Abstract
Ohno and Zagier (Indag Math 12:483–487, 2001) found that a generating function of sums of multiple polylogarithms can be written in terms of the Gauss hypergeometric function \({}_2F_1\). As a generalization of the Ohno and Zagier formula, Ihara et al. (Can J Math 76:1–17, 2022) showed that a generating function of sums of interpolated multiple polylogarithms of Hurwitz type can be expressed in terms of the generalized hypergeometric function \({}_{r+1}F_r\). In this paper, we establish q- and elliptic analogues of this result. We introduce elliptic q-multiple polylogarithms of Hurwitz type and study a generating function of sums of them. By taking the trigonometric and classical limits in the main theorem, we can obtain q- and elliptic generalizations of the Ihara, Kusunoki, Nakamura and Saeki formula.
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1 Introduction
The Gauss hypergeometric function \({}_2F_1\) is a special function with various interesting properties and important applications. The function \({}_2F_1\) is defined by
with
For \({\varvec{k}}=(k_1, \dots , k_l) \in {{\mathbb {Z}}}_{> 0}^l\) and \( a \in {{\mathbb {C}}}\) with \(|a|<1\), the multiple polylogarithm is defined by
In [20], Ohno and Zagier found that a generating function of sums of multiple polylogarithms can be expressed in terms of the Gauss hypergeometric function. Since then, various generalizations and analogous results have been obtained for the Ohno and Zagier formula. See for example [3,4,5, 8, 11, 14,15,16, 19, 21, 22].
In this paper, we focus on Ihara, Kusunoki, Nakamura and Saeki’s work [8]. They generalized the Ohno and Zagier’s result to interpolated multiple polylogarithms of Hurwitz type. The multiple polylogarithm of Hurwitz type is defined by
and its interpolated version is defined by
where we put \(c_0(\alpha )=\frac{1}{\alpha -1}\) and the sum on the right-hand side of (1.1) runs over all \(\widetilde{\varvec{k}}\) of the form \(\widetilde{\varvec{k}}=(k_1 \Box \cdots \Box k_l) \) in which each \(\Box \) filled by two candidates “,” or “\(+\)”. The expression \(\textrm{dep}\, \widetilde{\varvec{k}}\) denotes the number of the components of \(\widetilde{\varvec{k}}\).
It is shown in [8] that a generating function of sums of \(L_{\varvec{k}}(a,\alpha ,x)\) of fixed weight, depth and all i-heights (see Sect. 3.1) can be written in terms of the generalized hypergeometric function \({}_{r+1}F_r\). The function \({}_{r+1}F_r\) is defined by
The purpose of this paper is to establish q- and elliptic analogues of the Ihara, Kusunoki, Nakamura and Saeki’s result. To treat both cases, we introduce elliptic q-multiple polylogarithms of Hurwitz type, denoted by \(L_{\varvec{k}}(a,\alpha ,\beta ;p,q)\), and study a generating function of sums of \(L_{\varvec{k}}(a,\alpha ,\beta ;p,q)\). By taking the trigonometric and classical limits in the main theorem (Theorem 4.9), we can obtain q- and elliptic generalizations of Theorem 3.5 of [8]. In particular, we show that a generating function of q-multiple polylogarithms of Hurwitz type can be written in terms of the q-hypergeometric function \({}_{r+1}\phi _{r}\). (See Theorem 5.3.)
The rest of this paper is organized as follows. In Sect. 2, we define elliptic q-multiple polylogarithms of Hurwitz type and establish basic properties of them that will be used in the later sections. In Sect. 3, we introduce generating functions \(\Phi _j(a;p,q)\) of sums of \(L_{\varvec{k}}(a,\alpha ,\beta ;p,q)\) with fixed weight, depth and i-height and deduce q-difference equations satisfied by them. In Sect. 4, we solve the q-difference equation (3.1) satisfied by \(\Phi _{r-1}(a;p,q)\) by generalizing the idea used in the previous work [11]. In [11], we used Heine’s transformation formula ( [7])
In this paper, Kajihara’s transformation formula for multiple basic hypergeometric series [9], which is a generalization of (1.2), plays important roles. In Sect. 5, we establish q- and elliptic analogues of the result due to Ihara, Kusunoki, Nakamura and Saeki [8] described above. When \(\beta =1\), the q-analogue is essentially the same as Theorem 5.3 of Li-Wakabayashi [16].
