Lens Generalisation of \uptau-functions for Elliptic Painlevé Equation

# Lens Generalisation of \uptau-functions for the Elliptic Discrete Painlevé Equation

Andrew P. Kels (APK) Institute of Physics, University of Tokyo, Komaba, Tokyo 153-8902, Japan  and  Masahito Yamazaki (MY) Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, Kashiwa, Chiba 277-8583, Japan
###### Abstract.

We propose a new bilinear Hirota equation for -functions associated with the root lattice, that provides a “lens” generalisation of the -functions for the elliptic discrete Painlevé equation. Our equations are characterized by a positive integer in addition to the usual elliptic parameters, and involve a mixture of continuous variables with additional discrete variables, the latter taking values on the root lattice. We construct explicit -invariant hypergeometric solutions of this bilinear Hirota equation, which are given in terms of elliptic hypergeometric sum/integrals.

## 1. Introduction

In the literature many variations of the differential and discrete (difference) Painlevé equations has been found. These equations have been classified into rational, trigonometric and elliptic equations. At the top level of the hierarchy is the elliptic discrete Painlevé equation with affine Weyl group symmetry of type . This equation has been obtained from geometric considerations [1], and as a discrete system on the root lattice [2] (see [3, 4] for relation between the two approaches, and [5] for a comprehensive survey).

The goal of this paper is to propose a generalization of the ORG -function [2, 6] for the elliptic discrete Painlevé equation. The construction of this -function involves two copies of the root lattice and a positive integer parameter , and the resulting equations depend on the usual continuous variables, as well additional discrete variables on the root lattice. We propose a bilinear Hirota-type equation for the -function, and construct explicit solutions of the bilinear equation in terms of an elliptic hypergeometric sum/integral [7] for a general value of the integer parameter , which is fixed throughout the paper. The hypergeometric -functions of this paper are expected to provide a solution for a generalisation of the elliptic discrete Painlevé equation.

The results of this paper open up many future research directions. For example, it would be be interesting to find an explicit Hamiltonian form of the discrete Painlevé equation associated to the -function of this paper, and to explore the various degenerations of the equations. It would also be interesting to explore the geometric aspects of these equations along the lines of [1]. In another direction, the lens elliptic gamma function, which is a central function for this paper, first appeared in the study of four-dimensional supersymmetric gauge theories on a circle times the lens space [8]. This connection suggests that there exists an interpretation of the results of this paper in terms of supersymmetric gauge theories and associated integrable lattice models [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19].

The results of this paper were inspired by the work of Noumi [6], which gave details of the construction of the -function for the regular elliptic discrete Painlevé equation. However, it is rather non-trivial to generalise Noumi’s results to the case of , where additional discrete variables on the root lattice must be taken into consideration. For example, -invariance of the hypergeometric solution for the -function, required the derivation of a specific elliptic hypergeometric sum/integral transformation formula, that acts on both the continuous and discrete variables as an reflection. Moreover, even for the special case of , the bilinear equations and hypergeometric -function of this paper retain the dependence on the discrete variables, and hence are different from the corresponding equations of [6].

The rest of this paper is organized as follows. In Section 2, we provide definitions of the “lens” set of special functions, which generalise the special functions that appear in the theory of elliptic hypergeometric integrals. In Section 3, we define an elliptic hypergeometric sum/integral for constructing the hypergeometric -function, and present the relevant identities that it satisfies. In Section 4, we formulate the Hirota identities for the -function on the lattice, which are then decomposed into the -orbits in Section 5. In Section 6, we state the main theorem of this paper (Theorem 1), which provides an explicit -invariant, lens elliptic hypergeometric solution of the -function. The proof of the main theorem is provided in Section 7. In the Appendices, we respectively present the derivation of the sum/integral transformation, and provide a brief overview of the multiple Bernoulli polynomials.

Acknowledgements: The authors thank Yasuhiko Yamada for stimulation discussions, many useful suggestions and encouragement. APK also thanks Yang Shi for helpful discussions. APK is an overseas researcher under Postdoctoral Fellowship of Japan Society for the Promotion of Science (JSPS). MY was supported in part by World Premier International Research Center Initiative (WPI), MEXT, Japan, and by the JSPS Grant-in-Aid for Scientific Research No. 17KK0087.

## 2. Lens Theta Functions and Lens Elliptic Gamma Function

In this section, the definitions of the special functions are given that play a central role in this paper. Namely, these are the lens theta function, the lens elliptic gamma function, and the lens triple gamma function.

