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A study of elliptic gamma function and allies

Research in the Mathematical Sciences20185:39

https://doi.org/10.1007/s40687-018-0158-9

  • Received: 30 December 2017
  • Accepted: 4 May 2018
  • Published:

Abstract

We study analytic and arithmetic properties of the elliptic gamma function
$$\begin{aligned} \prod _{m,n=0}^\infty \frac{1-x^{-1}q^{m+1}p^{n+1}}{1-xq^mp^n}, \quad |q|,|p|<1, \end{aligned}$$
in the regime \(p=q\), in particular, its connection with the elliptic dilogarithm and a formula of S. Bloch. We further extend the results to more general products by linking them to non-holomorphic Eisenstein series and, via some formulae of D. Zagier, to elliptic polylogarithms.

Keywords

  • Theta function
  • Elliptic gamma function
  • Elliptic dilogarithm
  • Elliptic polylogarithm

1 Introduction

For complex z and \(\tau \) with \({\text {Im}}\tau >0\), set \(x=e^{2\pi iz}\) and \(q=e^{2\pi i\tau }\). Transformation properties of the so-called short theta function
$$\begin{aligned} \theta _0(z;\tau ):=\prod _{m=0}^\infty (1-x^{-1}q^{m+1})(1-xq^m) \end{aligned}$$
under the action of the modular group are well understood. In view of its transparent invariance under translation \(\tau \mapsto \tau +1\), the main source of the modular action originates from the \(\tau \)-involution
$$\begin{aligned} z\mapsto {\hat{z}}=\frac{z}{\tau }, \quad \tau \mapsto {\hat{\tau }}=-\,\frac{1}{\tau }. \end{aligned}$$
(1)
The related classical transformation of \(\theta _0(z;\tau )\) can be recorded as
$$\begin{aligned} q^{1/12}x^{-1/2}\theta _0(z;\tau ) =ie^{-\pi iz{\hat{z}}}{\hat{q}}^{1/12}{\hat{x}}^{-1/2}\theta _0({\hat{z}};{\hat{\tau }}) \end{aligned}$$
(2)
(see, for example, [3, Section 2]), where we define \({\hat{x}}=e^{2\pi i{\hat{z}}}\) and \({\hat{q}}=e^{2\pi i{\hat{\tau }}}\).
Less is known about modular properties of the related product
$$\begin{aligned} \theta _1(z;\tau ):=\prod _{m=0}^\infty \frac{(1-x^{-1}q^{m+1})^{m+1}}{(1-xq^m)^m}, \end{aligned}$$
which naturally comes as the \(\sigma =\tau \) specialisation of the elliptic gamma function
$$\begin{aligned} \Gamma (z;\tau ,\sigma ):=\prod _{m,n=0}^\infty \frac{1-x^{-1}q^{m+1}p^{n+1}}{1-xq^mp^n}, \quad \text {where}\; p=e^{2\pi i\sigma }, \end{aligned}$$
introduced by Ruijsenaars [5] (see also [3, 4]). Namely, we have
$$\begin{aligned} \theta _1(z;\tau ) =\theta _0(z;\tau )\Gamma (z;\tau ,\tau )=\Gamma (z+\tau ;\tau ,\tau ). \end{aligned}$$
A known functional equation of the elliptic gamma function [3, Theorem 4.1] represents an \({\text {SL}}_3({\mathbb {Z}})\) symmetry of \(\Gamma (z;\tau ,\sigma )\). The problem of determining its behaviour in the regime \(\sigma =\tau \) under \({\text {SL}}_2({\mathbb {Z}})\) transformations is specifically addressed in [2], where the (logarithm of the) infinite product is related to the elliptic dilogarithm via a formula of S. Bloch [1].

Our principal aim in this note is recasting analytic and arithmetic (modular) properties of the function \(\theta _1(z;\tau )\) and its relatives, in particular, linking them to non-holomorphic Eisenstein series and the elliptic dilogarithm. This programme is carried out in Sects. 24; it gives a new proof of Bloch’s formula and related results from [2]. In Sect. 5 we go further to discuss similar features of products that generalise ones for \(\theta _0\) and \(\theta _1\); their relationship with non-holomorphic Eisenstein series and formulae from [7] allow us to link them to elliptic polylogarithms.