2 q- and elliptic analogues of multiple polylogarithms of Hurwitz type
In this section, we introduce elliptic q-multiple polylogarithms of Hurwitz type and establish their basic properties.
2.1 q-Analogues of Apostol–Bernoulli Polynomials
Let q be a complex number with \(0<|q|<1\). For \(a \in {{\mathbb {C}}}\), we put
and define the theta function \(\theta (a;q)\) by
For \(a, \beta \in {{\mathbb {C}}}\), we define the Kronecker function \(F(a,\beta ;q)\) by
The Kronecker function has the following Laurent series expansion (see [23]):
For \(k \in {{\mathbb {Z}}}_{\ge 0}\), we define the function \(\psi _k (a, \beta ;q)\) by
By (2.1), the function \(\psi _k(a,\beta ;q)\) has the following Laurent series expansion:
For \(k=0\), we put \(\psi _0(a,\beta ;q)=-1\).
In [12], we showed that \(\psi _k(a;q):=\lim _{\beta \rightarrow 1} \psi _k(a,\beta ;q)\) can be regarded as q-analogues of periodic Bernoulli polynomials. We will show that the classical limits (the limits as \(q \rightarrow 1\)) of \(\psi _k(a,\beta ;q)\) are given by Apostol-Bernoulli polynomials introduced in [2]. The Apostol-Bernoulli polynomials \(B_k(x,w)\) are defined by
When \(w=1\), \(B_n(x,1):=B_n(x)\) are classical Bernoulli polynomials.
For \(-1<x<1, \ s \in {{\mathbb {R}}} \setminus {{\mathbb {Z}}}\) and \( k \in {{\mathbb {Z}}}_{\ge 0}\), we now put
where \(\{s \}\) denotes the fractional part of s. Then we have the following proposition:
Proposition 2.1
Let \(k \in {{\mathbb {Z}}}_{>0}\). Then, for \(0<t<k\) and \(s \in {{\mathbb {R}}} {\setminus } {{\mathbb {Z}}}\),
Proof
The proof is similar to that of Proposition 4.2 of [12] by using the Fourier series expansions of Apostol–Bernoulli polynomials given in [6, 17]. \(\square \)
2.2 Elliptic q-Multiple Polylogarithms of Hurwitz Type
Let p be a complex number satisfying \(0<|p|<1\). For \({\varvec{k}}=(k_i)_{i=1}^l \in {{\mathbb {Z}}}_{\ge 0}^l\), we define the elliptic q-multiple polylogarithm of Hurwitz type by
where \({{\mathbb {T}}}\) is the unit circle \( \{ z \in {{\mathbb {C}}}\mid |z|=1 \}\) traversed in the positive direction and \(t_i \ (i=1,\dots , l)\) are complex numbers satisfying the following conditions:
We put \(k:=\min \{k_i \mid k_i \ge 1\}\). Then the function \(L_{\varvec{k}}(a,\alpha ,\beta ;p,q)\) is holomorphic on the annulus \(|pq^k|<|a|<1\) as a function of the variable a.
Remark 2.2
-
(1)
When \(\beta =1\), the functions \(L_{\varvec{k}}(a,\alpha ,1;p,q)\) become elliptic q-multiple polylogarithms introduced and studied in [11, 13].
-
(2)
When \(k_1=0\), the function \(L_{(0,k_2,\ldots .k_l)}(a,\alpha ,\beta ;p,q)\) is constant as a function of a. This follows from a similar argument to that given in Remark 2.1 of [11].
Let \(T_a\) be the q-shift operator defined by
where f(a) is a function of the variable a. The functions \(L_{\varvec{k}}(a, \alpha , \beta ;p,q)\) satisfy the following q-difference relations:
Proposition 2.3
Let \({\varvec{k}}=(k_1,\ldots ,k_l) \in {{\mathbb {Z}}}_{\ge 0}^l\).
