array}{ll} a-x_1-1,&a-(t-1)x_2-1,\ 1\\ a-x_1,&a-tx_2 \end{array} ;1\right)-1\right\}. \end{align} $$

By substituting $a=0$ and $t=0$ into equations (4.3) and (4.4), respectively, we obtain the following:

(4.5) $$ \begin{align} 1+\sum_{i, j>0}\zeta(i+1, \{1\}_{j-1})x_1^{i}x_2^{j} &=1+ \frac{x_1x_2}{1-x_1} {}_{3}F_{2}\left( \begin{array}{ll} -x_1+1,&x_2+1,\ 1\\ -x_1+2,&2 \end{array} ;1\right) \end{align} $$
(4.6) $$ \begin{align} &= {}_{2}F_{1}\left( \begin{array}{l} -x_1, \ x_2\\ -x_1+1 \end{array} ;1\right)=\frac{\Gamma(1-x_1)\Gamma(1-x_2)}{\Gamma(1-x_1-x_2)}. \end{align} $$

The left-hand side of equation (4.5) should be symmetric in $x_1$ and $x_2$ based on the duality for MZVs of height 1; that is, $\zeta (i+1, \{1\}_{j-1})=\zeta (j+1, \{1\}_{i-1})$ for $i, j\in {\mathbb {N}}$ . The corresponding symmetry on the right-hand side of equation (4.5) is given by equation (4.6) or is given directly by the known transformation (see e.g., [Reference Gasper and Rahman6])

$$ \begin{align*} {}_{3}F_{2}\left( \begin{array}{l} b_1, b_2, b_3\\ c_1, c_2 \end{array} ;1\right)= \frac{\Gamma(c_2)\Gamma(c_1+c_2-b_1-b_2-b_3)}{\Gamma(c_2-b_1)\Gamma(c_1+c_2-b_2-b_3)} {}_{3}F_{2}\left( \begin{array}{l} b_1, \ c_1-b_2, \ c_1-b_2\\ c_1, c_1+c_2-b_2-b_3 \end{array} ;1\right) \end{align*} $$

for ${}_{3}F_{2}$ , where $\Re (c_2-b_1)>0$ and $\Re (c_1+c_2-b_1-b_2-b_3)>0$ when substituting

$$ \begin{align*} (b_1, b_2, b_3, c_1, c_2)=(1, 1-x_1, 1+x_2, 2, 2-x_1). \end{align*} $$

4.1.2 Application to t-values

The generating functions associated with the “t-values” defined below are described by hypergeometric functions ${}_{3}F_{2}$ as a simple application of Proposition 4.1.

Let $a=p/q\in {\mathbb {Q}}$ with $-1<a\le 0$ , where p and q are coprime integers, and $q>0$ . We introduce a variant of MZVs for an admissible index ${\boldsymbol k}\in {\mathbb {N}}^d$ as

$$ \begin{align*} t^{(p/q)}({\boldsymbol k})=t^{(p/q)}(k_1,\ldots, k_d)=\sum_{m_1>\cdots >m_d>0 \atop \forall m_i\equiv p\ (\mathrm{mod} q)} \frac{1}{m_1^{k_1}\cdots m_d^{k_d}}. \end{align*} $$

Note that

$$ \begin{align*} t^{(p/q)}({\boldsymbol k})&=\sum_{m_1>\cdots >m_d>0} \frac{1}{(qm_1+p)^{k_1}\cdots (qm_d+p)^{k_d}}\\ &=q^{-\mathrm{wt} ({\boldsymbol k})}\sum_{m_1>\cdots >m_d>0} \frac{1}{(m_1+\frac{p}{q})^{k_1}\cdots (m_d+\frac{p}{q})^{k_d}} =q^{-\mathrm{wt} ({\boldsymbol k})}\zeta({\boldsymbol k}; p/q). \end{align*} $$

By substituting $z=1$ , $a=p/q$ , $t=0$ , $qx_1=X_1$ , and $qx_2=X_2$ into Proposition 4.1, we obtain the following.

