Degenerate Sklyanin algebras, Askey-Wilson polynomials and Heun operators
Julien Gaboriaud, Satoshi Tsujimoto, Luc Vinet, Alexei Zhedanov
aa r X i v : . [ m a t h . QA ] A p r Degenerate Sklyanin algebras, Askey-Wilsonpolynomials and Heun operators
Julien Gaboriaud ∗ , Satoshi Tsujimoto † , Luc Vinet ‡ , Alexei Zhedanov § Centre de Recherches Math´ematiques, Universit´e de Montr´eal,P.O. Box 6128, Centre-ville Station, Montr´eal (Qu´ebec), H3C 3J7, Canada. Department of Applied Mathematics and Physics, Graduate School of Informatics,Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto 606-8501, Japan School of Mathematics, Renmin University of China, Beijing, 100872, China
May 15, 2020
Abstract
The q -difference equation, the shift and the contiguity relations of the Askey-Wilson poly-nomials are cast in the framework of the three and four-dimensional degenerate Sklyaninalgebras ska and ska . It is shown that the q -para Racah polynomials corresponding toa non-conventional truncation of the Askey-Wilson polynomials form a basis for a finite-dimensional representation of ska . The first order Heun operators defined by a degreeraising condition on polynomials are shown to form a five-dimensional vector space thatencompasses ska . The most general quadratic expression in the five basis operators andsuch that it raises degrees by no more than one is identified with the Heun-Askey-Wilsonoperator. Keywords:
Sklyanin algebras, Askey-Wilson operators and polynomials, q -para Racahpolynomials, Heun operators. Quite some time ago, it was shown [1, 2] that the Askey-Wilson difference operator could berealized as a quadratic expression in the generators of the degenerate Sklyanin algebra (ofdimension four). A little earlier Kalnins and Miller [3] used symmetry techniques to derive theorthogonality relation of the Askey-Wilson polynomials and identified to that end interestingladder operators. Over the years the application of the factorization method [4] to thesepolynomials and the study of their structure relations [5] brought attention to related elements.More recently advances have been made in the elaboration of the theory of q -Heun operatorsand the Heun-Askey-Wilson [6] and rational [7] Heun operators have been identified by focusing ∗ E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected] § E-mail: [email protected]
1n certain raising properties of their actions on appropriate spaces of functions. The purposeof this report is to stress the connections between these topics.The paper will develop as follows. In Section 2 we shall introduce three operators involving q -shifts that realize a three-dimensional degenerate Sklyanin algebra ska . These operators willbe ubiquitous; it will be observed that their linear combination is diagonal on special Askey-Wilson polynomials in base q and, following [1], that the most general quadratic expressionformed with them yields the full Askey-Wilson operator in base q . The degenerate Sklyaninalgebra ska obtained by Gorsky and Zabrodin will also be introduced. It has ska as asubalgebra and will be seen to admit a formal embedding of the Askey-Wilson algebra. InSection 3, it will be seen that the contiguity operators introduced by Kalnins and Miller intheir treatment of the Askey-Wilson polynomials all belong to a model of the degenerateSklyanin algebra ska . It will also be seen that the q -para Racah polynomials [8] support afinite-dimensional representation of the four-dimensional degenerate Sklyanin algebra. Section4 will indicate how the Askey-Wilson bispectral operators emerge in this context. We shallconsider first order q -difference