- (a):
-
For \(k_1 \ge 2\),
$$\begin{aligned} (1-\beta T_a)L_{\varvec{k}}(a, \alpha , \beta ;p,q)=L_{(k_1-1,k_2,\ldots ,k_l)} (a,\alpha , \beta ;p,q) . \end{aligned}$$ - (b):
-
For \(k_1=1\),
$$\begin{aligned} (1-\beta T_a)L_{\varvec{k}}(a, \alpha , \beta ;p,q)&=F(a,\alpha ;p)L_{(k_2,\ldots ,k_l)}(a,\alpha ,\beta ;p,q)\\&\quad +L_{(0,k_2,\ldots ,k_l)}(a,\alpha ,\beta ;p,q), \end{aligned}$$where we put
$$\begin{aligned} L_{(k_2,\ldots ,k_l)}(a,\alpha ,\beta ;p,q)=1 \end{aligned}$$for \(l=1\).
Proof
The proof is similar to that of Proposition 2.4 of [11]. \(\square \)
2.3 Trigonometric Limit of \(L_{\varvec{k}}(a,\alpha ,\beta ;p,q)\)
Hereafter we put
We define the depth of \({\varvec{k}}=(k_i)_{i=1}^l \in {{\mathbb {Z}}}_{ > 0}^l\) by \(\textrm{dep}\, {\varvec{k}}=l\).
For \({\varvec{k}} \in {{\mathbb {Z}}}_{>0}^l\), we define the functions \(L_{\varvec{k}}(a,\beta ;q)\) and \(L_{\varvec{k}}(a,\alpha ,\beta ;q)\) by
where the sum on the right-hand side of (2.3) runs over all \(\widetilde{\varvec{k}}\) of the form \(\widetilde{\varvec{k}}=(k_1 \Box \cdots \Box k_l) \) in which each \(\Box \) filled by two candidates “,” or “\(+\)”. Then we have
where \(L_{\varvec{k}}(a,\alpha ,x)\) is the interpolated multiple Hurwitz polylogarithm (1.1).
The trigonometric limit (the limit as \(p \rightarrow 0\)) of \(L_{\varvec{k}}(a,\alpha ,\beta ;p,q)\) is given by \(L_{\varvec{k}}(a,\alpha ,\beta ;q)\), as follows:
Proposition 2.4
For \({\varvec{k}} \in {{\mathbb {Z}}}_{>0}^l\), we have
Proof
The proof is similar to that of Proposition 2.2 of [11]. \(\square \)
2.4 Classical Limit of \(L_{\varvec{k}}(a,\alpha ,\beta ;p,q)\)
We put \(p=e^{2\pi \sqrt{-1} \tau }\) and denote the Kronecker function \(F(e^{2\pi \sqrt{-1} u}, e^{2\pi \sqrt{-1}v};p)\) by \(F(u,v;\tau )\). For \({\varvec{k}}=(k_i)_{i=1}^l \in {{\mathbb {Z}}}_{\ge 0}^l\) and \(u, v \in {{\mathbb {C}}}\) with \(0<\textrm{Im}\, u<\textrm{Im} \, \tau \), we define the function \(L_{\varvec{k}}(u,v,x;\tau )\) by
Then, by Proposition 2.1, the following proposition holds:
Proposition 2.5
Remark 2.6
When \(x=0\), the functions \((-1)^l L_{\varvec{k}}(u,v,0;\tau )\) are multiple elliptic polylogarithms introduced in [13].
3 Generating Function of Elliptic q-Multiple Polylogarithms of Hurwitz Type
In this section, we introduce generating functions of \(L_{\varvec{k}}(a,\alpha ,\beta ;p,q)\) and derive q-difference equations satisfied by them.
