Basic properties of non-stationary Ruijsenaars functions

For any variable number, a non-stationary Ruijsenaars function was recently introduced as a natural generalization of an explicitly known asymptotically free solution of the trigonometric Ruijsenaars model, and it was conjectured that this non-stationary Ruijsenaars function provides an explicit solution of the elliptic Ruijsenaars model. We present alternative series representations of the non-stationary Ruijsenaars functions, and we prove that these series converge. We also introduce novel difference operators called $\mathcal{T}$ which, as we prove in the trigonometric limit and conjecture in the general case, act diagonally on the non-stationary Ruijsenaars functions.


Introduction
The celebrated quantum Calogero-Moser-Sutherland systems [1] have natural relativistic generalizations discovered by Ruijsenaars [2]. The Ruijsenaars systems come in four kinds: rational, trigonometric, hyperbolic, and elliptic, with the latter case being the most general and reducing to the others in certain limits [2]. While the explicit solution of the trigonometric Ruijsenaars model is known since a long time: it is given by the celebrated Macdonald polynomials [3], and a construction of eigenfunctions of the hyperbolic model was completed recently [4], only partial results about the explicit solution in the general elliptic case exist [5]. Recently, one of us (S) conjectured an explicit solution of the elliptic Ruijsenaars model as a limit of special functions defined by explicit formal power series and called non-stationary Ruijsenaars functions [6]. In particular, it was shown in [6] that these functions reduce to the known solutions of the trigonometric Ruijsenaars model in the trigonometric limit; they have several remarkable symmetry properties; and they arise in a quantum field theory related to the elliptic Ruijsenaars system in a way that is a natural generalization of how the known solutions of the trigonometric Ruijsenaars model arise in a quantum field theory related to the trigonometric Ruijsenaars model (this is only a partial list of results in [6]). The validity of this conjecture was also tested by symbolic computer computations.
In this paper we prove some properties of the non-stationary Ruijsenaars functions which, as we hope, will be useful to find proofs of the conjectures in [6]. In particular, we give alternative representations of these functions which are simpler than the original definitions; we prove that the series defining these functions are absolutely convergent in a suitable domain; and we present novel difference operators, called T , which, as we conjecture, acts diagonally on the non-stationary Ruijsenaars functions (by this we mean that the latter are eigenfunctions of the former).

Prerequisites
We recall some known facts about the Macdonald polynomials [3] and certain special functions generalizing the Macdonald polynomials and constructed so as to solve the trigonometric Ruijsenaars model [8,9] (Section 2.1). We also recall the eigenvalue problem defining the elliptic Ruijsenaars model, and the definition of the non-stationary Ruijsenaars functions (Section 2.2).
The operators D ± N (x|q, t) are related by similarity transformations to the operators defining the trigonometric Ruijsenaars model [2].
As conjectured by one of us (S) [8] and proved by two of us (NS) [9], these eigenfunctions are naturally generalized to a special function f N (x|s|q, t) depending on another set of variables, s = (s 1 , . . . , s N ) ∈ C N , and determined by the following requirement, up to normalization: for λ ∈ C N , the function is a common eigenfunction of D ± N (x|q, t) with corresponding eigenvalue N j=1 s ±1 j ; if λ is a partition, then the function in (2) is equal to the Macdonald polynomial P λ (z; q, t), up to a known multiplicative factor independent of x [9]. The function f N (x|s|q, t) is called asymptotically free solution of the trigonometric Ruijsenaars model.
One remarkable property of this function is that it has a simple explicit series representation which converges absolutely in a suitable domain [9]: with M N the is the set of N × N strictly upper triangular matrices with nonnegative integer entries: and (3) is equivalent to (1.10)-(1.11) in [9]).
For later reference, we also define the function which, as proved in [9], has the following remarkably symmetry properties:

