A Study of Elliptic Gamma Function and Allies Vicen¸Tiu Pa¸Sol1 and Wadim Zudilin2,3,4*

Paşol and Zudilin Res Math Sci (2018) 5:39 https://doi.org/10.1007/s40687-018-0158-9

RESEARCH A study of elliptic gamma function and allies Vicen¸tiu Pa¸sol1 and Wadim Zudilin2,3,4*

*Correspondence: [email protected]; Abstract [email protected]; [email protected] We study analytic and arithmetic properties of the elliptic gamma function 2Department of Mathematics, ∞ 1 − x−1qm+1pn+1 IMAPP, Radboud University, P.O. , |q|, |p| < 1, Box 9010, 6500 GL Nijmegen, 1 − xqmpn The Netherlands m,n=0 Full list of author information is = available at the end of the article 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 Dedicated to Don Zagier, in admiration of his insights on them to non-holomorphic Eisenstein series and, via some formulae of D. Zagier, to modular, elliptic and elliptic polylogarithms. polylogarithmic functions. Keywords: Theta function, Elliptic gamma function, Elliptic dilogarithm, Elliptic polylogarithm

1 Introduction For complex z and τ with Im τ>0, set x = e2πiz and q = e2πiτ . Transformation properties of the so-called short theta function ∞ −1 m+1 m θ0(z; τ):= (1 − x q )(1 − xq ) m=0 under the action of the modular group are well understood. In view of its transparent invariance under translation τ → τ + 1, the main source of the modular action originates from the τ-involution

z 1 → ˆ = τ → τ =− z z τ , ˆ τ . (1)

The related classical transformation of θ0(z; τ) can be recorded as

1/12 −1/2 −πizzˆ 1/12 −1/2 q x θ0(z; τ) = ie qˆ xˆ θ0(zˆ;ˆτ)(2)

(see, for example, [3, Section 2]), where we deﬁnex ˆ = e2πizˆ andq ˆ = e2πiτˆ . Less is known about modular properties of the related product

∞ (1 − x−1qm+1)m+1 θ (z; τ):= , 1 (1 − xqm)m m=0 © The Author(s) 2018. This 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, 123 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.

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which naturally comes as the σ = τ specialisation of the elliptic gamma function ∞ −1 m+1 n+1 1 − x q p π σ (z; τ, σ ):= , where p = e2 i , 1 − xqmpn m,n=0 introduced by Ruijsenaars [5](seealso[3,4]). Namely, we have

θ1(z; τ) = θ0(z; τ)(z; τ, τ) = (z + τ; τ, τ). A known functional equation of the elliptic gamma function [3, Theorem 4.1] represents

an SL3(Z) symmetry of (z; τ, σ ). The problem of determining its behaviour in the regime σ = τ under SL2(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 θ1(z; τ) 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 θ0 and θ1; their relationship with non-holomorphic Eisenstein series and formulae from [7]allowustolinkthemto elliptic polylogarithms. For future record, notice that iterating the transformation (z, τ) → (z,ˆ τˆ)twicemaps (z, τ)to(−z, τ) and that

1 −1 θ1(−z; τ) = and θ0(−z; τ) =−x θ0(z; τ). (3) θ1(z; τ)

2 Period functions A natural way of measuring failure of weight k modular behaviour under the transformation (z, τ) → (z,ˆ τˆ) for a function f (z, τ) is through the period function k g(z, τ) = gk (z, τ):= f (z,ˆ τˆ) − τ f (z, τ).

Lemma 1 We have τ k g(z,ˆ τˆ) + (−1)k g(z, τ) = τ k f (−z, τ) − (−1)k f (z, τ) .

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

Proof We only use (z,ˆ τˆ) = (−z, τ)andττˆ =−1: τ k g(z,ˆ τˆ) − g(z, τ) = τ k f (−z, τ) − τˆk f (z,ˆ τˆ) + (−1)k f (z,ˆ τˆ) − τ k f (z, τ) = τ k f (−z, τ) − (−1)k f (z, τ) .

