1. Introduction

Integrable systems are reputed for the numerous interrelations which establish links among them. Miura and/or Bäcklund transformations are omnipresent in the integrability domain. Once a relation of this type is established between two equations, they cease to be two different ones but are, in fact, the same system under two different guises. Thus establishing relations between known integrable systems is of paramount importance. Painlevé equations, be they continuous or discrete, possess the same rich structure of relations.

This paper focuses on a particular class of discrete Painlevé equations [1], multiplicative ones which are associated to the affine Weyl group [2] E7(1). The main bulk of discrete Painlevé equations with E7(1) symmetry are obtained by deautonomisation of a QRT mapping [3] belonging to one of three families characterised by distinct canonical forms [4] of the A₁ QRT matrix. The first two and best known families, referred to as VI and VI′ respectively, correspond to the matrices

and

(The third family, recently discovered [5], will not be the object of the present study.) The canonical form of the mappings of the VI and VI′ families are

and respectively. The transformation from a form VI to a from VI′ is straightforward. Separating even and odd indices and thus splitting (1.3) into two equations, we replace the variable x_m for m even by q²/x_m. We obtain thus an equation of the form (1.4), but written as a QRT-asymmetric one, where moreover we have q = z². The same procedure, starting from (1.4) and replacing x_m for m even by z²/x_m leads to a QRT-asymmetric equation of the form (1.3). Given this observation one would expect all the discrete Painlevé equations in form VI, studied in [6], and in form VI′, studied in [7], to be intimately related. An obvious criterion as to which equations may be related is the periodic dependence of their various parameters: if two equations in VI and VI′ forms have the same periodic dependence one would expect them to be somehow related. If we take, for instance, equations (2.16) of [6] and (27) of [7], which both have a quaternary periodic dependence in their parameters, we find that the transformation we outlined above allows to transform one into the other. However a difficulty does exist. It stems from two equations, one in form VI and the other in form VI′ which have the same double ternary dependence and which cannot be related through the transformation introduced above. This paper originated from our aspiration to resolve this minor enigma.

2. A short digression

Since in this paper we will be heavily using the QRT formalism [3] for the writing of mappings in symmetric and asymmetric form it is useful to summarise here the conventions that govern it and their consequences for discrete Painlevé equations.

The starting point is a QRT-symmetric mapping relating the variable w over three adjacent points, i.e. w_n+1 = F(w_n, w_n−1). Typically a discrete Painlevé equation involves a variable introducing the secular dependence for which we can simply assume the form s_n = αn + β. Moreover a discrete Painlevé equation equation can have parameters with a periodic dependence of the independent variable. To this end we have introduced two useful periodic functions ϕ_k and χ_2k obeying the periodicity relations ϕ_k(n) = ϕ_k(n + k) (but where the constant solution of this relation is excluded) and χ_2k(n) = −χ_2k(n + k). Practically this gives the following forms for ϕ_k and χ_2k:

When dealing with multiplicative equations the expressions above must be understood as defining the logarithms of the periodic functions and thus the ϕ and χ we shall encounter in section 2, 3 and 4 are the exponentials of the right-hand sides of the above expressions.

Starting from a QRT-symmetric mapping one constructs an asymmetric one by separating the evolution for even and for odd values of the indices. One has thus two mappings one giving w_2n+1 as a function of w_2n and w_2n−1 and one giving w_2n+2 as a function of w_2n+1 and w_2n. The QRT convention is to introduce two distinct dependent variables corresponding to indices of different parity. One has thus x_m = w_2n and y_m = w_2n+1. Similarly for the secular dependence one introduces z_m = s_2n and ζ_m = s_2n+1. Given this notation one remarks that ζ_m = z_m+1/2.