For future record, notice that iterating the transformation \((z,\tau )\mapsto ({\hat{z}},{\hat{\tau }})\) twice maps \((z,\tau )\) to \((-z,\tau )\) and that
$$\begin{aligned} \theta _1(-z;\tau )=\frac{1}{\theta _1(z;\tau )} \quad \text {and}\quad \theta _0(-z;\tau )=-\,x^{-1}\theta _0(z;\tau ). \end{aligned}$$
(3)

2 Period functions

A natural way of measuring failure of weight k modular behaviour under the transformation \((z,\tau )\mapsto ({\hat{z}},{\hat{\tau }})\) for a function \(f(z,\tau )\) is through the period function
$$\begin{aligned} g(z,\tau )=g_k(z,\tau ):=f({\hat{z}},{\hat{\tau }})-\tau ^k f(z,\tau ). \end{aligned}$$

Lemma 1

We have
$$\begin{aligned} \tau ^k g({\hat{z}},{\hat{\tau }})+(-1)^k g(z,\tau ) =\tau ^k\bigl (f(-z,\tau )-(-1)^kf(z,\tau )\bigr ). \end{aligned}$$

Observe that the expression in the parentheses on the right-hand side measures the failure of k-parity of \(f(z,\tau )\).

Proof

We only use \(({\hat{{\hat{z}}}},{\hat{{\hat{\tau }}}})=(-z,\tau )\) and \(\tau {\hat{\tau }}=-\,1\):
$$\begin{aligned} \tau ^k g({\hat{z}},{\hat{\tau }})- g(z,\tau )&=\tau ^k\bigl (f(-z,\tau )-{{\hat{\tau }}}^kf({\hat{z}},{\hat{\tau }})\bigr )+(-1)^k\bigl (f({\hat{z}},{\hat{\tau }})-\tau ^k f(z,\tau )\bigr )\\&=\tau ^k\bigl (f(-z,\tau )-(-1)^kf(z,\tau )\bigr ). \end{aligned}$$
\(\square \)

The lemma and the parity relation for \(\ln \theta _1(z;\tau )\) in (3) imply the following.

Lemma 2

The function
$$\begin{aligned} T(z;\tau )=\tau \ln \theta _1(z;\tau )-\ln \theta _1({\hat{z}};{\hat{\tau }}) \end{aligned}$$
(4)
satisfies the functional equation
$$\begin{aligned} T({\hat{z}};{\hat{\tau }})=\tau ^{-1}T(z;\tau ). \end{aligned}$$
Furthermore, we can relate the function \(T(z;\tau )\) to the dilogarithm function
$$\begin{aligned} {\text {Li}}_2(x)=-\,\int _0^x\ln (1-t)\,\frac{{\mathrm d}t}{t}. \end{aligned}$$

Lemma 3

The function (4) admits the following representation:
$$\begin{aligned} T(z;\tau )&=\frac{\pi i(\tau -2z)(1+2\tau z-2z^2)}{12\tau } +z\ln \theta _0(z;\tau )\\&\quad -\frac{1}{2\pi i}\sum _{m=0}^\infty \bigl ({\text {Li}}_2(x^{-1}q^{m+1})-{\text {Li}}_2(xq^m)\bigr ). \end{aligned}$$