3.1 Sums of Fixed Weight, Depth, and i-Heights
For an index \( {\varvec{k}}=(k_1,\dots ,k_l) \in {{\mathbb {Z}}}_{> 0}^l\), we define the weight and i-height of \({\varvec{k}}\), respectively by
For nonnegative integers \(k, l, h_1, \dots , h_r \) and \( -2 \le j \le r-1\), we define the set of indices \(I_j(k,l,h_1,\dots ,h_r)\) by
for \(j \ge -1\) and
for \(j=-2\). We define the sum \(G_j(k,l,h_1,\dots ,h_r;a;p,q)\) by
Since \(G_{-2}(k,l,h_1,\dots ,h_r;a,\alpha ,\beta ;p,q)\) is constant as a function of the variable a, we omit the variable a to denote it:
For \(I_j(k,l,h_1,\dots ,h_r) =\emptyset \), we put
The functions \(G_j(k,l,h_1,\dots ,h_r;a;p,q)\) satisfy the following q-difference relations:
Proposition 3.1
- (a):
-
$$\begin{aligned}{} & {} (1-\beta T_a)G_{r-1}(k,l,h_1,\dots ,h_r;a,\alpha ,\beta ;p,q)\\{} & {} =G_{r-1}(k-1,l,h_1,\dots ,h_r;a,\alpha ,\beta ;p,q )\\{} & {} \quad +G_{r-2}(k-1,l,h_1,\dots ,h_{r-1},h_r-1;a,\alpha ,\beta ;p,q) \\{} & {} \quad -G_{r-1}(k-1,l,h_1,\dots ,h_{r-1},h_r-1;a,\alpha ,\beta ;p,q). \end{aligned}$$
- (b):
-
For \(0 \le j \le r-2\),
$$\begin{aligned}{} & {} (1-\beta T_a)(G_j(k,l,h_1,\dots ,h_r;a,\alpha ,\beta ;p,q)-G_{j+1}(k,l,h_1,\dots ,h_r;a,\alpha ,\beta ;p,q))\\{} & {} \quad =G_{j-1}(k-1,l,h_1,\dots ,h_j, h_{j+1}-1,h_{j+2},\dots ,h_r;a,\alpha ,\beta ;p,q)\\{} & {} \qquad -G_{j}(k-1,l,h_1,\dots ,h_j,h_{j+1}-1,h_{j+2},\dots ,h_r;a,\alpha ,\beta ;p,q). \end{aligned}$$ - (c):
-
$$\begin{aligned}{} & {} (1-\beta T_a)(G_{-1}(k,l,h_1,\dots ,h_r;a,\alpha ,\beta ;p,q)-G_0(k,l,h_1,\dots ,h_r;a,\alpha ,\beta ;p,q)) \\{} & {} \quad =F(a,\alpha ;p)G_{-1}(k-1,l-1,h_1,\dots ,h_r;a,\alpha ,\beta ;p,q)\\{} & {} \qquad +G_{-2}(k-1,l,h_1,\dots ,h_r;\alpha ,\beta ;p,q). \end{aligned}$$
Proof
The proof of this proposition is similar to that of Lemma 6 of [16]. Thus we omit the detailed proof. \(\square \)
3.2 Generating Functions of \(G_j\)
We now define the generating functions \(\Phi _j(a;p,q)\) of \(G_j\) by
We also write \(\Phi _{-2}(a;p,q)\) as \(\Phi _{-2}(p,q)\). Note that \(\Phi _{-2}(p,q)\) is constant as a function of the variable a.
The functions \(\Phi _j(a;p,q)\) satisfy the following q-difference system:
Proposition 3.2
- (a):
-
$$\begin{aligned}&(1-\beta T_a)\Phi _{r-1}(a;p,q)=x_1 \Phi _{r-1}(a;p,q)\\&+\frac{x_{r+2}}{x_{r+1}}(\Phi _{r-2}(a;p,q)-\Phi _{r-1}(a;p,q)-\delta _{r,1}), \end{aligned}$$
where \(\delta \) stands for Kronecker’s delta.