operators and identify the conditions for such operators to raiseby one the degree of polynomials in the symmetric variable x = z + z − . This will lead to afive dimensional vector space of operators. A basis will consist of one lowering operator, twothat stabilize polynomials of a given degree and two that are raising this degree by one. Thelowering and stabilizing operators will coincide with the operators realizing ska introduced inSection 2. A combination involving the two raising operators will give the realization of thefourth generator of ska beyond those of ska . The relations obeyed by these five operators willbe found in an Appendix. It will further be seen that the most general quadratic operator inthe five basis elements and not raising the degree by more than one is the Heun-Askey-Wilsonoperator [9]. This parallels the fact that in bispectral situations Heun operators could bedefined equivalently as raising operators or as bilinear expressions in the bispectral operators.As will be indicated in the conclusion this approach is paving the way to the definition ofSklyanin-like Heun algebras associated to different degenerations of the Askey-Wilson grid. It is well known that quantum algebras can be realized in terms of q -derivatives (see [10, 11]).In the case of U q ( su (2)) for example, the commutation relations[ ˆ B, ˆ C ] = ˆ A − ˆ D q − q − , [ ˆ A, ˆ D ] = 0 , ˆ A ˆ B = q ˆ B ˆ A, ˆ B ˆ D = q ˆ D ˆ B, ˆ C ˆ A = q ˆ A ˆ C, ˆ D ˆ C = q ˆ C ˆ D (2.1)are realized [12], [13] by takingˆ A ( ν ) = q − ν T + , ˆ B ( ν ) = z q − q − ) ( q ν T − − q − ν T + ) , ˆ C = 2( q − q − ) z ( T + − T − ) , ˆ D ( ν ) = q ν T − , (2.2)where in the case of finite dimensional representations ν is integer or half-integer (see below)and where T + and T − are the q -shift operators that act as follows on functions of z : T + f ( z ) = f ( qz ) , T − f ( z ) = f ( q − z ) . (2.3)We shall in the following look at models built with operators of the divided difference type.2 .1 The three-dimensional degenerate Sklyanin algebra ska Let p = ( a, b, c, α, β, γ ) be a set of parameters. The generalized three-dimensional Sklyaninalgebra ˆ S p as defined in [14] (see also [15]), is given by three generators u, v, y and the relations: uv − avu − αyy = 0 , vy − byv − βuu, yu − cuy − γvv = 0 . (2.4)Consider the operators Y = 1 z − z − ( T + − T − ) ,U = 1 z − z − ( zT + − z − T − ) ,V = 1 z − z − ( zT − − z − T + ) . (2.5)It is readily checked that they satisfy the following relations: V Y − qY V = 0 ,Y U − qU Y = 0 , [ U, V ] = ( q − q − ) Y . (2.6)Under the correspondence lowercase → uppercase , it is seen that Y , U , V realize a special caseof ˆ S p with a = 1 , b = c = q, α = ( q − q − ) , β = γ = 0 . (2.7)We shall henceforth denote this algebra by ska . As shown in [14], it corresponds to one ofthe situations (( a, b, c ) = (0 , ,
0) and β = γ = b − c = 0)) for which the generalized Sklyaninalgebra ˆ S p has a polynomial growth Hilbert series (PHS) and Koszul properties. The algebra ska thus defined possesses a quadratic Casimir elements Ω (2) :Ω (2) = uv + q − y (2.8)that takes the value 1 in the realization (2.5) which implies that U V is related to Y . Let us recall that the Askey-Wilson polynomials p n ( x ; a, b, c, d | q ) defined by a n p n ( x ; a, b, c, d | q )( ab, ac, ad ; q ) n = φ (cid:20) q − n , abcdq n − , az, az − ab, ac, ad (cid:12)(cid:12)(cid:12)(cid:12) q ; q (cid:21) (2.9)with x = z + z − are eigenfunctions of the operator L ( a,b,c,d ) q [16] L ( a,b,c,d ) q p n ( x ; a, b, c, d | q ) = λ n p n ( x ; a, b, c, d | q ) (2.10)with eigenvalues λ n = q − n (1 − q n )(1 − abcdq n − ) . (2.11)We use standard notation for the basic hypergeometric functions and q -shifted factorials [16].In base q r , the Askey-Wilson operator reads L ( a,b,c,d ) q r = A ( r ) ( z ) T r + − [ A ( r ) ( z ) + A ( r ) ( z − )] I + A ( r ) ( z − ) T r − (2.12)3ith A ( r ) ( z ) = (1 − az )(1 − bz )(1 − cz )(1 − dz )(1 − z )(1 − q r z ) (2.13)and where I is the identity operator.The Askey-Wilson algebra AW (3) [17] that encodes the bispectrality of the polynomials p n is realized by taking the generators K = L ( a,b,c,d ) q + (1 + q − abcd ) and K = x to find thatthe defining relations of AW (3)[ K , K ] q = K , [ K , K ] q = µK + ν K + ρ , [ K , K ] q = µK + ν K + ρ , (2.14)where [ A, B ] q = q / AB − q − / BA , are verified with the parameters µ , ν and ρ related tothose, a , b , c , d of the polynomials p n (see for instance [18]).Consider now the following general linear combination of Y , U and V : M ( α,β,γ ) = αY + βU + γV. (2.15)Using (2.5), we see that M ( α,β,γ ) = F ( z ) T + + F ( z − ) T − (2.16)where F ( z ) = γ (1 − az )(1 − bz )1 − z (2.17)with αγ = ( a + b ) , βγ = − ab. (2.18)Since F ( z ) + F ( z − ) = γ (1 − ab ) , (2.19)we observe that M ( α,β,γ ) = γ [ L ( a,b,q , − q ) q + (1 − ab )] . (2.20)It follows that the eigenfunctions of a linear combination of the operators Y, U, V such as M ( α,β,γ ) are special Askey-Wilson polynomials with the property of being “symmetric” whenlooked at from the dual perspective where variable and degree are exchanged; this is becausethe diagonal term in M ( α,β,γ ) is constant. Correspondingly, following [18], by taking K = 1 γ M ( α,β,γ ) and K = x (2.21)we find that the Askey-Wilson algebra relations (2.14) are satisfied with µ = 0 , ν = 1 , ρ = 0 ,ν = − ab ( q − q − ) , ρ = (1 − q − )( a + b )( ab + q ) . (2.22)We shall consider next quadratic expressions in the generators of ska .4 .3 The Askey-Wilson operator and ska An important observation [1] comes from considering the most general quadratic expression inthe operators { Y, U, V } representing ska . Let us go over this. Define as before another generallinear combination of these operators: M ( δ,ǫ,ζ ) = δY + ǫU + ζV = G ( z ) T + + G ( z − ) T − (2.23)where G ( z ) = ζ (1 − q − cz )(1 − q − dz )1 − z (2.24)with δζ = q − ( c + d ) , ǫζ = − q − cd. (2.25)The product M ( α,β,γ ) M ( δ,ǫ,ζ ) will take the form: M ( α,β,γ ) M ( δ,ǫ,ζ ) = F ( z ) G ( qz ) T + [ F ( z ) G ( q − z − ) + F ( z − ) G ( q − z )] I + F ( z − ) G ( qz − ) T − , (2.26)A straightforward computation shows that for the specific functions F ( z ) and G ( z ) given in(2.17) and (2.25), the following identity holds: F ( z ) G ( q − z − ) + F ( z − ) G ( q − z ) = − F ( z ) G ( qz ) − F ( z − ) G ( qz − ) + Γ (2.27)with Γ a constant given by Γ = γζ ( abcdq − − ab − cdq − + 1) . (2.28)Recalling the expression of A (2) ( z ) in (2.13), we see that F ( z ) G ( qz ) = γζA (2) ( z ) (2.29)and hence we write M ( α,β,γ ) M ( δ,ǫ,ζ ) = γζ [ L ( a,b,c,d ) q + ( abcdq − − ab − cdq − + 1) I ] . (2.30)We have thus obtained a factorization of the Askey-Wilson operator L ( a,b,c,d ) q as a product oftwo linear combinations of the generators in the representation (2.5) of the special generalizedSklyanin algebra ska .We also note that M ( α,β,γ ) M ( δ,ǫ,ζ ) = ( αY + βU + γV )( δY + ǫU + ζV ) (2.31)provides the most general quadratic expression in the three generators { Y, U, V } . Taking intoaccount the relations (2.5) between the generators and the expression of U V (2.47) (and