The lemma and the parity relation for ln θ1(z; τ)in(3) imply the following.

Lemma 2 The function

T(z; τ) = τ ln θ1(z; τ) − ln θ1(zˆ;ˆτ)(4)

satisﬁes the functional equation − T(zˆ;ˆτ) = τ 1T(z; τ). Paşol and Zudilin Res Math Sci (2018) 5:39 Page 3 of 11 39

Furthermore, we can relate the function T(z; τ) to the dilogarithm function x dt Li2(x) =− ln(1 − t) . 0 t Lemma 3 The function (4) admits the following representation:

πi(τ − 2z)(1 + 2τz − 2z2) T(z; τ) = + z ln θ (z; τ) 12τ 0 ∞ 1 − + − Li (x 1qm 1) − Li (xqm) . 2πi 2 2 m=0

Proof As shown in the proof of Theorem 5.2 in [3],

ln θ1(z; τ) = ln θ0(z; τ) + ln (z; τ, τ) θ (z; τ) =−πiλ(z; τ) + ln 0 θ0(zˆ;ˆτ) ∞ ∞ (ˆx−1qˆ)k xˆk + (ˆτ − zˆ) − zˆ k(1 − qˆk ) k(1 − qˆk ) k=1 k=1 ∞ ∞ 1 xˆk − (ˆx−1qˆ)k qˆk (ˆxk − (ˆx−1qˆ)k ) + − τˆ , 2πi k2(1 − qˆk ) k(1 − qˆk )2 k=1 k=1

where z3 2τ − 1 (τ − 1)(5τ − 1) (τ − 2)(2τ − 1) λ(z; τ) = − z2 + z − 3τ 2 2τ 2 6τ 2 12τ and the assumptions |xˆ|, |xˆ−1qˆ| < 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 ∞ ∞ 1 qˆk = qˆmk and = mqˆmk , 1 − qˆk (1 − qˆk )2 m=0 m=0 interchanging summation and summing over k,weobtain 1 z2 τ 1 z ln θ (z; τ) =−πi λ(z; τ) − + + − z + + 1 2 τ 6 6τ τ ∞ − + + zˆ ln 1 − xˆ 1qˆm 1 + ln (1 − xˆqˆm) m=0 ∞ − + − τˆ (m + 1) ln 1 − xˆ 1qˆm 1 − m ln (1 − xˆqˆm) m=0 ∞ 1 −1 m+1 m − Li2 xˆ qˆ − Li2 (xˆqˆ ) 2πi m=0 πi 2z(1 + z)(1 + 2z) = (1 + 2z) − + zˆ ln θ (zˆ;ˆτ) − τˆ ln θ (zˆ;ˆτ) 12 τ 2 0 1 ∞ 1 − + − Li xˆ 1qˆm 1 − Li (xˆqˆm) . 2πi 2 2 m=0 39 Page 4 of 11 Paşol and Zudilin Res Math Sci (2018) 5:39

(This formula can be alternatively derived from logarithmically diﬀerentiating identity (2) with respect to τ and further integrating the result with respect to z.) Substituting (z/τ, −1/τ) for (z, τ) translates the result into

πi(τ − 2z)(1 + 2τz − 2z2) τ ln θ (z; τ) − ln θ (zˆ;ˆτ) = + z ln θ (z; τ) 1 1 12τ 0 ∞ 1 − + − Li x 1qm 1 − Li (xqm) , 2πi 2 2 m=0

the desired relation.

3 Non-holomorphic modularity Denote − = τ = z z ∈ R A A(z, ): τ − τ , so that zτ − zτ ˆ = ˆ τ = ∈ R A A(z, ˆ): τ − τ and z = Aτ − Aˆ . Deﬁne

∞ m m+A B (A)/3 A / (1 − xq ) q 3 θ (z; τ) τ = B3(A) 3 = 0 Q(z; ): q + − , (5) − −1 m+1 m 1 A θ (z; τ) m=0 1 x q 1

= 3 − 3 2 + 1 − =− where B3(t): t 2 t 2 t is the third Bernoulli polynomial, B3(1 t) B3(t), and

F+(z; τ):= ln Q(zˆ;ˆτ) − τ ln Q(z; τ),F−(z; τ):= ln Q(zˆ;ˆτ) − τ ln Q(z; τ).