- (b):
-
For \(0 \le j \le r-2\),
$$\begin{aligned}&(1-\beta T_a)(\Phi _j(a;p,q)-\Phi _{j+1}(a;p,q))\\&=\frac{x_{j+3}}{x_{j+2}}(\Phi _{j-1}(a;p,q)-\Phi _j(a;p,q)-\delta _{j,0}). \end{aligned}$$ - (c):
-
$$\begin{aligned} (1-\beta T_a)(\Phi _{-1}(a;p,q)-\Phi _0(a;p,q))&=1-\beta +x_2F(a,\alpha ;p)\Phi _{-1}(a;p,q)\\&\,\,\quad +x_1\Phi _{-2}(p,q). \end{aligned}$$
Proof
The proof is similar to that of Proposition 7 of [16]. \(\square \)
We will now consider solving the q-difference system in Proposition 3.2 in a different way. We observe the following proposition:
Proposition 3.3
-
(1)
The functions \(\Phi _j(a;p,q) (-1 \le j \le r-1)\) can be written as a linear combination of \((1-\beta T_a)^i \Phi _{r-1}(a;p,q) \ (i=0,\ldots ,r)\) as follows:
$$\begin{aligned}&\Phi _j(a;p,q)=\frac{1}{x_{r+2}} \left\{ \sum _{i=0}^{r-2-j} (x_{r+2-i}-x_1 x_{r+1-i})(1-\beta T_a)^i\right. \\&\qquad \qquad \qquad \qquad \left. +x_{j+3} (1-\beta T_a)^{r-1-j} \right\} \Phi _{r-1} (a;p,q)+\delta _{j,-1}. \end{aligned}$$ -
(2)
The function \(\Phi _{r-1}(a;p,q)\) satisfies the following higher order q-difference equation:
$$\begin{aligned}{} & {} \{ (1-\beta T_a)^{r+1}-(x_1+x_2F(a,\alpha ;p))(1-\beta T_a)^r \nonumber \\{} & {} \qquad -F(a,\alpha ;p)\sum _{j=0}^{r-1}(x_{r+2-j}-x_1x_{r+1-j})(1-\beta T_a)^j \} \Phi _{r-1}(a;p,q)\nonumber \\{} & {} \quad =x_{r+2}F(a,\alpha ;p)+\frac{x_1x_{r+2}}{x_2}\Phi _{-2}(p,q). \end{aligned}$$(3.1)
Proof
We put \(y_0=\Phi _{r-1}(a;p,q), \ y_j=\Phi _{r-j-1}(a;p,q)-\Phi _{r-j}(a;p,q) \ (1 \le j \le r)\). Then Proposition 3.2 implies
By substituting the expressions (3.2) into
we obtain the claim (1). The proof of the claim (2) is similar to that of Corollary 8 (iii) of [16]. \(\square \)
Proposition 3.3 implies solving the q-difference system reduces to the Eq. (3.1).
4 Solving the q-Difference Equation (3.1)
In this section, we will solve the q-difference equation (3.1) based on a perturbation approach and the variation of parameters, which are also used in [11]. We first expand \(\Phi _{r-1}(a;p,q)\), \(\Phi _{-2}(p,q) \) and \( F(a,\alpha ;p)\) into power series of the parameter p in (3.1). We define the coefficients \(\phi _{0,n}(a;q), \ \phi _{-2,n}(q)\) of \(\Phi _{r-1}(a;p,q), \ \Phi _{-2}(p,q)\), respectively by
The coefficients of \(F(a,\alpha ;p)\) are given by the following:
Lemma 4.1
We define the Laurent polynomials \(s_n(a,\alpha )\) by
where \(\sum _{d \mid n}\) denotes the sum running over all positive divisors of n. Then, for \(|p|<|a|<1\) and \( |p|<|\alpha |<1\), we have
Proof
See Proposition 3.2 of [11]. \(\square \)
By (4.1), (4.2) and (4.3), comparing the coefficient of \(p^n\) on both sides of (3.1) yields the following non-homogeneous q-difference equation:
where we put
We will solve the equation (4.4) by using the method of variation of parameters. To do this, we introduce the q-hypergeometric function \({}_{r+1}\phi _r\). For \(n \in {{\mathbb {Z}}}_{\ge 0}\), we define the q-shifted factorial \((a;q)_n\) by
and denote the product \( (a_1;q)_n (a_2;q)_n \cdots (a_{m};q)_n\) by \((a_1,a_2,\dots ,a_m;q)_n\). The q-hypergeometric function \({}_{r+1}\phi _r \begin{bmatrix} \begin{matrix} a_1,\dots , a_{r+1} b_1, \dots , b_r \end{matrix} ;q,z \end{bmatrix}\) is defined by
In the following, we also use the notation
when one of the \(b_1,\dots ,b_{r+1}\) is q.