Proof

As shown in the proof of Theorem 5.2 in [3],
$$\begin{aligned} \ln \theta _1(z;\tau )&=\ln \theta _0(z;\tau )+\ln \Gamma (z;\tau ,\tau )\\&=-\,\pi i\lambda (z;\tau )+\ln \frac{\theta _0(z;\tau )}{\theta _0({\hat{z}};{\hat{\tau }})}\\&\quad +({\hat{\tau }}-{\hat{z}})\sum _{k=1}^\infty \frac{({\hat{x}}^{-1}{\hat{q}})^k}{k(1-{\hat{q}}^k)} -{\hat{z}}\sum _{k=1}^\infty \frac{{\hat{x}}^k}{k(1-{\hat{q}}^k)}\\&\quad +\frac{1}{2\pi i}\sum _{k=1}^\infty \frac{{\hat{x}}^k-({\hat{x}}^{-1}{\hat{q}})^k}{k^2(1-{\hat{q}}^k)} -{\hat{\tau }}\sum _{k=1}^\infty \frac{{\hat{q}}^k({\hat{x}}^k-({\hat{x}}^{-1}{\hat{q}})^k)}{k(1-{\hat{q}}^k)^2}, \end{aligned}$$
where
$$\begin{aligned} \lambda (z;\tau )=\frac{z^3}{3\tau ^2}-\frac{2\tau -1}{2\tau ^2}\,z^2+\frac{(\tau -1) (5\tau -1)}{6\tau ^2}\,z-\frac{(\tau -2)(2\tau -1)}{12\tau } \end{aligned}$$
and the assumptions \(|{\hat{x}}|,|{\hat{x}}^{-1}{\hat{q}}|<1\) are made to ensure convergence. (The latter can be dropped in the final result by appealing to the analytic continuation in z.) Recalling the transformation (2), using
$$\begin{aligned} \frac{1}{1-{\hat{q}}^k}=\sum _{m=0}^\infty {\hat{q}}^{mk} \quad \text {and}\quad \frac{{\hat{q}}^k}{(1-{\hat{q}}^k)^2}=\sum _{m=0}^\infty m{\hat{q}}^{mk}, \end{aligned}$$
interchanging summation and summing over k, we obtain
$$\begin{aligned} \ln \theta _1(z;\tau )&=-\,\pi i\biggl (\lambda (z;\tau )-\frac{1}{2}+\frac{z^2}{\tau }+\frac{\tau }{6}-z+\frac{1}{6\tau }+\frac{z}{\tau }\biggr )\\&\quad +\,{\hat{z}}\sum _{m=0}^\infty \bigl (\ln \left( 1-{\hat{x}}^{-1}{\hat{q}}^{m+1}\right) +\ln \left( 1-{\hat{x}}{\hat{q}}^m\right) \bigr )\\&\quad -{\hat{\tau }}\sum _{m=0}^\infty \bigl ((m+1)\ln \left( 1-{\hat{x}}^{-1}{\hat{q}}^{m+1}\right) -m\ln \left( 1-{\hat{x}}{\hat{q}}^m\right) \bigr )\\&\quad -\frac{1}{2\pi i}\sum _{m=0}^\infty \bigl ({\text {Li}}_2\left( {\hat{x}}^{-1}{\hat{q}}^{m+1}\right) -{\text {Li}}_2\left( {\hat{x}}{\hat{q}}^m\right) \bigr )\\&=\frac{\pi i}{12}\biggl ((1+2z)-\frac{2z(1+z)(1+2z)}{\tau ^2}\biggr ) +{\hat{z}}\ln \theta _0\left( {\hat{z}};{\hat{\tau }}\right) -{\hat{\tau }}\ln \theta _1\left( {\hat{z}};{\hat{\tau }}\right) \\&\quad -\frac{1}{2\pi i}\sum _{m=0}^\infty \bigl ({\text {Li}}_2\left( {\hat{x}}^{-1}{\hat{q}}^{m+1}\right) -{\text {Li}}_2\left( {\hat{x}}{\hat{q}}^m\right) \bigr ). \end{aligned}$$
(This formula can be alternatively derived from logarithmically differentiating identity (2) with respect to \(\tau \) and further integrating the result with respect to z.) Substituting \((z/\tau ,-1/\tau )\) for \((z,\tau )\) translates the result into
$$\begin{aligned} \tau \ln \theta _1(z;\tau )-\ln \theta _1\left( {\hat{z}};{\hat{\tau }}\right)&=\frac{\pi i(\tau -2z)(1+2\tau z-2z^2)}{12\tau } +z\ln \theta _0(z;\tau )\\&\quad -\frac{1}{2\pi i}\sum _{m=0}^\infty \bigl ({\text {Li}}_2\left( x^{-1}q^{m+1}\right) -{\text {Li}}_2(xq^m)\bigr ), \end{aligned}$$
the desired relation. \(\square \)