Let us illustrate this by producing the asymmetric form of the symmetric mapping

Using the conventions explained in the previous paragraph we find for n = 2m, n = 2m + 1 respectively

Up to this point the transcription is straightforward. However in the case of discrete Painlevé equations one must also be able to take care of the periodic functions. The periodic function ϕ₂(n) is simply equal to γ(−1)ⁿ. Thus whenever the equation involves this function the latter will enter as γ in the even-index equation (2.2a) in the example above) and as −γ in the odd-index one (2.2b) in the example). The periodic function χ₂(n) is identical to ϕ₂(n). For the function ϕ₃ we consider the four points ϕ₃(2n − 1), ϕ₃(2n), ϕ₃(2n + 1), and ϕ₃(2n + 2). Using the property ϕ₃(n) = ϕ₃(n + 3) we can rewrite these four quantities as ϕ₃(2n − 4), ϕ₃(2n), ϕ₃(2n − 2), and ϕ₃(2n + 2) and the transcription becomes now easy: φ₃(m − 2), φ₃(m), φ₃(m − 1), and φ₃(m + 1). In fact, the periodicity property can be used for all ϕs of odd periods. While ϕ_k(2n) and ϕ_k(2n + 2) are simply ϕ_k(m) and ϕ_k(m + 1), when k = 2p + 1 we have ϕ_k(2n + 1) = ϕ_k(2n + 1 − k) = ϕ_k(2n − 2p) which can be written as φ_k(m − p). The case of even k is more complicated. We start with the remark that the ϕ_k for even k can be decomposed in terms of a ϕ of lower index and a χ based on the identity ϕ_2p(n) = ϕ_p(n) + χ_2p(n). Since we already discussed the case of ϕ₂ it suffices to study the transcription of the first few χs. The function χ₄(n) is represented in the QRT-asymmetric form by two different functions χ₂(m) and χ^2(m) for the even and odd-index equation respectively. A short analysis shows that χ₆(n) is transcribed as ϕ₃(m) + c in the even-index equation and ϕ₃(m − 1) − c in the odd-index one. The function χ₈(n) becomes two different functions χ₄(m) and χ^4(m) for the even- and odd-index equation respectively. The pattern becomes now clear. For χ_2(2p+1) we must introduce ϕ_2p+1(m) + c in the even-index equation and ϕ_2p+1(m − p) − c in the odd-index one. The χ_4p on the other hand become two different functions χ_2p and χ^2p.

Using the procedure described above one can transcribe any QRT-symmetric discrete Painlevé equation into a QRT-asymmetric one in a straightforward way.

3. The case of the two E7(1)-associated equations

The case that motivated the present study is that of two discrete Painlevé equations which have a double ternary periodicity, i.e. one involving two different periodic functions ϕ₃(n) and φ^3. This is a feature appearing in equations associated to E7(1) and which does not exist for any of the “lower” groups. The first equation was obtained in [6] under the form

where s_n has a secular dependence of the form logs_n = αn+β plus a periodic dependence involving the two different period-3 functions ϕ₃(n) and φ^3(n). The two quantities a_n and b_n obey the relation ab = 1 at the autonomous limit and are such that each of the products as² and bs² involve one of the ϕ₃(n), φ^3(n). Finally c is a pure constant. The second equation was obtained in [7] and has the form

Here s_n has a secular dependence of the form logs_n = αn + β and a periodic dependence involving a period-3 function ϕ₃(n). The functions d_n and g_n have the same ternary periodicity involving a function φ^3(n) as well as an even-odd dependence, which is also present in f_n, h_n. Neglecting the period-3 dependence we have simply d_nf_n = 1 and h_ng_n = 1 and moreover g_n+1 = d_n.

Before addressing the question of whether (3.1) and (3.2) are related in full generality by the simple transformation presented in the previous paragraph, we can simply ask here whether it is possible to transform (3.1) into (3.2), and vice versa, in the case where the ternary dependence is neglected and we keep just the secular one. We start from (3.2) and write it in asymmetric, in the QRT sense, form as

where ζ(m) = z(m + 1/2). Similarly starting from (3.1) we obtain the QRT-asymmetric form where ρ(m) = p(m + 1/2). We start with equation (3.3), perform the transformation x_n = X_n/z_n, y_n = ζ_n/Y_n and use the relation pn=zn2. We transform thus the form-VI′ equation (3.3) into

A simple glance at the right-hand sides of (3.5) suffices to convince oneself that the terms do not match those of (3.4). Thus even when one considers just the secular dependence one cannot relate the two form-VI and form-VI′ equations.

Does the result above mean that there is no relation whatsoever between equations (3.1) and (3.2)? As we shall show in the next section this is not the case: the two equations have a common ancestor, namely an equation with E8(1) symmetry from which they can be obtained through two different limiting procedures.