Non-stationary Ruijsenaars functions
The analogue of the operators in (1) for the elliptic Ruijsenaars model depends on a further complex parameter, p such that |p| < 1: with the theta function θ(z; p) ≡ (z; The non-stationary Ruijsenaars function f gl N (x, p|s, κ|q, t) is a conjectured eigenfunction of a deformation of the operators in (6), depending on a further complex parameter, κ, and reducing to the operators in (6) in the limit κ → 1 [6].
Remark 2.2. To explain the scaling just mentioned, we point out one important technical point: in Definition 2.1, Equations (9) and (10) below, and Equations (40) and (41) in Section 4, we use balanced coordinates x B , p B , s B , κ B and t B (written without the subscript B for simplicity), whereas elsewhere in the paper we use unbalanced coordinates x U , p U , s U , κ U and t U (also written without subscript U) related to the balanced coordinates as follows, Thus, the scaling just described can be understood as a transformation from balanced to unbalanced coordinates.
The main conjecture in Ref. [6] is that eigenfunction of the operator in (6) can be obtained by dividing this rescaled function f gl N (p δ/N x, p 1/N |κ δ/N s, κ 1/N |q, q/t) by a (known) factor α(p 1/N |κ δ/N s, κ 1/N |q, t) and taking the limit κ → 1; see Conjecture 1.14 in [6]. One important open problem is to find the operator depending on κ having these rescaled non-stationary Ruijsenaars functions as eigenfunctions and reducing to the Macdonald-Ruijsenaars operator in (6) in the limit κ → 1. 1 At this point, this operator is only known in limiting cases: the non-relativistic limit q → 1 where the Ruijsenaars systems reduce to the non-stationary elliptic Calogero-Sutherland system [6], and the limit t, p → 0 with fixed p/t leading to the affine Toda system [6]. We stress that the non-stationary T -operators introduced in this paper do not reduce to the elliptic Macdonald-Ruijsenaars operators in the limit κ → 1: the T -operators are of a different kind, and they are new even in the trigonometric limit; only the affine Toda limit of the non-stationary T -operator was known before [6].
A natural generalization of the function in (4) is and, as conjectured in [6], it has the following symmetry properties generalizing the ones in (5).
Conjecture 2.1. The functions in (9) satisfy ϕ gl N (x, p|s, κ|q, t) = ϕ gl N (s, κ|x, p|q, t) (bispectral duality), 3 Results on the non-stationary Ruijsenaars function We give alternative series representations of the non-stationary Ruijsenaars functions (Section 3.1) and prove convergence of these series in a suitable domain (Section 3.2).

Alternative series representations
Our first result makes manifest that the non-stationary Ruijsenaars function in (7) is a natural generalization of the asymptotically free solutions of the trigonometric Ruijsenaars model in (3). For that, we extend the variables ; as we will see, the pertinent extension is provided by the parameters p and κ, respectively -see (12b).
We first introduce a natural generalization of the function in (3) to infinitely many variables.
Definition 3.1. Forx = (x 1 , x 2 , . . .) ands = (s 1 , s 2 , . . .) two sets of infinitely many complex variables and q, t complex parameters, let withM N the set of infinite, N-periodic, strictly upper triangular matrices with nonnegative integer entries which are non-zero only in a finite strip away from the diagonal: and in the sense of formal power series.
Note that, by the condition θ ik = θ i+N,k+N , a matrix θ ∈M N is fully determined by the matrix elements θ ik for 1 ≤ i ≤ N and 1 ≤ k < ∞. Moreover, matrices in M N can be naturally identified with matrices θ inM N by setting θ ik = 0 if i > N, or k > N, or both.
To state out result we use the N-vector δ ≡ (δ 1 , . . . , δ N ) with δ i = N − i, and the notation p δ/N x and κ δ/N s for the N-vectors with components (p δ/N x) i = p (N −i)/N x i and (κ δ/N s) i = κ (N −i)/N s i , respectively (i = 1, . . . , N). As explained in Remark 2.2, this can be understood as a transformation going from balanced to unbalanced coordinates. (7) is related to the function in (11) as follows,