3 Non-holomorphic modularity

Denote
$$\begin{aligned} A=A(z,\tau ):=\frac{z-\overline{z}}{\tau -\overline{\tau }}\in {\mathbb {R}}, \end{aligned}$$
so that
$$\begin{aligned} {\hat{A}}=A({\hat{z}},{\hat{\tau }}):=\frac{z\overline{\tau }-\overline{z}\tau }{\tau -\overline{\tau }}\in {\mathbb {R}} \end{aligned}$$
and \(z=A\tau -{\hat{A}}\). Define
$$\begin{aligned} Q(z;\tau ):=q^{B_3(A)/3}\prod _{m=0}^\infty \frac{\left( 1-xq^m\right) ^{m+A}}{\left( 1-x^{-1} q^{m+1}\right) ^{m+1-A}} =\frac{q^{B_3(A)/3}\theta _0(z;\tau )^A}{\theta _1(z;\tau )}, \end{aligned}$$
(5)
where \(B_3(t):=t^3-\frac{3}{2}t^2+\frac{1}{2}t\) is the third Bernoulli polynomial, \(B_3(1-t)=-\,B_3(t)\), and
$$\begin{aligned} F_+(z;\tau ):=\ln Q({\hat{z}};{\hat{\tau }})-\tau \ln Q(z;\tau ), \quad F_-(z;\tau ):=\ln \overline{Q({\hat{z}};{\hat{\tau }})}-\tau \ln \overline{Q(z;\tau )}. \end{aligned}$$
It follows then from Lemma 1 and the parity relations (3) that
$$\begin{aligned} \tau F_+({\hat{z}};{\hat{\tau }})-F_+(z;\tau )&=\tau \bigl (\ln Q(-z;\tau )+\ln Q(z;\tau )\bigr )\\&=\frac{2\pi i}{3}(B_3(-A)+B_3(A))\tau ^2+2\pi iA z\tau -\pi i A\tau \\&=-\,\pi iA\bigl (2(A\tau -z)+1\bigr )\tau =-\,\pi i A(2{\hat{A}}+1)\tau \end{aligned}$$
and
$$\begin{aligned} \tau F_-({\hat{z}};{\hat{\tau }})-F_-(z;\tau )&=\tau \bigl (\ln \overline{Q(-z;\tau )}+\ln \overline{Q(z;\tau )}\bigr )\\&=-\,\frac{2\pi i}{3}(B_3(-A)+B_3(A))\tau \overline{\tau }-2\pi iA\overline{z}\tau +\pi i A\tau \\&=\pi iA\bigl (2(A\overline{\tau }-\overline{z})+1\bigr )\tau =\pi iA(2{\hat{A}}+1)\tau . \end{aligned}$$
We summarise our finding in the following claim.

Lemma 4

We have
$$\begin{aligned} \tau F_+\left( {\hat{z}};{\hat{\tau }}\right) -F_+(z;\tau )&=-\,\pi i A\left( 2{\hat{A}}+1\right) \tau ,\\ \tau F_-\left( {\hat{z}};{\hat{\tau }}\right) -F_-(z;\tau )&=\phantom {+}\pi iA\left( 2{\hat{A}}+1\right) \tau . \end{aligned}$$

Lemma 3 leads to the following expansions of the functions \(F_+\) and \(F_-\).

Theorem 1

We have
$$\begin{aligned} F_+(z;\tau )&=S(z,\tau )-\frac{1}{2\pi i}\,L(z,\tau ),\\ F_-(z;\tau )&=-\,\frac{2\pi i\overline{\tau }(\tau -\overline{\tau })}{3}B_3(A)+\overline{S(z,\tau )}+\frac{1}{\pi }\, \overline{U(z,\tau )}+\frac{1}{2\pi i}\,\overline{L(z,\tau )}, \end{aligned}$$
where
$$\begin{aligned} L(z,\tau )&:=\sum _{m=0}^\infty \bigl ({\text {Li}}_2\left( x^{-1}q^{m+1}\right) -{\text {Li}}_2(x q^m)\bigr ),\\ U(z,\tau )&:=\sum _{m=0}^\infty \bigl (\ln |x^{-1}q^{m+1}|\,{\text {Li}}_1\left( x^{-1}q^{m+1}\right) -\ln |xq^{m}|\,{\text {Li}}_1(x q^{m})\bigr ),\\ S(z,\tau )&:=\frac{-\,\pi i}{12}(2A-1)\left( 6z^2-12 A\tau z+6z+8 A^2\tau ^2-2 A\tau ^2-6A\tau +1\right) . \end{aligned}$$