4. From E8(1) to E7(1) through limiting procedures

Our starting point will be the multiplicative E8(1)-associated equation which has the form

Taking the limit x → ∞ while keeping z finite leads to the equation

where the right-hand side S must now be understood as the limit of the right-hand side of (4.1) where we have taken x as well as certain of the parameters which appear in R(x_n) to infinity. We remark readily that the equation obtained is of the form VI′.

In practical calculations, using the explicit form of R(x) may turn out to be cumbersome and it is preferable in this case to resort to the ancillary representation we have introduced in [8]. As we showed there, we can introduce the ancillary dependent variable ξ by

and use it to rewrite the right-hand side of (4.1) as

The P(ξ) and Q(ξ) are given by

where the Ani are eight quantities which, in principle, depend on the independent variable n. As we had shown in [8] they obey the relation

Before proceeding to the calculations aiming at relating (3.1) and (3.2) we introduce another auxiliary variable q_n = q₀λⁿ which will be used in order to represent the secular dependence of the various parameters we are going to work with.

In our exhaustive investigation of symmetric E8(1)-associated equations we have, among others, derived a single equation, 4.3.1 in [9], which has a double ternary dependence, just like equations (3.1) and (3.2). The equation was obtained in additive form but its transcription to multiplicative form is straightforward. In the latter case its parameters are:

where ϕ₂(n) = γ⁽⁻¹⁾ⁿ. The right-hand side of this equation is a ratio of two quadratic polynomials which means that it was obtained after two simplifications obtained by assuming An7An8=1 and An5An6=1. It has the form where σ₁ is the sum of the four Aⁱ, σ₂, σ₃ the sum of all the products of two and three Aⁱ respectively and σ₄ the product of all four Aⁱ. As we have seen, in order to obtain an equation of form VI′ we must take x_n → ∞. Clearly, if we wish that the right-hand side remain quadratic, some of the Aⁱ’s must also go to infinity with some going to zero so as to preserve (4.6). Given the form of the Aⁱ in (4.7) this can be done in two different ways. The first possibility is to take both c and b to infinity (while keeping the ratio c/b finite and free). We find in this case the right-hand side

Thus is exactly the right-hand side of (3.2) (calling x what is called there w) where s_n = q_nϕ₃(n) (as in equation 4.3.1), dn=φ2(n)φ^3(n)c/b, gn=φ^3(n)c/(bφ2(n)), f_n = ϕ₃(n)/ϕ₂(n) and h_n = ϕ₂(n). Thus by taking the limit x_n → ∞ together with b and c we obtained precisely equation (3.2). However another possibility of limit does exist. We can balance the limit x_n → ∞ by taking the γ of the ϕ₂(n) to infinity. In that case it is more convenient to split the equation according to the parity of the indices and write it in QRT-asymmetric form.

and

Taking the limit γ → ∞ we obtain for the even-index equation. In order to use the QRT notations we introduce s_m = z_2m and ζ_m = z_2m+1. Since z_n = q_nϕ₃(n) we can define φ₃(m) = ϕ₃(2n) and rewrite s_m = q_2mφ₃(m) which allows us to write ζ_m = q_2m+1ϕ₃(2m + 1) = q_2m+1ϕ₃(2m − 2) = q_2m+1φ₃ = (m − 1). The right-hand side is

where ϕ^3(m)=φ^3(2m). Similarly for the odd-index right-hand side we find where φ^3(2m+1)=φ^3(2m−2) and thus equal to ϕ^3(m−1). Since the resulting equation is in VI′ form we must transform it to a form VI by introducing the transformation x_m = X_m/s_m, y_m = ζ_m/Y_m. We obtain thus the system

This is just equation (3.1), written in form (3.4).

Thus, by taking two different limits of the same E8(1)-associated equation we obtained the two E7(1)-related equations with double ternary dependence.

5. More limits

Having introduced the two possible ways of taking limits in E8(1)-associated equations in order to reduce them to E7(1)-associated ones, one can ask the question whether the case of the two equations analysed in the previous section is unique or more similar cases do exist. It turns out that the latter is indeed the case.

We start from the equations derived in [9] and identified the equations where at least one constant and one ϕ₂(n) were present in their parameters. Four such equations are obtained, the ones labeled 4.2.1, 4.2.5, 5.2.1 and 5.2.5 in [9]. In what follows we shall present their different limits. (As pointed out in the previous section, the results of [9] concerned additive equations but their transcription to multiplicative ones is straightforward).