It follows then from Lemma 1 and the parity relations (3) that τF+(zˆ;ˆτ) − F+(z; τ) = τ ln Q(−z; τ) + ln Q(z; τ) π 2 i 2 = (B3(−A) + B3(A))τ + 2πiAzτ − πiAτ 3 =−πiA 2(Aτ − z) + 1 τ =−πiA(2Aˆ + 1)τ

and τF−(zˆ;ˆτ) − F−(z; τ) = τ ln Q(−z; τ) + ln Q(z; τ) 2πi =− (B3(−A) + B3(A))ττ − 2πiAzτ + πiAτ 3 = πiA 2(Aτ − z) + 1 τ = πiA(2Aˆ + 1)τ.

We summarise our ﬁnding in the following claim.

Lemma 4 We have τF+ (zˆ;ˆτ) − F+(z; τ) =−πiA 2Aˆ + 1 τ, τF− (zˆ;ˆτ) − F−(z; τ) = πiA 2Aˆ + 1 τ.

Lemma 3 leads to the following expansions of the functions F+ and F−. Paşol and Zudilin Res Math Sci (2018) 5:39 Page 5 of 11 39

Theorem 1 We have 1 F+(z; τ) = S(z, τ) − L(z, τ), 2πi 2πiτ(τ − τ) 1 1 F−(z; τ) =− B (A) + S(z, τ) + U(z, τ) + L(z, τ), 3 3 π 2πi where ∞ −1 m+1 m L(z, τ):= Li2 x q − Li2(xq ) , m=0 ∞ −1 m+1 −1 m+1 m m U(z, τ):= ln |x q | Li1 x q − ln |xq | Li1(xq ) , m=0 − πi S(z, τ):= (2A − 1) 6z2 − 12Aτz + 6z + 8A2τ 2 − 2Aτ 2 − 6Aτ + 1 . 12

Proof For F+ substitute the expression of T(z; τ) from Lemma 3 into the computation

F+(z; τ) = ln Q (zˆ;ˆτ) − τ ln Q(z; τ) 2πi = B (Aˆ )ˆτ − B (A)τ 2) + Aˆ ln θ (zˆ;ˆτ) − Aˆ + z ln θ (z; τ) 3 3 3 0 0 + τ ln θ1(z; τ) − ln θ1 (zˆ;ˆτ) .

This leads to the formula 1 F+(z; τ) = S(z, τ) − L(z, τ) 2πi with 2πi τ τˆ 1 S(z, τ) = B (Aˆ )ˆτ − B (A)τ 2 + Aˆ πi − + zzˆ − − z + zˆ 3 3 3 6 6 2 πi + (τ − 2z)(1 + 2τz − 2z2), 12τ and the latter simpliﬁes to the expression given in the statement of Theorem 1 by elemen- tary manipulation. For F− we proceed as follows. We have ∞ 2πiτB (A) − + ln Q(z; τ) = 3 − (m + 1 − A)Li (x 1qm 1) − (m + A)Li (xqm) . 3 1 1 m=0 Multiply this expression by τ − τ = 2i Im τ and use A(τ − τ) = 2i Im z to get 2πiτ(τ − τ)B (A) 1 (τ − τ)lnQ(z; τ) = 3 − U(z, τ). 3 π Now, notice

(τ − τ)lnQ(z; τ) = F−(z; τ) − F+(z; τ)

to deduce the expression for F− as in the theorem.

A consequence of this expansion is the invariance of F+(z; τ) + F−(z; τ) F(z; τ):= = ln |Q(zˆ;ˆτ)|−τ ln |Q(z; τ)| 2 under translation τ → τ + 1. 39 Page 6 of 11 Paşol and Zudilin Res Math Sci (2018) 5:39

Lemma 5 We have F+(z; τ + 1) − F+(z; τ) =− F−(z; τ + 1) − F−(z; τ) .