Proof

For \(F_+\) substitute the expression of \(T(z;\tau )\) from Lemma 3 into the computation
$$\begin{aligned} F_+(z;\tau )&=\ln Q\left( {\hat{z}};{\hat{\tau }}\right) -\tau \ln Q(z;\tau )\\&=\frac{2\pi i}{3}\bigl (B_3({\hat{A}}){\hat{\tau }}-B_3(A)\tau ^2)+{\hat{A}}\ln \theta _0 \left( {\hat{z}};{\hat{\tau }}\right) -\left( {\hat{A}}+z\right) \ln \theta _0(z;\tau )\\&\quad +\tau \ln \theta _1(z;\tau )-\ln \theta _1\left( {\hat{z}};{\hat{\tau }}\right) . \end{aligned}$$
This leads to the formula
$$\begin{aligned} F_+(z;\tau ) =S(z,\tau )-\frac{1}{2\pi i}\,L(z,\tau ) \end{aligned}$$
with
$$\begin{aligned} S(z,\tau )&=\frac{2\pi i}{3}\bigl (B_3({\hat{A}}){\hat{\tau }}-B_3(A)\tau ^2\bigr ) +{\hat{A}}\pi i\biggl (\frac{\tau }{6} -\frac{{\hat{\tau }}}{6}+z{\hat{z}}-\frac{1}{2}-z+ {\hat{z}}\biggr )\\&\quad +\frac{\pi i}{12\tau }(\tau -2z)(1+2\tau z-2z^2), \end{aligned}$$
and the latter simplifies to the expression given in the statement of Theorem 1 by elementary manipulation.
For \(F_-\) we proceed as follows. We have
$$\begin{aligned} \ln Q(z;\tau )=\frac{2\pi i\tau B_3(A)}{3}-\sum _{m=0}^\infty \bigl ((m+1-A){\text {Li}}_1(x^{-1}q^{m+1})-(m+A){\text {Li}}_1(x q^{m})\bigr ). \end{aligned}$$
Multiply this expression by \(\tau -\overline{\tau }=2i{\text {Im}}\tau \) and use \(A(\tau -\overline{\tau })=2i{\text {Im}}z\) to get
$$\begin{aligned} (\tau -\overline{\tau })\ln Q(z;\tau )=\frac{2\pi i\tau (\tau -\overline{\tau }) B_3(A)}{3}-\frac{1}{\pi }\, U(z,\tau ). \end{aligned}$$
Now, notice
$$\begin{aligned} \overline{(\tau -\overline{\tau })\ln Q(z;\tau )}=F_-(z;\tau )-\overline{F_+(z;\tau )} \end{aligned}$$
to deduce the expression for \(F_-\) as in the theorem. \(\square \)
A consequence of this expansion is the invariance of
$$\begin{aligned} F(z;\tau ):=\frac{F_+(z;\tau )+F_-(z;\tau )}{2} =\ln |Q({\hat{z}};{\hat{\tau }})|-\tau \ln |Q(z;\tau )| \end{aligned}$$
under translation \(\tau \mapsto \tau +1\).

Lemma 5

We have
$$\begin{aligned} F_+(z;\tau +1)-F_+(z;\tau )=-\,\bigl (F_-(z;\tau +1)-F_-(z;\tau )\bigr ). \end{aligned}$$

Proof

The functions \(L(z,\tau )\) and \(U(z,\tau )\) (hence their complex conjugates) are clearly invariant under translation \(\tau \mapsto \tau +1\). The result follows from noticing that
$$\begin{aligned} 2 {\text {Re}}S(z,\tau )+ \frac{2\pi i\overline{\tau }(\tau -\overline{\tau })}{3}\,B_3(A)&=\frac{-\,\pi i (\tau -\overline{\tau })^2 A(1-A)(1-2A)}{6}\\&=\frac{-\,\pi i (\tau -\overline{\tau })^2}{3}\,B_3(A) \end{aligned}$$
is also invariant under the transformation. \(\square \)

We summarise the results in this section as follows.