In this paper we use the two complex parameters , that satisfy

 \rm Im(σ),\rm Im(τ)>0. (1)

Our equations will also depend on an additional integer parameter

 r=1,2,… (2)

In this paper, pairs , of continuous and discrete variables, are distinguished with the notation , and , respectively.

### 2.1. Lens Theta Functions

The two lens theta functions , are defined by [17, 7]

 θτ(z,z;σ,τ):=eϕτ(z,z;σ,τ)θ(e−2πize2πiτz|e2πiτr),θσ(z,z;σ,τ):=eϕσ(z,z;σ,τ)θ(e2πize2πiσz|e2πiσr),\omit\raisebox6.0pt$z∈C$,$z∈Z$, (3)

where for , is the regular theta function

 θ(z|q)=(z;q)∞(qz−1;q)∞,(z;q)∞=∞∏j=0(1−zqj), (4)

and the normalization factors are given by

 ϕτ(z,z;σ,τ):=πi6r(3(r+1−2z)(2z+1)−(r2−1)(σ−τ−1)−6z(r−z)(τ+1)),ϕσ(z,z;σ,τ):=−πi6r(3(r−1−2z)(2z−1)−(r2−1)(σ−τ−1)+6z(r−z)(σ−1)). (5)

For , the lens theta functions (3) reduce to regular theta functions

 θτ(z,z;σ,τ)|r=1=θ(e2πiz|e2πiτ),θσ(z,z;σ,τ)|r=1=θ(e2πiz|e2πiσ). (6)

Note that the theta functions and defined in (3), each have non-trivial dependence on both of the parameters , and , through the normalization functions (5).

For brevity, the lens theta functions (3) will typically be denoted by

 θτ(z,z):=θτ(z,z;σ,τ),θσ(z,z):=θσ(z,z;σ,τ), (7)

with implicit dependence on the two parameters and .

Furthermore, a shorthand notation will be used throughout this paper, where in the argument of a function denotes that both the and factors should be taken as a product, e.g.

 θσ(xj±xk,xj±xk)=θσ(xj+xk,xj+xk)θσ(xj−xk,xj−xk). (8)
###### Proposition 1.

The lens theta functions satisfy ( indicates that an identity holds for either or ):

(1) (periodicity) For ,

 θτ,σ(z+2kr,z)=θτ,σ(z,z),θτ,σ(z,z+kr)=θτ,σ(z,z). (9)

(2) (inversion)

 θτ,σ(−z,−z)=−θτ,σ(z,z)e−2πir(z−z). (10)

(3) (recurrence relation) For ,

 θτ(z+nτ,z+n)=θτ(z,z)e−nπir(2z+(n−1)τ+r−2z−n+1),θσ(z+nσ,z−n)=θσ(z,z)e−nπir(2z+(n−1)σ+r−2z+n−1). (11)

(4) (quasi-periodicity) For ,

 θτ(z+rnτ,z)=θτ(z,z)e−nπi(2z+τ(rn−1)+1),θσ(z+rnσ,z)=θσ(z,z)e−nπi(2z+σ(rn−1)+1). (12)

(5) (three-term relation) For , and , or ,

 e2πir(xk−xk)θτ,σ(xj±xk,xj±xk)θτ,σ(xi±z,xi±z)+e2πir(xi−xi)θτ,σ(xk±xi,xk±xi)θτ,σ(xj±z,xj±z)+e2πir(xj−xj)θτ,σ(xi±xj,xi±xj)θτ,σ(xk±z,xk±z)=0. (13)
###### Proof.

These identities simply follow from the definitions (3), and similar identities that hold for the regular theta function , defined in (4). ∎

### 2.2. Lens Elliptic Gamma Function

The lens elliptic gamma function [8, 16, 17, 18] is defined here by

 Γ(z,z;σ,τ):=eϕe(z,z;σ,τ)γσ(z,z;σ,τ)γτ(z,z;σ,τ),z∈C,z∈Z, (14)

where and , are the following infinite products

 γσ(z,z;σ,τ) :=∞∏j,k=01−e−2πize−2πiσze2πi(σ+τ)(j+1)e2πiσr(k+1)1−e2πize2πiσze2πi(σ+τ)je2πiσrk, (15) γτ(z,z;σ,τ) :=∞∏j,k=01−e−2πize−2πiτ(r−z)e2πi(σ+τ)(j+1)e2πiτr(k+1)1−e2πize2πiτ(r−z)e2πi(σ+τ)je2πiτrk, (16)

and the normalisation function is given by [17, 18, 7]

 ϕe(z,z;σ,τ) :=πiz(z−r)(6x−3σ−3τ+(1−σ+τ)(r−2z))6r. (17)

The functions (15), (16), are symmetric with respect to the following shifts

 γσ(z+kσ,z−k;σ,τ)=γσ(z,z;σ,τ),γτ(z+kτ,z+k;σ,τ)=γτ(z,z;σ,τ), (18)

for .