Proposition 4.2

$$ \begin{align*} \sum_{i, j>0}t^{(p/q)}(i+1, \{1\}_{j-1})X_1^{i-1}X_2^{j-1} = \frac{1}{(p+q-X_1)(p+q)} {}_{3}F_{2}\kern-1pt\left( \!\!\kern-1pt\begin{array}{ll} \frac{p+q-X_1}{q},&\frac{p+q+X_2}{q},\ 1\\ \frac{p+2q-X_1}{q},&\frac{p+2q}{q} \end{array} ;1\kern-1pt\right). \end{align*} $$

For example, cases $q\le 4$ are given by

$$ \begin{align*} \sum_{i, j>0}t^{(-1/2)}(i+1, \{1\}_{j-1})X_1^{i-1}X_2^{j-1} &= \frac{1}{1-X_1} {}_{3}F_{2}\left( \begin{array}{ll} \frac{1-X_1}{2},&\frac{1+X_2}{2},\ 1\\[4pt] \frac{3-X_1}{2},&\frac{3}{2} \end{array} ;1\right), \\[4pt] \sum_{i, j>0}t^{(-1/3)}(i+1, \{1\}_{j-1})X_1^{i-1}X_2^{j-1} &= \frac{1}{2(2-X_1)} {}_{3}F_{2}\left( \begin{array}{ll} \frac{2-X_1}{3},&\frac{2+X_2}{3},\ 1\\[4pt] \frac{5-X_1}{3},&\frac{5}{3} \end{array} ;1\right), \\[4pt] \sum_{i, j>0}t^{(-2/3)}(i+1, \{1\}_{j-1})X_1^{i-1}X_2^{j-1} &= \frac{1}{1-X_1} {}_{3}F_{2}\left( \begin{array}{ll} \frac{1-X_1}{3},&\frac{1+X_2}{3},\ 1\\[4pt] \frac{4-X_1}{3},&\frac{4}{3} \end{array} ;1\right), \\[4pt] \sum_{i, j>0}t^{(-1/4)}(i+1, \{1\}_{j-1})X_1^{i-1}X_2^{j-1} &= \frac{1}{3(3-X_1)} {}_{3}F_{2}\left( \begin{array}{ll} \frac{3-X_1}{4},&\frac{3+X_2}{4},\ 1\\ \frac{7-X_1}{4},&\frac{7}{4} \end{array} ;1\right), \\[4pt] \sum_{i, j>0}t^{(-3/4)}(i+1, \{1\}_{j-1})X_1^{i-1}X_2^{j-1} &= \frac{1}{1-X_1} {}_{3}F_{2}\left( \begin{array}{ll} \frac{1-X_1}{4},&\frac{1+X_2}{4},\ 1\\[4pt] \frac{5-X_1}{4},&\frac{5}{4} \end{array} ;1\right). \end{align*} $$

4.2 The case of ${\boldsymbol k}=(n, \ldots , n)$

Letting $n\in {\mathbb {N}}$ with $n\ge 2$ , when $r=n-1$ , we substitute $x_1=\cdots =x_n=0$ into equations (3.9) and (3.10).

Theorem 4.3 For $n\ge 2$ , we have

(4.7) $$ \begin{align} \sum_{d=1}^{\infty} \mathrm{Li}^t_{(\{n\}_{d})}(z; a)x_{n+1}^d &= \frac{x_{n+1}}{\mathcal{P}_1(a)}\,z^{a}\, {}_{n+1}F_{n}\left(\!\!\!\begin{array}{lcl} a+\alpha_{1},&\hspace{-2ex}\ldots, &\hspace{-1.5ex} a+\alpha_{n},\ 1\\ a+\beta_1+1,&\hspace{-1.5ex} \ldots,&\hspace{-1.5ex} a+\beta_{n}+1 \end{array}\!\!;z\right) \notag \\[4pt]&=\frac{x_{n+1}}{\mathcal{P}_2(a-1)}\,z^{a-1} \left\{{}_{n+1}F_{n}\left( \!\!\!\begin{array}{lcl} a+\alpha_{1}-1,&\hspace{-2ex}\ldots, &\hspace{-1.5ex} a+\alpha_{n}-1,\ 1\\ a+\beta_1,&\hspace{-1.5ex} \ldots,&\hspace{-1.5ex} a+\beta_{n} \end{array}\!\!;z\right)-1\!\right\}, \end{align} $$