Theorem 3.2. The non-stationary Ruijsenaars function in
on the right-hand side by the rules 2 (The proof is by straightforward computations given in Appendix A.) In the following, it is sometimes convenient to use a notation for the functions f N,∞ that emphasizes that the argumentsx ands are fixed by x, s, p and κ: and the identifications in (12b) on the RHS in (13c).
Theorem 3.2 makes manifest the following important result in [6]: After suitably scaling the variables, the non-stationary Ruijsenaars function reduces the asymptotically free solution of the trigonometric Ruijsenaars model, f N (x|s|q, t) in (3), in the limit p → 0; in particular, it becomes independent of κ in this limit: Proof. By Theorem 3.2, (14) is equivalent to but this is obvious from definitions: by (12b), (x k /x i ) → 0 for k > N as p → 0; therefore, the sum over θ ∈M N on the RHS in (11a) collapses to a sum over θ ∈ M N in this limit; obviously, for θ ∈ M N , the coefficients c N,∞ (θ|s|q, t) in (11c) do not depend on s i>n and are identical with the coefficients c N (θ|s|q, t) in (3c).
We prove Theorem 3.2 by a direct computation in Appendix A. This proof uses an alternative representation of the function f N,∞ (x;s|q, t) which is interesting in its own right: Lemma 3.5. The formal power series in (11) can be written as with P N the set of all N-partitions λ = (λ (1) , λ (2) , . . . , λ (N ) ), λ (i) a partition of arbitrary length for i = 1, . . . , N, and Proof. Straightforward computations, using that θ ik = λ It is interesting to note that is a natural generalization of the function in (4) due to the following implication of Theorem 3.2.
Fact 3.1. The function in (9) is related to the one in (16) as follows, on the right-hand side by the rules in (12b). Moreover, the conjectures in (10) are equivalent to: this is proved in Appendix D, Lemma D.2. .

Convergence
We prove that the non-stationary Ruijsenaars functions f N (x, p|s, κ|q, t) in Definitions 3.1 and 3.3 are absolutely convergent in a certain domain of variables and parameters.
Then, there exists a constant ρ > 0 such that the formal power series in Definitions 3. 1

and 3.3 is absolutely convergent in the domain
Remark 3.7. In our proof, we actually show convergence for any ρ < 1/C 1 C 2 where Remark 3.8. We believe that it is possible to refine this convergence results. In particular, we believe that there are regions of convergence where s i /s j , 1 ≤ i < j ≤ N, are real and q and κ have non-trivial imaginary parts.
Proof of Theorem 3.6. Our strategy of proof is to show that our assumptions imply simple upper bounds on the terms appearing in the series in (15): With that, absolute convergence follows from the comparison test: the series in (15) is of the form λ∈P N a λ with |a λ | ≤ α |λ| for all λ ∈ P N , and the series λ∈P N α |λ| converges absolutely for |α| < 1.
The first estimate in (22) is a simple consequence of the conditions in (20): since x i+N = px i for all i ≥ 1, these conditions are equivalent to which clearly implies the result.
The proof of the second estimate in (22) is more involved and, for this reason, we supplement our somewhat descriptive arguments in the main text below by a detailed argument in Appendix B.
We observe that C N,∞ (λ|s|q, t) in (15b) is a product of fractions (1 − q l au)/(1 − q l u) with a = t/q in the first group of products and a = q/t in the second group, l ∈ Z, and u = s j /s i for i = 1, . . . , N and j ≥ i; moreover, s j+ℓN = κ ℓ s j for ℓ ∈ Z ≥1 . Such a fraction can be estimated in a simple way: If j − i is not an integer multiple of N, we can estimate this further using (to see that the latter inequality holds, write z = |z|e iϕ and note that (23) is equivalent to which is obvious). Since we assume that q and κ both are real, for all integers l, ℓ, we get a simple universal bound for these fractions: for all integers l. However, this bound does not work for j = i + ℓN with ℓ ∈ Z ≥0 since, in these cases, q l u = q l s j /s i = q l κ ℓ is real. However, one can check that, in all these latter cases, either l ≤ 0 and ℓ > 0, or l < 0 and ℓ ≥ 0, and thus, by our assumptions, z ≡ q l u = q l κ ℓ always satisfies either |z| ≥ min(|q| −1 , |κ|) > 1 (if |q| < 1 and |κ| > 1) or |z| ≤ max(|q|, |κ| −1 ) < 1 (if |q| > 1 and |κ| < 1); we therefore can use the inequality to get simple universal bounds for the cases j = i + ℓN with ℓ ∈ Z ≥0 as well (we spell our details of this argument in Appendix B.2.2). We thus get estimates with different upper bounds, C 1 and C 2 , for all fractions in the first and second groups of products on the RHS in (15b), respectively. The arguments above allow to compute the constants C 1 and C 2 and give the results in (21); the interested reader can find details of this computation in Appendix B.
Inserting these bounds into (15b) we obtain computing telescoping products in the second step and using N i=1 k≥1 λ (i) k = |λ| in the last step. This proves the second estimate in (22).
To conclude, we give an elementary computation proving that the series λ∈P N α |λ| for |α| < 1 is absolutely convergent: summing repeatedly geometric series.