The normalisation function (17), has a useful factorisation in terms of the multiple Bernoulli polynomial (173), as [18, 7]

 ϕe(z,z;σ,τ)=2πi(R2(z,0;σ−12,τ+12)−R2(z,z;σ−12,τ+12))=2πi(R2(z,0;σ,τ)+R2(0,z;12,−12)−R2(z,z;σ,τ)), (19)

where

 R2(z,z;σ,τ):=R(z+zσ;rσ,σ+τ)+R(z+(r−z)τ;rτ,σ+τ), (20)

and

 R(z;σ,τ): =B3,3(z;σ,τ,−1)+B3,3(z−1;σ,τ,−1)12. (21)

For , the lens elliptic gamma function (14) reduces to the regular elliptic gamma function [20], which is denote here by ,

 Γ(z,z;σ,τ)|r=1=Γ1(z;σ,τ)=∞∏j,k=01−e2πi(−z+(j+1)σ+(k+1)τ)1−e2πi(z+jσ+kτ). (22)

In terms of the regular elliptic gamma function (22), the functions (15), and (16), are simply

 γσ(z,z;σ,τ)=Γ1(z+σz;rσ,σ+τ),γτ(z,z;σ,τ)=Γ1(z+τ(r−z);rτ,σ+τ). (23)

Similarly to the lens theta functions, the lens elliptic gamma function (14) will typically be denoted as

 Γ(z,z):=Γ(z,z;σ,τ), (24)

with implicit dependence on the two parameters , and .

###### Proposition 2.

The lens elliptic gamma function (14) satisfies

(1) (periodicity) For ,

 Γ(z+2kr,z)=Γ(z,z),Γ(z,z+kr)=Γ(z,z). (25)

(2) (inversion)

 Γ((σ+τ)−z,−z)Γ(z,z)=1. (26)

(3) (recurrence relation) For ,

 Γ(z+nσ,z−n)=Γ(z,z)n−1∏j=0θτ(z+jσ,z−j), (27) Γ(z+nτ,z+n)=Γ(z,z)n−1∏j=0θσ(z+jτ,z+j). (28)
###### Proof.

These identities can be verified by direct computation. A proof of the -periodicity in (25) previously appeared in Appendix C of [18]. For (27), (28), the normalisation of the lens theta functions (5) are in fact chosen to satisfy

 ϕe(z+σ,z−1;σ,τ)−ϕe(z,z;σ,τ)=ϕτ(z,z;σ,τ),ϕe(z+τ,z+1;σ,τ)−ϕe(z,z;σ,τ)=ϕσ(z,z;σ,τ). (29)

Due to the relations (18), only the factor of on the left hand side of (27), contributes to the infinite product part of the theta function , while only the factor of on the left hand side of (28), contributes to the infinite product part of the theta function . ∎

###### Remark 1.

Although for any , the -periodicity of in (25) comes from the normalisation factor in (19).

### 2.3. Lens Triple Gamma Functions

The lens triple gamma functions and , are defined here by

 Γ+σ(z,z;σ,τ,ω):=eϕ+(z,z;σ,τ,ω)γ+σ(z,z;σ,τ,ω),Γ+στ(z,z;σ,τ,ω,μ):=Γ+σ(z,z;σ,τ,ω)γ+τ(z,z;σ,τ,μ),\omit\raisebox6.0pt$z∈C$,$z∈Z$, (30)

where

 γ+σ(z,z;σ,τ,ω):=gσ(z,z;σ,τ,ω)gσ(σ+τ+ω−z,r+1−z;σ,τ,ω),γ+τ(z,z;σ,τ,ω):=gτ(z,z;σ,τ,ω)gτ(σ+τ+ω−z,r−z;σ,τ,ω), (31)

and

 gσ(z,z;σ,τ,ω):=∞∏k1,k2,k3=0(1−e2πixe2πiσze2πiσrk1e2πi(σ+τ)k2e2πi(σ+ω)k3),gτ(z,z;σ,τ,ω):=∞∏k1,k2,k3=0(1−e2πixe2πiσ(r−z)e2πiτrk1e2πi(σ+τ)k2e2πiωk3). (32)