The homogeneous q-difference equation corresponding to (4.4) can be solved by using the function \({}_{r+1}\phi _r\), as follows:
Proposition 4.2
The solutions of the homogeneous q-difference equation
which have the form of
are given by the following:
where we put
Furthermore, \(\lambda =\lambda _i \ (i=1, \dots , r+1) \) are the solutions of the equation
and \(\rho _i \ (i=1,\dots , r+1) \) are the parameters determined by the following:
Proof
We put
By putting \(f(a)=a^{\lambda }\sum _{n=0}^{\infty } w_n a^n, \ w_0=1\) in (4.5) and then comparing the coefficients gives
and
for \(n \ge 1\). Since
we have
for \(\lambda =\lambda _i\) and thus obtain the proposition. \(\square \)
We now assume that the vector
is expressed, in terms of the vector
and the matrix
as follows:
Then we have the following:
Proposition 4.3
For \(i=1,\dots , r+1\), \(c_{n,i}(a)\) satisfies the following q-difference equation:
In the following, we put
Proposition 4.3 will be deduced from the following two lemmas:
Lemma 4.4
The determinant \(W(a) =\begin{vmatrix} \Psi (a) \end{vmatrix} \) can be expressed as follows:
Lemma 4.5
Let \(D_i(a)\) be the determinant of order r obtained by removing the r-th row and i-th column of \(\Psi (a)\):
Then \(D_i(a) \) can be expressed as follows:
Lemmas 4.4 and 4.5 can be proved by using the following Kajihara’s transformation formula for multiple basic hypergeometric series [9] (see also [10]):
Proposition 4.6
(Theorem 1.1 of [9]) For
we have
where \(\Delta ({\varvec{x}})=\Delta (x_1,\dots ,x_m)\) denotes the product of differences of \(x_1,\dots x_m\):
Proof of Lemma 4.4
The determinant W(a) can be written as follows:
Thus, by putting
in Proposition 4.6, we obtain the lemma. \(\square \)
Remark 4.7
In the proof of Lemma 4.4, we have used the case \(n=0\) of Proposition 4.6. This special case is due to Milne [18].
Proof of Lemma 4.5
We define the parameters \(z_1,\dots , z_r\) and \({\bar{\rho }}_1, \dots , {\bar{\rho }}_r\) by
and the functions \(g_j(a) \) by
Then we have
Thus, by putting
in Proposition 4.6, we have
which completes the proof. \(\square \)
We are now in a position to prove Proposition 4.3.
Proof of Proposition 4.3
We define the vector \(\textbf{r}_n(a)\) by
and matrix A by
Then the q-difference equation (4.4) can be represented as
Substituting (4.6) into (4.8) gives
By observing
the left-hand side of (4.9) can be written, as follows:
Thus we have
By Cramer’s rule, it holds that
Thus, by applying Lemmas 4.4 and 4.5, we finish the proof. \(\square \)
Proposition 4.3 implies that \(\Phi _{r-1}(a) \) satisfies the following integral equation:
Proposition 4.8
We put \({\widetilde{F}}(a,\alpha ;p):=F(a,\alpha ;p)-s_0(a,\alpha )\) and define the function K(a, b; q) by
Then, for \(|pq|<|a|<1\), we have the following:
Proof
By definition, the right-hand side of (4.7) can be expressed as
Meanwhile, by (4.6), we can put
By substituting these expressions into (4.7), we obtain
Thus it holds that
By observing
we find that
and thus obtain the proposition. \(\square \)
Let us consider solving the integral equation (4.10) in terms of formal power series in p. We rewrite the q-difference equation (3.1) as
The coefficients \(\xi _j(a)\) can be explicitly expressed, as follows:
For integers i, j, n with \(0 \le i \le r, \ 0 \le j \le r, \ n \ge 1 \), we define \(K_{i,j}(a,b;p,q)\) as
and \(K_{j}^{(n)}(a,b;p,q)\) as
Under the condition
\(K_{j}^{(n)}(a,b;p,q)\) is holomorphic as a function of b on the domain given by
It is observed that \(K_{j}^{(n)}(a,b;p,q)\) is a power series in p expressed as
For \(j=0,\dots , r\), we define \(\Gamma _j(a,b;p,q)\) as formal power series in p given by
Since
for \(N \ge 1\) and \(L \ge N-1\), we have
in \({{\mathbb {C}}}[[p]]\). For integers i with \( 0 \le i \le r \), we define \(h_i (a;p,q) \) by
Then \(h_i(a;p,q)\) are holomorphic on the domain given by \(|pq^{1-i}|<|a|<\min \{1, \ |w^{-1}|\}\).