Theorem 2

The weight 1 period function
$$\begin{aligned} F(z;\tau )&=\ln |Q({\hat{z}};{\hat{\tau }})|-\tau \ln |Q(z;\tau )| \\&=\frac{1}{2\pi }\sum _{m=0}^\infty \bigl (\ln |x^{-1}q^{m+1}|\,\overline{{\text {Li}}_1\left( x^{-1}q^{m+1}\right) } -\ln |xq^{m}|\,\overline{{\text {Li}}_1(x q^{m})}\,\bigr )\\&\quad -\frac{\pi i(\tau -\overline{\tau })^2}{6}\,B_3(A) -\frac{1}{2\pi i}\,{\text {Im}}\sum _{m=0}^\infty \bigl ({\text {Li}}_2(x^{-1}q^{m+1})-{\text {Li}}_2(x q^m)\bigr ) \end{aligned}$$
of \(\ln |Q(z;\tau )|\) satisfies
$$\begin{aligned} \tau F({\hat{z}};{\hat{\tau }})=F(z;\tau ) \quad \text {and}\quad F(z;\tau )=F(z;\tau +1). \end{aligned}$$
In other words, it behaves like a Jacobi form of weight 1 on \({\text {SL}}_2({\mathbb {Z}})\).

4 Elliptic dilogarithm

Theorem 2 provides a natural link between the period function \(F(z;\tau )\) and the elliptic dilogarithm [7]
$$\begin{aligned} D(q;x):=\sum _{m\in {\mathbb {Z}}}D(xq^m)=\sum _{m=0}^\infty \bigl (D(xq^m)-D(x^{-1}q^{m+1})\bigr ) \end{aligned}$$
together with its companion
$$\begin{aligned} J(q;x):=\sum _{m=0}^\infty \bigl (J(xq^m)-J(x^{-1}q^{m+1})\bigr ) +\frac{\log ^2|q|}{3}\,B_3\biggl (\frac{\log |x|}{\log |q|}\biggr ), \end{aligned}$$
where
$$\begin{aligned} D(x):=\ln |x|\,\arg (1-x)+{\text {Im}}{\text {Li}}_2(x) =-\,\ln |x|\,{\text {Im}}{\text {Li}}_1(x)+{\text {Im}}{\text {Li}}_2(x) \end{aligned}$$
denotes the Bloch–Wigner dilogarithm and
$$\begin{aligned} J(x):=\ln |x|\,\ln |1-x|=-\,\ln |x|\,{\text {Re}}{\text {Li}}_1(x) \end{aligned}$$
its companion. Namely, the expansion in the theorem can be stated as
$$\begin{aligned} F(z;\tau )=\frac{1}{2\pi i}\bigl (D(q;x)+iJ(q;x)\bigr ). \end{aligned}$$
(6)
This is essentially the result discussed in [2, Section 1].
Viewing now z as an element of the lattice \({\mathbb {R}}+{\mathbb {R}}\tau \), so that A and \({\hat{A}}\) in the representation \(z=-\,{\hat{A}}+A\tau \) are fixed, we find out that the \(\tau \)-derivative
$$\begin{aligned} \frac{1}{2\pi i}\,\frac{{\mathrm d}}{{\mathrm d}\tau }\ln Q(z;\tau ) =q\frac{{\mathrm d}}{{\mathrm d}q}\ln Q(z;\tau ) \end{aligned}$$
is the Eisenstein series
of weight 3, where the notation \(\sum '\) indicates omitting the term \(m=n=0\) from the summation. Integrating we obtain
implying
This is equation (7) in [2]. Since \({\hat{z}}=z/\tau =A-{\hat{A}}/\tau =A+{\hat{A}}{\hat{\tau }}\), it follows that
implying
The latter is a (non-holomorphic) modular form of weight 1, and combined with equation (6) is the formula of Bloch mentioned previously.