V U )in terms of Y provided by the value of the Casimir, the product M ( α,β,γ ) M ( δ,ǫ,ζ ) can bereduced to: M ( α,β,γ ) M ( δ,ǫ,ζ ) = βǫU + γζV + ( αδ − βζq − − γǫq ) Y + ( αǫq + βδ ) U Y + ( αζq − + γδ ) V Y + ( βζ + γǫ ) I . (2.32)We thus recover (with a different parametrization) the result of Gorsky and Zabrodin [1]according to which the Askey-Wilson q -difference operator is a quadratic expression in thegenerators of ska . (As a matter of fact this result is presented in [1] in the context of thefour-dimensional degenerate Sklyanin algebra to which we shall turn in a moment.)5 emark 2.1 The idea of obtaining operators of interest, like the Askey-Wilson one, asquadratic expressions in the generators of fundamental algebras has precedents. Of note is theidentification of the Askey-Wilson algebra as a coideal subalgebra of U q ( sl (2)) [19] and the useof the realization (2.1) to obtain the difference operator of the big q -Jacobi polynomials [20] asa generator in this embedding. Returning to the factorization formula, since γ and ζ only occur in the global factor, we mayset γ = ζ = 1. Summing up we thus have: M ( α,β, M ( δ,ǫ, = L ( a,b,c,d ) q + ( abcdq − − ab − cdq − + 1) I (2.33)with α = ( a + b ) , β = − ab, (2.34) δ = q − ( c + d ) , ǫ = − q − cd. (2.35)With the eigenvalues λ n of L ( a,b,c,d ) q given by (2.11) with q replaced by q , it is straightforwardto see that the Askey-Wilson polynomials with base q correspondingly verify h M ( α,β, M ( δ,ǫ, i p n ( x ; a, b, c, d | q ) = ρ n p n ( x ; a, b, c, d | q ) (2.36)with ρ n = q − n (1 − abq n )(1 − cdq n − ) . (2.37) Remark 2.2
If we were to consider two linear combinations M (¯ α, ¯ β, and M (¯ δ, ¯ ǫ, of Y , U and V where the roles of the pairs of parameters ( a, b ) and ( c, d ) are exchanged with respect to M ( α,β, and M ( δ,ǫ, , namely if we were to take ¯ α = q − ( a + b ) = q − α, ¯ β = − q − ab = q − β, (2.38)¯ δ = ( c + d ) = qδ, ¯ ǫ = − cd = q ǫ, (2.39) we would obtain again a factorization of the Askey-Wilson operator L ( a,b,c,d ) q of similar form M (¯ δ, ¯ ǫ, M (¯ α, ¯ β, = L ( a,b,c,d ) q + ( abcdq − − abq − − cd + 1) I . (2.40) This is because L ( a,b,c,d ) q is invariant under the permutation of the parameters and the exchangesof the operators M with the q -shifts of the parameters given in (2.38) and in (2.39) simplyamount to permuting the pairs ( a, b ) and ( c, d ) in the constant term of the rhs of (2.33) . Thisis in line with the fact that M ( α,β, M ( δ,ǫ, − M (¯ δ, ¯ ǫ, M (¯ α, ¯ β, = ( − ab + cd )(1 − q − ) I (2.41) as is easily checked using the relations (2.6) as well as (2.47) . Conversely, M ( α,β, M ( δ,ǫ, − M (¯ δ, ¯ ǫ, M (¯ α, ¯ β, = ( β − q ǫ )(1 − q − )Ω (2) (2.42) with the M s taken as the linear combinations of the generators, can be seen to package (ab-stractly) the relations between y, u and v (2.4) with parameters (2.7) . .4 The four-dimensional degenerate Sklyanin algebra ska The four-dimensional degenerate Sklyanin algebra ska was obtained in [1] as a limit of theelliptic algebra originally introduced by Sklyanin [12] (see [21] for a mathematically orientedreview). It is presented in terms of four generators A , B , C , D obeying the following homoge-neous quadratic relations: DC = qCD, CA = qAC, [ A, D ] = ( q − q − ) C , [ B, C ] = A − D q − q − ,AB − qBA = qDB − BD = − q − q − DC − CA ) . (2.43)This algebra possesses two Casimir elements:Ω = AD + ( q − q − ) q C , Ω = q − A + qD ( q − q − ) + BC + q + q − C . (2.44)We note that the subalgebra generated by { A, C, D } is isomorphic to ska .It was observed [1] that the degenerate Sklyanin algebra contracts to U q ( su (2)); indeed, ifone sets A = ǫ ˆ A , B = ˆ B , C = ǫ ˆ C , D = ǫ ˆ D and let ǫ go to zero we see that the relations(2.43) reduce to (2.1). In keeping with