where

$$ \begin{align*} \mathcal{P}_1(X)=X^{n}-tx_{n+1}=\prod_{j=1}^n(X+\beta_j), \quad \mathcal{P}_2(X)=X^{n}-(t-1)x_{n+1}=\prod_{j=1}^n(X+\alpha_j). \end{align*} $$

As an alternative proof, equation (4.7) can be proven by computing the coefficients of the hypergeometric series on the right-hand side. Although the computation is slightly complicated, it is worth noting. For $m\ge 1$ , the following holds:

$$ \begin{align*} \prod_{j=1}^n \frac{[a+\alpha_j-1]_m}{[a+\beta_j]_m} &=\prod_{i=0}^{m-1} \prod_{j=1}^n\frac{a+\alpha_j+i-1}{a+\beta_j+i} =\prod_{i=0}^{m-1} \frac{{\mathcal P}_2(a+i-1)}{{\mathcal P}_1(a+i)}\\[4pt] &=\prod_{i=0}^{m-1} \frac{(a+i-1)^n-(t-1)x_{n+1}}{(a+i)^n-tx_{n+1}}\\[4pt] &=\frac{(a-1)^n-(t-1)x_{n+1}}{(a+m-1)^n-tx_{n+1}}\prod_{i=0}^{m-2} \left(1+\frac{x_{n+1}}{(a+i)^n-tx_{n+1}}\right). \end{align*} $$

Hence, we have

$$ \begin{align*} \mbox{RHS of equation (23) } &=\sum_{m>0}\frac{x_{n+1}}{(a+m-1)^n-tx_{n+1}} \prod_{i=0}^{m-2} \left(1+\frac{x_{n+1}}{(a+i)^n-tx_{n+1}}\right)z^{m+a-1}\\ &=\sum_{m>0}\sum_{\ell=1}^{m} \sum_{m-1=m_1>\cdots >m_{\ell} \ge0} \prod_{s=1}^{\ell} \frac{x_{n+1}}{(a+m_s)^n-tx_{n+1}} z^{m+a-1}\\ &=\sum_{\ell>0}\sum_{m_1>\cdots >m_{\ell} \ge0} \prod_{s=1}^{\ell} \frac{x_{n+1}}{(a+m_s)^n-tx_{n+1}} z^{m_1+a}\\ &=\sum_{\ell>0}\sum_{m_1>\cdots >m_{\ell} \ge0} \prod_{s=1}^{\ell} \sum_{d_s>0} \frac{t^{d_s-1}x_{n+1}^{d_s}}{(a+m_s)^{nd_s}}z^{m_1+a}\\ &=\sum_{\ell>0}\sum_{d\ge \ell}\sum_{d_1+\cdots +d_{\ell}=d}\sum_{m_1>\cdots >m_{\ell} \ge0} \prod_{s=1}^{\ell} \frac{1}{(a+m_s)^{nd_s}}z^{m_1+a}t^{d-\ell}x_{n+1}^{d}\\ &=\sum_{d>0}\sum_{\ell=1}^{d}\sum_{d_1+\cdots +d_{\ell}=d}\mathrm{Li}_{(nd_1,\ldots, nd_{\ell})}(z; a) t^{d-\ell}x_{n+1}^{d}\\ &=\sum_{d>0}\mathrm{Li}^t_{(\{n\}_d)}(z; a)x_{n+1}^{d}=\mbox{LHS of (23)}. \end{align*} $$

Acknowledgment

The authors would like to thank the anonymous referees for providing their valuable comments, which were used to improve this paper.

Footnotes

The first author is supported by JSPS KAKENHI Grant Number 18K03260.

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