T -operators
For fixed N ∈ Z ≥1 , we define an operator T which acts diagonally on the asymptotically free solution of the trigonometric Ruijsenaars model (Section 4.1). We also present a natural non-stationary generalization of this operator which, as we conjecture, acts diagonally on the corresponding non-stationary Ruijsenaars function (Section 4.2).

Trigonometric case
We find it convenient to work with formal power series.
with β = ln(t)/ ln(q), let Clearly, the operator T N (x|q, t) is complicated: it has the same complexity as the function f N (x|s|q, t); cf. (3a). Still, it is interesting since, different from the elliptic Macdonald-Ruijsenaars operators in (6), we know its natural generalization to the non-stationary case; see Section 4.2.
The following is our main result in this section. (1):

Proposition 4.2. The T -operator in (27) is well-defined, it commutes with the trigonometric Macdonald-Ruijsenaars operators in
] for all λ ∈ C N , and it acts diagonally on the asymptotically free solutions of the trigonometric Ruijsenaars model in (3): (note that ln(s i )/ ln(q) = λ i + β(N − i)).
(A proof based on results in the rest of this section can be found in Appendix C.) Our proof of Proposition 4.2 is based on the following convenient representation of the T -operator.

Proof of Lemma 4.3.
We use that C[[x 2 /x 1 , .., x N /x N −1 ]] is spanned by (a subset of) monomials x µ with µ ∈ Z N . For fixed λ ∈ C N , we compute the action of q 1 2 ∆ on x λ x µ , µ ∈ Z N : ]. This and the definition in (27) give , and using (3a) and the definition in (32) we obtained (30).
We note that χ N (x|y|q, t) = χ N (y|x|q, q/t) (this is proved in Appendix C, Lemma C.1); inserting this in (30) and backtracking, one obtains the following alternative representation of the T -operator:

Non-stationary case
We present a non-stationary generalization of the T -operator.
Remark 4.6. To make the p-dependence of this operator manifest, one can write it as using the definitions in (13c) and Lemma D.2 in Appendix D.
By comparing with (11), it is clear that the operator in (34) is a natural non-stationary generalization of the trigonometric T -operators in (27); however, there is one important new feature: the shift operator T κ,p acting on p.
We propose the following generalization to Proposition 4.2; this conjecture is a complement to the ones in [6].