The normalisation function is defined by (c.f. the expression (19) for in terms of )

 ϕ+(z,z;σ,τ,ω):=2πi(T2(z,0;σ−12,τ+12,ω)−S2(z,z;σ−12,τ+12,ω)), (33)

where

 S2(z,z;σ,τ,ω):=S(z+zσ;rσ,σ+τ,ω+σ)+S(z+(r−z)τ;rτ,σ+τ,ω−τ),T2(z,z;σ,τ,ω):=S(z+zσ;rσ,σ+τ,ω)+S(z+(r−z)τ;rτ,σ+τ,ω), (34)

and

 S(z;σ,τ,ω):=B4,4(z;σ,τ,−1,ω)+B4,4(z−1;σ,τ,−1,ω)48. (35)

In the last equation, is the multiple Bernoulli polynomial (174), defined through the generating function (172).

###### Proposition 3.

The functions , , defined in (31), satisfy

(1) (shift symmetry)

 γ+σ(z+σ,z−1;σ,τ,ω)=γ+σ(z,z;σ,τ,ω),γ+τ(z+τ,z+1;σ,τ,ω)=γ+τ(z,z;σ,τ,ω). (36)

(2) (inversion)

 γ+σ(σ+τ+ω−z,r+1−z;σ,τ,ω)=γ+σ(z,z;σ,τ,ω),γ+τ(σ+τ+ω−z,r−z;σ,τ,ω)=γ+τ(z,z;σ,τ,ω). (37)

(3) (recurrence relation)

 γ+σ(ω+z,1+z;σ,τ,ω)=γσ(z,z;σ,τ)γ+σ(z,z;σ,τ,ω),γ+τ(ω+z,z;σ,τ,ω)=γτ(z,z;σ,τ)γ+τ(z,z;σ,τ,ω). (38)
###### Proof.

These relations essentially follow from the definitions given in (31). ∎

###### Corollary 1.

The lens triple gamma functions (30) satisfy

(1) (inversion)

 Γ+σ(σ+τ+ω−z,r+1−z;σ,τ,ω)=Γ+σ(z,z;σ,τ,ω), (39)

(2) (recurrence relation)

 Γ+σ(ω+z,1+z;σ,τ,ω)=eϕe(z,z;σ,τ)γσ(z,z;σ,τ)Γ+σ(z,z;σ,τ,ω),Γ+στ(σ+τ−z,r−z;σ,τ,ω,μ)=Γ(z,z;σ,τ)Γ+στ(z,z;σ,τ,ω,μ), (40)

where , and are defined respectively in (19), and (15), and is the lens elliptic gamma function (14).

###### Proof.

The relations (39), (40) follow from the relations given in Proposition 3, and also the following relations satisfied by the normalisation function (33)

 ϕ+(σ+τ+ω−z,r+1−z;σ,τ,ω)=ϕ+(z,z;σ,τ,ω),ϕ+(ω+z,z+1;σ,τ,ω)=ϕ+(z,z;σ,τ,ω)+ϕe(z,z,σ,τ), (41)

where is the normalisation function for the lens elliptic gamma function given in (19). ∎

Note that unlike the lens theta and elliptic gamma functions, the lens triple gamma functions (30) are not -periodic in , and even for there remains a dependence on the integer variable .

## 3. Elliptic Hypergeometric Sum/Integral and W(e7)

### 3.1. Elliptic Hypergeometric Sum/Integral

A central role in this paper is played by the following sum/integral, defined in terms of the lens elliptic gamma function (14), by

 I(x,x;σ,τ)=λ(σ,τ)2r−1∑z=0∫[0,1]dz∏7j=0Γ(xj±z,xj±~z)Γ(±2z,±2~z), (42)

where , , and . Notice that in contrast to the previous section, here we allow to have either integer, or half-integer components. The discrete summation variable is chosen so that the second argument of each factor of the lens elliptic gamma functions appearing in (42) is an integer, and is defined by

 ~z:=z+(r+((r+1)mod2))(x1mod1)=⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩z(x∈Z8),z+r+12(x∈(Z+12)8,r even ),z+r2(x∈(Z+12)8,r odd ). (43)

The prefactor in (42) is given by

 λ(σ,τ)=(e2πirσ;e2πirσ)∞(e2πirτ;e2πirτ)∞. (44)

The condition may be relaxed, by deforming the contour connecting the points , and , such that the respective poles of the integrand of (42) do not cross over the contour [7].