1. a)

   Equation 4.2.1 The parameters of this equation are

   zn=qnφ5(n+1)2φ5(n−1)2/φ5(n)An1=qn3φ2(n)/φ5(n)An2=qn2φ^2(n)φ5(n)cAn3=qn2φ5(n)/(φ^2(n)c)An4=qnφ5(n+1)2φ5(n−1)2/(φ5(n)φ2(n)).(5.1)

   Note that two different period-2 functions appear here ϕ₂(n) = γ⁽⁻¹⁾ⁿ and φ^2(n)=δ(−1)n. In order to balance the limit x_n → ∞ we must take the γ of the ϕ₂(n) to infinity as well as either of the c or the δ of φ^2(n). It turns out that the two latter choices are perfectly equivalent leading to the same final equation.

   Once the equation is VI′ form is obtained it is interesting to convert it into a form VI, using the transformations we outlined in the Introduction. The end result is a QRT-symmetric equation of the form

   (XnXn+1−qn2qn+12φ52(n+1)φ52(n−1)/φ52(n−2))(XnXn−1−qn2qn−12φ52(n+1)φ52(n−2)/φ52(n+2))(XnYn−1)(XnYn−1−1)=(Xn−φ52(n+1)φ52(n−1)qn4/φ52(n))(Xn−cφ52(n+1)φ52(n−1)ϕ2(n)qn3)(Xn−cφ52(n+1)φ52(n−1)ϕ2(n)/(qnφ52(n)))(Xn−1),(5.2)

   where φ₂(n) must be understood as the limit of φ^2(n)/φ2(n) when both γ and δ go to infinity. This is an E7(1)-associated equation which, at least to the authors knowledge, has not been previously derived.

2. b)

   Equation 4.2.5 Before giving the parameters of this equation it is convenient to introduce the variable Z_n, a variable simply related to z_n by Z_n = z_nz_n+1. We can now write the parameters as

   Zn=qnqn+1φ6(n)An1=qn2qn−1φ2(n)φ6(n+2)c/φ6(n+3)An2=qn2qn+1φ2(n)φ6(n−2)φ6(n−3)/cAn3=qn−1φ6(n)φ6(n+1)φ6(n−1)/(φ2(n)φ6(n−2)c)An4=qn+1φ6(n−1)c/(φ2(n)φ6(n+2)φ6(n+1)).(5.3)

   We remark that, here, it is possible to balance the limit x_n → ∞ either by taking c to infinity or by taking the γ of the ϕ₂(n) to infinity. Taking c to infinity results to an equation in a QRT-symmetric equation of form VI′. Its precise expression is

   qn+1qn2qn−1φ6(n)φ6(n−1)(xn+1qnqn+1φ6(n)−xn)(xn−xn−1qnqn−1φ6(n−1))(xn+1−qnqn+1φ6(n)nxn)(xnqnqn−1φ6(n−1)−xn−1)(xn−qnqn+1φ6(n−2)φ6(n−3)φ2(n))(xn−qn−1φ6(n)φ6(n+1)φ6(n−1)/(φ2(n)φ6(n−2)))(xn−φ6(n−3)/(φ6(n+2)φ2(n)qn2qn−1))(xn−φ2(n)φ6(n+1)φ6(n+2)/(φ6(n−1)qn+1)).(5.4)

   When we take γ to infinity we obtain a QRT-asymmetric equation in VI′ form. However it is possible to cast it is form VI in which case the equation becomes QRT-symmetric. We are not going to give the intermediate calculations (which are straightforward but rather tedious) and present directly the form of the resulting equation:

   (Xn+1Xn−Zn)(XnXn−1−Zn−1)(Xn+1Xn−1)(XnXn−1−1)=(Xn−An)(Xn−Bn)(Xn−Cn)(Xn−Dn).(5.5)

   The various quantities appearing in (5.5) are given by the expressions: logZ_nalphan + β − 2γ(−1)ⁿ + ϕ₃(n + 1) + χ₆(n) + χ₆(n − 1), logA_n = αn + β + (−1)ⁿ(γn + δ) + ϕ₃(n) + χ₆(n) and B_n = Z_nZ_n+4/A_n+5, C_n = Z_n+2/A_n+3, D_n = A_n+2/Z_n+1. (Note that in these expressions we have used parameters and functions without making any attempt to link them to those of the initial equation, lest the result become inextricably complicated.)