Conjecture 4.1. The non-stationary T -operator in (34) has a well-defined diagonal action on the non-stationary Ruijsenaars function in Definitions 2.1 and 3.3:
In the rest of this section, we present two generalizations of results about the trigonometric T -operators: (i) the constant-term representation of the T -operator in Lemma 4.3, (ii) the alternative representation in (33) obtained with the duality in (10). We also rephrase Conjecture 4.1 in terms of the non-stationary Ruijsenaars functions as defined in [6].
One can adapt the proof Lemma 4.3 to obtain the following constant-term representation of the T -operator in (34): with setting y i+N = uy i for all i ≥ 1, f N,∞ (x, p|y, u|q, t) in (11), and [· · · ] 1;y,u , short for [[· · · ] 1,y ] 1,u , the constant term in y ∈ C N and u ∈ C.
Proof. This is proved by a straightforward generalization of the arguments given in the proof of Lemma 4.3; the only new ingredient is T κ,p p n = (κp) n = 1 1 − κp/u u n 1,u (n ∈ Z ≥0 ), and therefore Moreover, similarly as for the trigonometric T -operator, the conjectured duality in (18) implies χ N,∞ (x, p|y, u|q, t) = χ N,∞ (y, u|x, p|q, q/t) (38) and the following alternative representation of this T -operator To conclude, we rephrase Conjecture 4.1 using balanced coordinates.    (7),

.1 is equivalent to the following diagonal action of the non-stationary T -operator in (40) on the non-stationary Ruijsenaars function in
Proof. This is implied by Theorem 3.2, using that the shift operator T κ,p commutes with the following operator, Φ, switching from unbalanced to balanced coordinates: and noting that Theorem 3.2 implies (Φf N,∞ )(x, p|s, κ|q, t) = f gl N (x, p|s, κ|q, t).

Final remarks
The main conjecture in [6] can be tested systematically using a perturbative solution of the elliptic Ruijsenaars model that generalizes the perturbative solution of the elliptic Calogero-Sutherland (eCS) model in [10]. We plan to present this elsewhere.
As already mentioned, one important outstanding problem is to find κ-deformations of the elliptic Macdonald-Ruijsenaars operators in (6) that have the non-stationary Ruijsenaars functions as eigenfunctions. As conjectured in [6], the limit q → 1 of this hypothetical non-stationary Ruijsenaars model is a known non-stationary eCS model depending on parameters β, p and κ related to the non-stationary Ruijsenaars parameters as follows, t = q β and κ = q −κ . 3 Recently, a rigorous construction of integral representations of eigenfunctions of the non-stationary eCS model for κ = β was presented [11]. We hope that, by combining the latter results with recent results on the non-stationary Ruijsenaars functions for the corresponding special value of κ [12], it will be possible to prove the main conjecture in [6] in the non-stationary eCS limit q → 1 and for κ = β. Another possible strategy to prove the conjecture in [6] for q → 1 and general κ-values is to try to generalize the perturbative solution of the non-stationary Lamé equation in [13]. The elliptic Ruijsenaars model is invariant under the exchange p ↔ q [2]. 4 The nonstationary Ruijsenaars functions do not have this property manifestly; we plan to report elsewhere on how this duality is recovered from the non-stationary Ruijsenaars function.
It was suggested more than 20 years ago that the elliptic Ruijsenaars model has a doubleelliptic generalization with remarkable duality properties [14,15], and recently an explicit formula for an operator defining such a model was conjectured [16]. It would be interesting to obtain a better understanding of the relation between the non-stationary Ruijsenaars functions and this double elliptic system recently proposed in [17].
Since the non-stationary T -operator proposed in this paper contains a factor q 1 2 ∆ T κ,p , its eigenvalue equation can be regarded as a q-deformed heat equation. We mention the work of Felder and Varchenko on the q-deformed KZB heat equation [18,19] which seems related; it would be interesting to understand this relation in detail.