Proof The functions L(z, τ)andU(z, τ) (hence their complex conjugates) are clearly invariant under translation τ → τ + 1. The result follows from noticing that

2πiτ(τ − τ) − πi(τ − τ)2A(1 − A)(1 − 2A) 2ReS(z, τ) + B (A) = 3 3 6 − πi(τ − τ)2 = B (A) 3 3

is also invariant under the transformation.

We summarise the results in this section as follows.

Theorem 2 The weight 1 period function

F(z; τ) = ln |Q(zˆ;ˆτ)|−τ ln |Q(z; τ)| ∞ 1 − + = ln |x 1qm 1| Li x−1qm+1 − ln |xqm| Li (xqm) 2π 1 1 m=0 ∞ 2 πi(τ − τ) 1 − + − B (A) − Im Li (x 1qm 1) − Li (xqm) 6 3 2πi 2 2 m=0

of ln |Q(z; τ)| satisﬁes

τF(zˆ;ˆτ) = F(z; τ) and F(z; τ) = F(z; τ + 1).

In other words, it behaves like a Jacobi form of weight 1 on SL2(Z).

4 Elliptic dilogarithm Theorem 2 provides a natural link between the period function F(z; τ) and the elliptic dilogarithm [7] ∞ − + D(q; x):= D(xqm) = D(xqm) − D(x 1qm 1) m∈Z m=0 together with its companion ∞ 2 − + log |q| log |x| J(q; x):= J(xqm) − J(x 1qm 1) + B , 3 3 log |q| m=0 where

D(x):= ln |x| arg(1 − x) + Im Li2(x) =−ln |x| Im Li1(x) + Im Li2(x) denotes the Bloch–Wigner dilogarithm and

J(x):= ln |x| ln |1 − x|=−ln |x| Re Li1(x) its companion. Namely, the expansion in the theorem can be stated as

1 F(z; τ) = D(q; x) + iJ(q; x) . (6) 2πi Paşol and Zudilin Res Math Sci (2018) 5:39 Page 7 of 11 39

This is essentially the result discussed in [2, Section 1]. Viewing now z as an element of the lattice R+Rτ, so that A and Aˆ in the representation z =−Aˆ + Aτ are ﬁxed, we ﬁnd out that the τ-derivative 1 d d ln Q(z; τ) = q ln Q(z; τ) 2πi dτ dq is the Eisenstein series i e2πi(mAˆ +nA) 4π 3 (mτ + n)3 m,n∈Z of weight 3, where the notation indicates omitting the term m = n = 0 from the summation. Integrating we obtain

1 e2πi(mAˆ +nA) ln Q(z; τ) = 4π 2 m(mτ + n)2 m,n∈Z implying

1 ln |Q(z; τ)|= ln Q(z; τ) + ln Q(z; τ) 2 1 π ˆ + 1 1 = e2 i(mA nA) − 8π 2 m(mτ + n)2 m(mτ + n)2 m,n∈Z 1 π ˆ + imIm τ (m Re τ + n) = e2 i(mA nA) 2π 2 m(mτ + n)2(mτ + n)2 m,n∈Z i Im τ e2πi(mAˆ +nA)(m Re τ + n) = . 2π 2 |mτ + n|4 m,n∈Z

This is equation (7) in [2]. Since zˆ = z/τ = A − Aˆ /τ = A + Aˆ τˆ, it follows that

i Imτ ˆ e2πi(−mA+nAˆ )(m Reτ ˆ + n) ln |Q(zˆ;ˆτ)|= 2π 2 |mτˆ + n|4 m,n∈Z τ 2πi(nAˆ −mA) − τ /|τ|2 + = i Im e ( m(Re ) n) 2π 2|τ|2 |n − m/τ|4 m,n∈Z τ 2πi(nAˆ −mA) |τ|2 − τ = i Im e (n m Re ) 2π 2 |nτ − m|4 m,n∈Z τ 2πi(mAˆ +nA) |τ|2 + τ = i Im e (m n Re ) 2π 2 |mτ + n|4 m,n∈Z τ 2πi(mAˆ +nA) τ + τ + τ + τ = Im e ((m Re n) i (m n)Im ) 2π 2 |mτ + n|4 m,n∈Z