The main theorem of this paper is as follows:
Theorem 4.9
Assume that the following condition holds:
We consider \(\Phi _{r-1}(a;p,q)\) and \(h_i(a;p,q) \ (i=0,\ldots ,r)\) as formal power series in p by (4.1) and (4.3). Then we have the following equality:
Proof
We first show that the following identity holds for \( 0 \le i \le r\) and \(|pq^{1-i}|<|a|<\min \{1, \ |w^{-1}| \}\):
When \(i=0\), (4.17) follows immediately from (4.10) and the definition of \(K_{0,j}(a,b;p,q)\).
We now assume that (4.17) holds for some i with \(0\le i \le r-1\). When \(|pq^{-i}|<|a|<1\), (4.17) implies
Since
for functions f(a, b) and g(a, b) of a and b, we have
By applying (4.13) to \((1-\beta T_z)^{r+1} \Phi _{r-1}(tz;p,q)\), we find that (4.17) holds for \(i+1\), which completes the proof of (4.17) for \(i=0,\ldots ,r\).
We next assume that the condition
holds and show that the following identity holds for \(n \in {{\mathbb {Z}}}_{>0}\):
When \(n=1\), (4.19) follows immediately from (4.17) with \(i=0\). We assume that (4.19) holds for some n. The condition (4.18) allows us to apply (4.17) to \((1-\beta T_{z})^j\Phi _{r-1}(tz;p,q)\) in (4.19) for n. By (4.17) and the induction hypothesis, we have
Since
it holds that
which implies that (4.19) holds for \(n+1\). Thus we find that (4.19) holds for all \(n \in {{\mathbb {Z}}}_{>0}\).
Finally we consider (4.19) as the equality of formal power series in p. By (4.15), letting \(n \rightarrow \infty \) in (4.19) gives (4.16). Thus we finish the proof of the theorem. \(\square \)
Remark 4.10
By the definition of the functions \(K_{j}^{(n)}(a,b;p,q)\), the formal power series \(\Gamma _i(a,b;p,q) \ (i=0,\ldots ,r)\) satisfy the following integral equations:
5 Trigonometric and Classical Limits of the Main Theorem
In this section, we examine what Theorem 4.9 can be reduced to by taking the trigonometric and classical limits.
5.1 Trigonometric Limit
For \(-1 \le j \le r-1\), we put
and define the function \(\Phi _j(a;q)\) by
Then, by Proposition 2.4, we have
for \(j \ge -1\). The limit of \(\Phi _{-2}(p,q)\) as \( p \rightarrow 0\) is given by the following:
Lemma 5.1
Proof
For \(t \in {{\mathbb {C}}}\) satisfying \(|p|<|at|<1\) and \( |q|<|t^{-1}|<1 \), we have
Thus we obtain
Hence the limit of \(\Phi _{-2}(p,q)\) can be calculated as
\(\square \)
As the trigonometric degeneration of (4.16), we obtain the following:
Theorem 5.2
Proof
By Lemma 5.1, comparing the constant terms of the formal power series in p on both sides of (4.16) yields
By observing
we obtain the theorem. \(\square \)
By applying Kajihara’s transformation (Proposition 4.6) to Theorem 5.2, Theorem 5.2 is equivalent to the following theorem:
Theorem 5.3
Proof
By applying Proposition 4.6 with
to \({}_{r+1}{\hat{\phi }}_r \begin{bmatrix} \begin{matrix} q^{\lambda _i} \rho _1, \dots , q^{\lambda _i}\rho _{r+1} \\ q^{\lambda _i-\lambda _1+1},\dots ,q^{\lambda _i-\lambda _{r+1}+1} \end{matrix} ;q, wa \end{bmatrix}\) on the right-hand side of (5.1), we have
where we have used the identity
We now recall Lagrange’s interpolation formula: For a polynomial p(x) of degree less than or equal to r,
By putting \(p(x)=x^r, \ \ y_i=q^{\nu _i-\lambda _i} \ (1 \le i \le r+1)\) in (5.3) and then divide the both sides by \(\prod _{k=1}^{r+1} (x-q^{\nu _k-\lambda _k})\), we have
Setting \(x=q^{-1}\) in (5.4) gives
Thus we find that
By putting
in Proposition 4.6, we see that
where we have used the identity (5.2) again. Hence we obtain the theorem. \(\square \)
Since
Theorem 5.3 with \(\beta =1\) is essentially the same as Theorem 9 of [16].