A Alternative series representation
We prove Theorem 3.2.
We start with details complementing the concise proof of Lemma 3.5 in the main text (Appendix A.1). The main part of the proof is in Appendix A.2.
A.1 Details on Lemma 3.5 One can check that the following two formulas provide a correspondence between multipartitions λ = (λ (1) , . . . , λ (N ) ) in P N and matrices θ = (θ ik ) ∞ i,k=1 inM N that is one-to-one: inserting a telescoping product in the second step, using (43b) in the third, and computing a telescoping product in the fourth. This proves the result.

A.2 Proof of Theorem 3.2
We show by direct computations that the function on the LHS in (14) is equal to the function f N,∞ (x|s|q, t) in (15) with x i+N = px i and s i+N = κs i , for all i ≥ 1. This, together with Lemma 3.5, proves the result.
We compute the function on the LHS in (14) using (7a)-(7b): By Definition 2.1 of the non-stationary Ruijsenaars functions, the variables (p δ/N x) i above are extended from i = 1, . . . , N to all i ≥ 1 by the rule (p δ/N x) i+N = (p δ/N x) i , whereas . . , N and k ∈ Z ≥1 , and thus α . Renaming indices (α, β) → (k, i), we thus can write the function on the LHS in (14) as Thus, to complete the proof, we have to show thatC N (λ; s|q, t, κ) in (45) is equal to C N,∞ (λ;s|q, t) in (15b) for s i+N = κs i (i ≥ 1). For that, we compute the Nekrasov factors in (7b), partially specializing to the variables we need: We note that the constraints on b in the first product is solved by b = a + j + ℓN − i with ℓ an arbitrary integer ≥ χ(i > j), using the definition χ(i > j) = 1 for j < i and 0 otherwise; similarly, the constraints on β in the second product is solved by β = α + i + ℓ ′ N − j − 1 with arbitrary integer ℓ ′ ≥ χ(j ≥ i). We thus can write these Nekrasov factors as We now specialize further to the arguments of interest to us: For these arguments, the manifest κ-dependence disappears: We now take the product of these Nekrasov factors over i, j = 1, . . . , N, change variables j +ℓN → j in the first group of products and i + ℓ ′ N → i in the second group, and obtain where we changed variables a → k = a + j and α → k = α + i − 1 in the last step. We find it convenient to write this result as swapping variable names i ↔ j in the second group of products. We insert this into (45) to obtainC To proceed, we use the identity (see Appendix D, Lemma D.1): applying this to the factors in the second group of products To complete the proof thatC N (λ; s|q, t, κ) in (45) is identical with C N,∞ (λ|s|q, t) in (15b), we swap the order of the two groups of products and compute the overall power of (q/t): cancelling the factor (t/q) |λ| . This proves the identity in (12a) with f N,∞ (x|s|q, t) in (15) and x i+N = px i , s i+N = κs i (i ≥ 1). This, together with Lemma 3.5, implies the result.

B Estimates
We give a complementary proof of the second estimate in (22), to compute the upper bounds C 1,2 in Theorem 3.6.