For , the elliptic hypergeometric sum/integral (42) previously appeared as part of a key identity (star-star relation) for the integrability of multi-spin lattice models [14, 7], and is a 2-parameter extension of the left hand side of the elliptic beta sum/integral formula that was proven in [17]. It has also previously been studied with respect to , and transformations proven by the authors [7] (where the transformation was previously proven by Spiridonov [21], and the cases of the transformations were previously proven by Rains [22]).

### 3.2. Contiguity Relation

Define the shift operator (), that acts on the continuous variables , and discrete variables , as

 Tτ,kf(x0,…,x7,x0,…,x7)=f(x0,…,xk+τ,x7,x0,…,xk+1,…,x7). (45)
###### Proposition 4.

The elliptic hypergeometric sum/integral (42) satisfies the three-term relation

 (e2πir(xk−xk)θσ(x2±x3,x2±x3)Tτ,i+e2πir(xi−xi)θσ(x3±x1,x3±x1)Tτ,j+e2πir(xj−xj)θσ(x1±x2,x1±x2)Tτ,k)I(x,x)=0, (46)

for any triple .

###### Proof.

By (28), the integrand

 Δ(z,z;x,x):=∏7j=0Γ(xj±z,xj±~z)Γ(±2z,±2~z), (47)

satisfies

 Tτ,kΔ(z,z;x,x)=θσ(xk±z,xk±z)Δ(z,z;x,x), (48)

with respect to the shift operator . Note that by the choice of in (43), , and , are always integers. Next by the three-term relation (13), the equation

 (e2πir(xk−xk)θσ(xj±xk,xk±xk)Tτ,i+e2πir(xi−xi)θσ(xk±xi,xk±xi)Tτ,j+e2πir(xj−xj)θσ(xi±xj,xi±xj)Tτ,k)Δ(z,z;x,x)=0, (49)

holds for . By commuting the shift operator with the integral, we obtain (46). ∎

### 3.3. W(e7) Transformation

###### Proposition 5.

For , , and with the restriction

 7∑i=0xi≡2(σ+τ),7∑i=0xi=2r, (50)

the sum/integral (42) satisfies

 I(x,x)=I(~x,~x)∏0≤i

where the transformed variables and are given by

 ~xi ={xi+σ+τ2−12∑3j=0xj(i=0,1,2,3),xi+σ+τ2−12∑7j=4xj(i=4,5,6,7), (52) ~xi ={xi+r2−12∑3j=0xj(i=0,1,2,3),xi+r2−12∑7j=4xj(i=4,5,6,7). (53)

The special case of Proposition 5 for , and with the additional restrictions that , and are even integers, is due to Spiridonov [21]. Proposition 5 is proven with the use of a variation of the elliptic beta sum/integral formula [17] in Appendix A.

Note that the variables of the elliptic hypergeometric sum/integral (42), transform in the formula (51) under the action of a reflection for an element of the Weyl group . This property is particularly important for the construction of the -function from the sum/integral (42) (see Section 6).

## 4. E8\uptau-function

In this section we will consider the properties of the root lattice of which are used to define our -function. Many of the properties and definitions are essentially based on the case of Noumi [6].

### 4.1. E8 Root Lattice

We denote the root lattice of by , and the Weyl group by . The root lattice is more explicitly given as a -span of the vectors

 ±vi±vj,(0≤i

where , is the orthonormal basis with respect to the canonical symmetric bilinear form on the root lattice , namely . Note also that , for .

###### Definition 1.

A set of vectors in is called a -frame if the following two conditions are satisfied:

(1) ,

(2) .

Notice that this definition implies that the set of vectors

 {±ai±aj∣∣(0≤i

is contained in the root lattice and forms a root lattice of type . In the following sections we will mostly work with the -frame for .

### 4.2. E8\uptau-function

For a pair , where , and , we define

 [Z]=[(z,z)]:=eπir(−z+z)θσ(z,z), (56)

where is the lens theta function defined in (3).