implying

(Im τ)2 e2πi(mAˆ +nA)(mτ + n) ln |Q(zˆ;ˆτ)|−τ ln |Q(z; τ)|= . 2π 2 |mτ + n|4 m,n∈Z The latter is a (non-holomorphic) modular form of weight 1, and combined with equation (6) is the formula of Bloch mentioned previously. 39 Page 8 of 11 Paşol and Zudilin Res Math Sci (2018) 5:39

Theorem 3 (Bloch’s formula [1,2,7]) For z = Aτ − A,ˆ we have

1 F(z; τ) = D(q; x) + iJ(q; x) 2πi (Im τ)2 e2πi(mAˆ +nA)(mτ + n) = . 2π 2 |mτ + n|4 m,n∈Z

5 General weight A natural generalisation of the product in (5)is

∞ / + + k − + − k + − k Bk+2(A) (k 2) m (m A) 1 m 1 ( 1) (m 1 A) Qk (z; τ):= q (1 − xq ) (1 − x q ) , (7) m=0

where k = 0, 1, 2, ... and Bk (t) stands for the kth Bernoulli polynomial. Then Q0(z; τ) is an arithmetic normalisation of the short theta function θ0(z; τ) (a Siegel modular unit) and Q1(z; τ) coincides with (5). Following the earlier recipe, deﬁne

k−2 F+(z; τ) = Fk,+(z; τ):= ln Qk (zˆ;ˆτ) − τ ln Qk (z; τ), k−2 F−(z; τ) = Fk,−(z; τ):= ln Qk (zˆ;ˆτ) − τ ln Qk (z; τ) τ = 1 τ + τ and Fk (z; ): 2 Fk,+(z; ) Fk,−(z; ) . Then from Lemma 1 we deduce the following generalisation of Lemma 4.

Lemma 6 We have, for k ≥ 1,

τ k F+(zˆ;ˆτ) + (−1)k F+(z; τ) = (−1)k πiAk (2Aˆ + 1)τ k ,

τ k F−(zˆ;ˆτ) + (−1)k F−(z; τ) =−(−1)k πiAk (2Aˆ + 1)τ k .

Proof Apply Lemma 1 and the relation

k k k+1 Bk+2(−t) − (−1) Bk+2(t) = (−1) (k + 2)t .

We further use that the τ-derivative of ln Qk (z; τ) is an Eisenstein series.

Lemma 7 For k ≥ 1,

(−1)k k! e2πi(mAˆ +nA) ln Qk (z; τ) = , (2πi)k+1 m(mτ + n)k+1 m,n∈Z where z =−Aˆ + Aτ.

Proof Consider Q˜ k (A, Aˆ ; τ):= Qk (Aτ − Aˆ ; τ) as a function of real variables A, Aˆ and complex variable τ.Theτ-derivative 1 d d G + (A, Aˆ ; τ):= ln Q (A, Aˆ ; τ) = q ln Q (A, Aˆ ; τ) k 2 2πi dτ k dq k is seen to be the Eisenstein series (−1)k+1(k + 1)! e2πi(mAˆ +nA) Ek+2(A, Aˆ ; τ):= (2πi)k+2 (mτ + n)k+2 m,n∈Z Paşol and Zudilin Res Math Sci (2018) 5:39 Page 9 of 11 39

of weight k + 2. This is true for k = 1 (see Sect. 4), while for k ≥ 1 we observe the functional equation ∂ ∂ Ek+3(A, Aˆ ; τ) = Ek+2(A, Aˆ ; τ). ∂Aˆ ∂τ

The equality Gk+2(A, Aˆ ; τ) = Ek+2(A, Aˆ ; τ) then follows by induction on k using the fact that the constant terms of both functions at τ =∞(or q = 0) agree. Integrating we obtain

(−1)k k! e2πi(mAˆ +nA) ln Qk (A, Aˆ ; τ) = . (2πi)k+1 m(mτ + n)k+1 m,n∈Z Since both sides continuously depend on A and Aˆ , the formula remains valid also for

ln Qk (z; τ).