B.1 Complementary proof of the second estimate in (22)
We prove that, under the assumptions in Theorem 3.6, the following estimates hold true for the fractions appearing in the formula (15b) for C N,∞ (λ|s|q, t), for all N-partitions λ = (λ (1) , . . . , λ (N ) ) ∈ P N , with C 1 and C 2 in (21). This and (15b) imply the estimate in (25) which, by the computation in (25), is equivalent to the second estimate in (22).
We observe all estimates in (46a)-(46b) are of the form in (46a) and in (46b). We prove (46a)-(46b) using three different kinds of estimates: , a ∈ C, q, κ ∈ R with either |q| < 1 and |κ| > 1 or |q| > 1 and |κ| < 1. Then the following estimates hold true, (a) for all l ∈ Z and u ∈ C \ {R}: (c) for all m ∈ Z ≥0 , ℓ ∈ Z ≥0 : (48c) (The proof is given in Appendix B.2.) Case A: For j − i / ∈ NZ ≥0 , we can use the estimate in (48a): Since s j+N = κs j and κ is real, we have | sin arg(s j /s i )| = | sin arg(s j+N /s i )| = | sin arg(s i /s j )| for all j ≥ i; since | sin arg(s j /s i )| ≥ σ for all 1 ≤ i < j ≤ N by assumption, | sin arg(s j /s i )| ≥ σ for all 1 ≤ i ≤ N and j ≥ i such that j − i = NZ ≥0 , and we get for all cases in (47a)-(47b). This proves that the estimates in (46a)-(46b) for all and for all cases j − i / ∈ NZ ≥0 .
We consider the remaining cases for (46a) and (46b) below in Cases B and C, respectively.
Case B: For j − i ∈ NZ ≥1 , we have s j /s i = s i+ℓN /s i = κ ℓ for some ℓ ∈ Z ≥1 , and we can use the estimate in (48b): We check that all cases in (46a) for j − i ∈ NZ ≥1 are covered by this: all l in (47a) for j = i + ℓN can be written as (recall that λ is a partition. This proves that (46a) holds true if for all cases j − i ∈ NZ ≥1 .
Case C: For j − i ∈ NZ ≥0 , we have s j /s i = s i+ℓN /s i = κ ℓ for some ℓ ∈ Z ≥0 , and we can use the estimate in (48c): We check that all cases in (46b) for j − i ∈ NZ ≥0 are covered by this: all l in can be written as This proves that (46b) holds true if for all cases j − i ∈ NZ ≥0 .

C Proof of Proposion 4.2
We prove Proposion 4.2 using Lemma 4.3 in the main text.
We note that the action of the Macdonald-Ruijsenaars operators in (1) on functions x λ f (x) can be written as [9] with the modified Macdonald-Ruijsenaars operators (this can be proved by simple computations which we skip).
We also need properties of the function χ N (x|y|q, t) in (32) which we summarize as follows.
Lemma C.1. The function χ N (x|y|q, t) satisfies the following duality relation, Moreover, with Proof. The definitions in (4) and (32) imply The factor in the parenthesis on the RHS is manifestly invariant under the transformation (x, y, t) → (y, x, q/t); the function ϕ N (x|y|q, t) has this invariance by (5). This proves (54).
The first identity in (55) is implied by E ± N (x|s|q, t)f N (x|s|q, t) = e 1 (s ±1 )f N (x|s|q, t) proved in [9]; the second follows from the first and the duality in (54). Equation (52) and Lemma 4.3 imply that the result we want to prove: The latter is obviously implied by the following two identities: first, and second, We first prove (59) in three steps, using the shorthand notation in (56). We start with proved by the following computation (we insert definitions and change the summation variable n i ± 1 → n i in the third equality), which is proved by LHS = e 1 (x ∓1 )e 1 (y ±1 ) − E ± N (x|y|q, t)E ∓ N (y|x|q, q/t) χ N (x|y|q, t) = e 1 (y ±1 )e 1 (x ∓1 ) − E ∓ N (y|x|q, q/t)E ± N (x|y|q, t) χ N (x|y|q, t) = RHS using (55) and [E ± N (x|y|q, t), E ∓ N (y|x|q, q/t)] = 0; the latter is verified by a simple computation using the definition in (53). Third, which is proved similarly as (61): We are now ready to prove (59): we insert (61) into the LHS in (59), use (62) and (63), and obtain the RHS in (59).

C.2 Eigenfunction property
The eigenfunctions x λ f N (x|s|q, t) of D ± N (x|s|q, t) are unique, and (28) therefore implies that x λ f N (x|s|q, t) also are eigenfunction of T N (x|q, t).

D Identities
For the convenience of the reader, we prove two well-known identities which we need.
for all a, b, q ∈ C.