Theorem 3

(Bloch’s formula [1, 2, 7]) For \(z=A\tau -{\hat{A}}\), we have

5 General weight

A natural generalisation of the product in (5) is
$$\begin{aligned} Q_k(z;\tau ):=q^{B_{k+2}(A)/(k+2)}\prod _{m=0}^\infty (1-xq^m)^{(m+A)^k}(1-x^{-1}q^{m+1})^{(-1)^k(m+1-A)^k}, \end{aligned}$$
(7)
where \(k=0,1,2,\dots \) and \(B_k(t)\) stands for the kth Bernoulli polynomial. Then \(Q_0(z;\tau )\) is an arithmetic normalisation of the short theta function \(\theta _0(z;\tau )\) (a Siegel modular unit) and \(Q_1(z;\tau )\) coincides with (5). Following the earlier recipe, define
$$\begin{aligned} F_+(z;\tau )&=F_{k,+}(z;\tau ):=\ln Q_k({\hat{z}};{\hat{\tau }})-\tau ^{k-2}\ln Q_k(z;\tau ),\\ F_-(z;\tau )&=F_{k,-}(z;\tau ):=\ln \overline{Q_k({\hat{z}};{\hat{\tau }})}-\tau ^{k-2}\ln \overline{Q_k(z;\tau )} \end{aligned}$$
and \(F_k(z;\tau ):=\frac{1}{2}\bigl (F_{k,+}(z;\tau )+F_{k,-}(z;\tau )\bigr )\). Then from Lemma 1 we deduce the following generalisation of Lemma 4.

Lemma 6

We have, for \(k\ge 1\),
$$\begin{aligned} \tau ^kF_+({\hat{z}};{\hat{\tau }})+(-1)^kF_+(z;\tau )&=\phantom {+}(-1)^k \pi iA^k(2{\hat{A}}+1)\tau ^k,\\ \tau ^kF_-({\hat{z}};{\hat{\tau }})+(-1)^kF_-(z;\tau )&=-\,(-1)^k \pi iA^k(2{\hat{A}}+1)\tau ^k. \end{aligned}$$

Proof

Apply Lemma 1 and the relation
$$\begin{aligned} B_{k+2}(-t)-(-1)^k B_{k+2}(t)=(-1)^k(k+2)t^{k+1}. \end{aligned}$$
\(\square \)

We further use that the \(\tau \)-derivative of \(\ln Q_k(z;\tau )\) is an Eisenstein series.

Lemma 7

For \(k\ge 1\),
where \(z=-\,{\hat{A}}+A\tau \).

Proof

Consider \({\tilde{Q}}_k(A,{\hat{A}};\tau ):=Q_k(A\tau -{\hat{A}};\tau )\) as a function of real variables \(A,{\hat{A}}\) and complex variable \(\tau \). The \(\tau \)-derivative
$$\begin{aligned} G_{k+2}(A,{\hat{A}};\tau ) :=\frac{1}{2\pi i}\,\frac{{\mathrm d}}{{\mathrm d}\tau }\ln Q_k(A,{\hat{A}};\tau ) =q\frac{{\mathrm d}}{{\mathrm d}q}\ln Q_k(A,{\hat{A}};\tau ) \end{aligned}$$
is seen to be the Eisenstein series
of weight \(k+2\). This is true for \(k=1\) (see Sect. 4), while for \(k\ge 1\) we observe the functional equation
$$\begin{aligned} \frac{\partial }{\partial {\hat{A}}}E_{k+3}(A,{\hat{A}};\tau ) =\frac{\partial }{\partial \tau }E_{k+2}(A,{\hat{A}};\tau ). \end{aligned}$$
The equality \(G_{k+2}(A,{\hat{A}};\tau )=E_{k+2}(A,{\hat{A}};\tau )\) then follows by induction on k using the fact that the constant terms of both functions at \(\tau =\infty \) (or \(q=0\)) agree.
Integrating we obtain
Since both sides continuously depend on A and \({\hat{A}}\), the formula remains valid also for \(\ln Q_k(z;\tau )\). \(\square \)
As in our computation in Sect. 4 we obtain
and
Thus,
where
(8)
for positive integers a and b.