   Both equations (5.4) and (5.5) are new, derived here for the first time (with, the usual caveat, that the novelty statement concerns the present authors and their knowledge of discrete Painlevé equations).

3. c)

   Equation 5.2.1 Here we have six Ani with expressions

   zn=qnφ3(n)An1=qn2φ2(n)a/φ3(n)An2=qn2/(φ2(n)φ3(n)a)An3=qnφ3(n)φ^2(n)bAn4=qnφ3(n)φ^2(n)/bAn5=qnφ3(n)c/φ^2(n)An6=qnφ3(n)/(φ^2(n)c)(5.6)

   Clearly there are four different ways one can balance the limit x_n → ∞: taking all three a, b, c to infinity or the γ of ϕ₂(n) and b, c or the δ of φ^2(n) and a or both γ and δ to infinity. All four limits lead to the same equation which is given in a QRT-symmetric VI′ form in the case (a, b, c) → ∞ or can be converted to such a form through a simple gauge choice. Without entering into further details we present the form obtained in the straightforward case when all three a, b, c go to infinity. We find

   (xn+1znzn+1−xn)(xn−xn−1znzn−1)(xn+1−znzn+1xn)(xnznzn−1−xn−1)=(xn−zn+1zn−1/φ2(n)(xn−zn/(cφ^2(n)))(xn−czn/φ^2(n))(xn−1/(zn+1zn−1φ2(n))(xn−b/(znφ^2(n)))(xn−1/(bznφ^2(n)),(5.7)

   where b, c must be understood as the ratios b/a, c/a when all three parameters go to infinity. Equation (5.7) is a discrete Painlevé equation already derived in [7].

4. d)

   Equation 5.2.5 Here again we have six Ani. Their expressions are

   Zn=qn3qn+13φ5(n)φ5(n+1)An1=qn5φ5(n)φ5(n+1)2φ5(n+2)2a/φ2(n)An2=qn5qn−12φ2(n)/(φ5(n)a)An3=qnqn+14φ5(n)φ5(n−1)2φ5(n−2)2a/φ2(n)An4=qn−2φ5(n)/(φ5(n+2)2φ5(n−2)2φ2(n)a)An5=qn3φ2(n)φ5(n)bAn6=qn3φ2(n)φ5(n)/b(5.8)

   We remark that the limit x_n → ∞ can be balanced either by taking both a, b to infinity or by taking to infinity the γ of ϕ₂(n). In the first case we obtain a QRT-symmetric discrete Painlevé equation in form VI′:

   ZnZn−1(xn+1Zn−xn)(xn−xn−1Zn−1)(xn+1−Znxn)(xnZn−1−xn−1)=(xn−φ2(n)qn5qn−12/φ5(n)(xn−bφ2(n)φ5(n)qn3)(xn−φ5(n)qn−2/(φ2(n)φ5(n+2)2φ5(n−2)2))(xn−b/(φ2(n)phi5(n)qn3))(xn−φ2(n)/(Φ(n)qn5))(xn−φ2(n)Φ(n)/(qnqn+14)),(5.9)

   where Φ(n) = ϕ₅(n)ϕ₅(n + 1) ²ϕ₅(n + 2) ². Again, this is an equation which does not appear among those previously derived. Next we turn to the limit obtained when we take the γ of ϕ₂(n) to infinity. In this case the equation obtained has a QRT-asymmetric form. However by transforming it into a VI form we find that the resulting discrete Painlevé equation can be cast into a QRT-symmetric form. Without entering into the tedious (but straightforward) details we give the final form of the equation:

   (Xn+1Xn−Zn2)(XnXn−1−Zn−12)(Xn+1Xn−1)(XnXn−1−1)=(Xn−aqn4qn+14φ5(n)/Φ(n))(Xn−aqn8φ5(n)Φ(n))(Xn−qn2qn−12φ5(n−1)2φ5(n)4φ5(n+1)2/a)(Xn−b)(Xn−1/b)(Xn−aφ5(n)2/(qn2qn−12)).(5.10)

   Again, this is a new, to the authors’ knowledge, discrete Painlevé equation.