We will show that Theorem 5.3 can be considered to be a q-generalization of Theorem 3.5 of [8]. For nonnegative integers \(k.k,h_1,\ldots ,h_r\), we put
and define the function \(\Phi _{r-1}(a)\) by
We put
Then, by (2.4), we have
For \(n \in {{\mathbb {Z}}}_{\ge 0}\), we define the shifted factorial \((a)_n\) by
and denote the product \( (a_1)_n (a_2)_n \cdots (a_{m})_n\) by \((a_1,a_2,\dots ,a_m)_n\). The generalized hypergeometric function \({}_{r+1}F_r \begin{bmatrix} \begin{matrix} a_1,\dots , a_{r+1} \\ b_1, \dots , b_r \end{matrix} ;z \end{bmatrix}\) is defined by
The following limit formula holds:
We also use the notation
when one of the \(b_1,\dots ,b_{r+1}\) is 1.
Let \({\widetilde{\rho }}_i \) and \( {\widetilde{\lambda }}_i \ (i=1,\ldots ,r+1)\) be complex numbers defined by the following relations:
By Theorems 5.3, (5.6) and (5.7), we have the following theorem.
Theorem 5.4
Theorem 5.4 is essentially the same as Theorem 3.5 of [8]. Meanwhile, by Theorem 5.2, we obtain the following theorem.
Theorem 5.5
By using the Gauss summation formula (Theorem 2.2.2 of [1])
and the Pfaff–Saalschutz identity (Corollary 3.3.5 of [1])
the formula (5.10) with \(r=1, \ c_0(\alpha )=-1, x=1\) and \(a=1\) can be written as follows:
The expression (5.11) is given in [4].
5.2 Classical Limit
Let u be a complex variable with \(0<\textrm{Im}\, u<\textrm{Im}\, \tau \). For \(-2 \le j \le r-1\), we put
and define the function \(\Phi _j(u;\tau )\) by
We also write \(\Phi _{-2}(u;\tau )\) as \(\Phi _{-2}(\tau )\). Then, by Proposition 2.5, we have
where \({\widetilde{x}}_1,\ldots ,{\widetilde{x}}_{r+2}\) are parameters defined as (5.5). By proposition 3.3 (2) and (5.12), \(\Phi _{r-1}(u;\tau )\) satisfies the following differential equation:
We now put
where \({\widetilde{\rho }}_i \) and \({\widetilde{\lambda }}_i \ (i=1,\ldots ,r+1)\) are parameters defined as (5.8) and (5.9).
For integers k, j, n with \(0 \le k \le r, \ 0 \le j \le r\) and \( n \ge 1 \), we define the functions \(K_{i,j}(u,s;\tau )\) and \(K_{j}^{(n)}(u,s;\tau )\) as follows:
It is observed that \(K_{j}^{(n)}(a,b;p,q)\) is a power series in p expressed as
We also put
for \(j=0,\dots , r\).
For integers i with \( 0 \le i \le r \), we define the functions \(h_i (u;\tau ) \) by
By Theorem 4.9, we obtain the following theorem:
Theorem 5.6
We consider \(\Phi _{r-1}(u;\tau )\) and \(h_i(u;\tau ) \ (i=0,\ldots ,r)\) as formal power series in p. Then we have the following equality:
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Kato, M. Generating Function of q- and Elliptic Multiple Polylogarithms of Hurwitz Type. Math Phys Anal Geom 27, 9 (2024). https://doi.org/10.1007/s11040-024-09480-1
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DOI: https://doi.org/10.1007/s11040-024-09480-1
Keywords
- Multiple polylogarithms
- Interpolated multiple Hurwitz polylogarithms
- q-Analogue
- Elliptic analogue
- Generating function
- Elliptic q-multiple polylogarithms