Finally, observe that the non-holomorphic Eisenstein series (8) can be identified with the elliptic polylogarithms using a formula of Zagier [7, Proposition 2]. This leads to the following general result.

Theorem 4

For \(k\ge 1\) and \(z=A\tau -{\hat{A}}\), we have
$$\begin{aligned} \ln |Q_k({\hat{z}};{\hat{\tau }})|-\tau ^k\ln |Q_k(z;\tau )| =\frac{i\,k!}{(4\pi {\text {Im}}\tau )^k}\sum _{j=1}^k\tau ^{k-j}{\text {Im}}(\tau ^j)\,D_{j+1,k-j+1}(q;x), \end{aligned}$$
where
$$\begin{aligned} D_{a,b}(q;x) =\sum _{m=0}^\infty \bigl (D_{a,b}(xq^m)+(-1)^{a+b}D_{a,b}(x^{-1}q^{m+1})\bigr ) +\frac{(4\pi {\text {Im}}\tau )^{a+b-1}}{(a+b)!}\,B_{a+b}(A) \end{aligned}$$
and
$$\begin{aligned} D_{a,b}(x)&=(-1)^{a-1}\sum _{\ell =a}^{a+b-1}2^{a+b-\ell -1}\left( {\begin{array}{c}\ell -1\\ a-1\end{array}}\right) \frac{(-\ln |x|)^{a+b-\ell -1}}{(a+b-\ell -1)!}\,{\text {Li}}_\ell (x)\\&\quad +(-1)^{b-1}\sum _{\ell =b}^{a+b-1}2^{a+b-\ell -1}\left( {\begin{array}{c}\ell -1\\ b-1\end{array}}\right) \frac{(-\ln |x|)^{a+b-\ell -1}}{(a+b-\ell -1)!}\,\overline{{\text {Li}}_\ell (x)}. \end{aligned}$$

6 Conclusion

This final (and very short!) part is devoted to highlighting some directions for further research.

In spite of generalisability of the story in Sects. 24 to the function
$$\begin{aligned} F_k(z;\tau ) =\ln |Q_k({\hat{z}};{\hat{\tau }})|-\tau ^k\ln |Q_k(z;\tau )|, \end{aligned}$$
where \(k\ge 1\) and the product \(Q_k(z;\tau )\) is defined in (7), the case \(k=1\) remains the only one, which is invariant under translation \(\tau \mapsto \tau +1\). At the same time, Lemma 6 implies the transformation
$$\begin{aligned} \tau ^kF_k({\hat{z}},{\hat{\tau }})=(-1)^{k-1}F_k(z,\tau ) \quad \text {for}\; k=1,2,\cdots . \end{aligned}$$
This consideration does not exclude, however, a possibility for modified products (7) and related functions \(F_k\) to exist such that the latter ones have true modular behaviour for each \(k\ge 1\). It sounds to us a nice problem to determine such modular objects.

Several arithmetic problems related to the case \(k=1\) (originating from the elliptic gamma function) are still open. Our personal favourites include connection of (5) with the Mahler measure and mirror symmetry; see, for example, observation in [6].

Notes

Declarations

Acknowledgements

We thank the anonymous referees for their careful reading of the manuscript and reporting valuable feedback. The first author was partially supported by a grant of Romanian Ministry of Research and Innovation, CNCS – UEFISCDI, Project Number PN-III-P4-ID-PCE-2016-0157, within PNCDI III7. The second author is partially supported by Laboratory of Mirror Symmetry NRU HSE, RF Government Grant, Ag. No. 14.641.31.0001.

Conflict of interest

On behalf of all authors, the corresponding author Wadim Zudilin states that there is no conflict of interest.

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Simion Stoilow Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania
(2)
Department of Mathematics, IMAPP, Radboud University, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands
(3)
School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan, NSW, 2308, Australia
(4)
Laboratory of Mirror Symmetry and Automorphic Forms, National Research University Higher School of Economics, 6 Usacheva str., 119048 Moscow, Russia

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