As in our computation in Sect. 4 we obtain − k ( 1) k! 2πi(mAˆ +nA) 1 1 ln |Qk (z; τ)|= e − 2(2πi)k+1 m(mτ + n)k+1 m(mτ + n)k+1 m,n∈Z k 2πi(mAˆ +nA) k (−1) k! e (τ − τ) − = (mτ + n)j(mτ + n)k j 2(2πi)k+1 (mτ + n)k+1(mτ + n)k+1 m,n∈Z j=0

k τ k 2πi(mAˆ +nA) =−i k!Im e (2π)k+1 (mτ + n)k−j+1(mτ + n)j+1 j=0 m,n∈Z

and

k ik k!Imτ e2πi(−mA+nAˆ ) ln |Qk (zˆ;ˆτ)|=− (2π)k+1|τ|2 (n − m/τ)j+1(n − m/τ)k−j+1 j=0 m,n∈Z k ik k!Imτ e2πi(mAˆ +nA)τ k−j τ j =− . (2π)k+1 (mτ + n)k−j+1(mτ + n)j+1 j=0 m,n∈Z

Thus,

k Fk (z; τ) = ln |Qk (zˆ;ˆτ)|−τ ln |Qk (z; τ)| k k 2πi(mAˆ +nA) i k!Imτ − e = τ k j(τ j − τ j) (2π)k+1 (mτ + n)j+1(mτ + n)k−j+1 j=0 m,n∈Z

k k i k! k−j j j = τ (τ − τ )Dj+1,k−j+1(q; x) 2(2π)k (τ − τ)k j=1 k ik! k−j j = τ Im(τ ) Dj+1,k−j+1(q; x), (4π Im τ)k j=1

where

(τ − τ)a+b−1 e2πi(mAˆ +nA) Da,b(q; x):= (8) 2πi (mτ + n)a(mτ + n)b m,n∈Z 39 Page 10 of 11 Paşol and Zudilin Res Math Sci (2018) 5:39

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 ≥ 1 and z = Aτ − A,ˆ we have k k ik! k−j j ln |Qk (zˆ;ˆτ)|−τ ln |Qk (z; τ)|= τ Im(τ ) Dj+1,k−j+1(q; x), (4π Im τ)k j=1 where ∞ π τ a+b−1 m a+b −1 m+1 (4 Im ) D (q; x) = D (xq ) + (−1) D (x q ) + B + (A) a,b a,b a,b (a + b)! a b m=0 and

+ − ab 1 a+b− −1 − + − − − 1 (− ln |x|) D (x) = (−1)a 1 2a b 1 Li (x) a,b a − 1 (a + b − − 1)! =a + − ab 1 a+b− −1 − + − − − 1 (− ln |x|) + (−1)b 1 2a b 1 Li (x). b − 1 (a + b − − 1)! =b

6Conclusion 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

k Fk (z; τ) = ln |Qk (zˆ;ˆτ)|−τ ln |Qk (z; τ)|,

where k ≥ 1 and the product Qk (z; τ)isdeﬁnedin(7), the case k = 1 remains the only one, which is invariant under translation τ → τ + 1. At the same time, Lemma 6 implies the transformation

k k−1 τ Fk (z,ˆ τˆ) = (−1) Fk (z, τ) for k = 1, 2, ... . This consideration does not exclude, however, a possibility for modiﬁed products (7)and

related functions Fk to exist such that the latter ones have true modular behaviour for each k ≥ 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].

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

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. Paşol and Zudilin Res Math Sci (2018) 5:39 Page 11 of 11 39

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

Received: 30 December 2017 Accepted: 4 May 2018 Published online: 28 September 2018

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