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. 2–4; 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. 2–4 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].