the representation theory [12] of the Sklyanin algebra,the finite dimensional representations of its degenerate version are characterized by an integeror half-integer ν and are of dimension (2 ν + 1). We know from [1] that these can be realizedby associating A , B , C , D to the following q -difference operators (we shall not distinguish herethe abstract algebra element from its realization): A = q − ν U, C = 2( q − q − ) Y, D = q ν V,B = 12( q − q − )( z − z − ) (cid:2) q ν ( z T − − z − T + ) − q − ν ( z T + − z − T − ) − ( q + q − )( T + − T − ) (cid:3) , (2.45)where U , V , Y are as in (2.5). In this realization the Casimir elements Ω and Ω take thefollowing values: Ω = 1 , Ω = q ν +1 + q − ν − ( q − q − ) . (2.46)In light of (2.44) and (2.45), the former relation restates the already observed fact that U V isrelated to Y , namely that U V = 1 − q − Y . (2.47)In the realization (2.45), the contraction from the degenerate Sklyanin algebra to U q ( su (2)amounts to taking z very large. It is quickly seen that in this limit the divided differenceoperators { A, B, C, D } given above reduce to the { ˆ A, ˆ B, ˆ C, ˆ D } of (2.2).Now using the variable x = z + z − , it is readily found that B can be expressed as B = 12( q − q − ) (cid:2) q − ν ( q − xU − U x ) + q ν ( qxV − V x ) − ( q + q − ) Y (cid:3) (2.48)7n terms of the operators (2.5) realizing ska . We shall now indicate how x can be expressed asa formal power series in terms of A, B, C, D by inverting (2.48). In light of the commutationrelation [
U, V ] = ( q − q − ) Y given in (2.5) and the Casimir relation (2.47) we have U V = 1 − q − Y and V U = 1 − qY . (2.49)It follows that V has an inverse V − given by the formal power series in Y expressed as follows: V − = U (1 − qY ) − = (1 − q − Y ) − U. (2.50)Using the relations U x − qxU = − ( q − q − ) Y and xV − qV x = q ( q − q − ) Y , we arrive at x = q − ν (cid:20) B + (cid:18) q + q − q − q − + q ν − q − ν (cid:19) Y (cid:21) (1 − q − ν V − U ) − V − . (2.51)As indicated before the Askey-Wilson operator L ( a,b,c,d ) q and x generate the Askey-Wilsonalgebra. Within the realization in terms of divided difference operators, we saw that L ( a,b,c,d ) q according to (2.33) is obtained as a quadratic expression in the generators of the subalgebra ska of ska and just found as per (2.51) that x is in the completion of the latter algebra. We cantherefore assert that the Askey-Wilson algebra can be formally embedded in this realizationof ska . In [3], Kalnins and Miller presented an elegant derivation of the weight function of the Askey-Wilson polynomials which is based on symmetry techniques. We here wish to point out thattheir approach can actually be cast in the framework of degenerate Sklyanin algebras. Centralto the treatment in [3] are certain contiguity and ladder operators that will prove familiar. Inorder to facilitate comparison with the original reference we shall adopt essentially the samenotation; we shall however use q as the base.Kalnins and Miller begin their considerations by observing that the Askey-Wilson polyno-mials satisfy the following contiguity relation µ ( a,b,c,d ) p n ( x ; a, b, c, d | q ) = q − n (1 − abq n − ) p n ( x ; aq − , bq − , cq, dq | q ) (3.1)if µ ( a,b,c,d ) is the following operator: µ ( a,b,c,d ) = 1( z − z − ) (cid:0) − z − (1 − aq − z )(1 − bq − z ) T + + z (1 − aq − z − )(1 − bq − z − ) T − (cid:1) . (3.2)It is further observed that µ ( cq,dq,aq − ,bq − ) p n ( x ; aq − , bq − , cq, dq | q ) = q − n (1 − cdq n ) p n ( x ; a, b, c, d | q ) . (3.3)We may proceed from here to derive the weight function by requesting that it be such that µ ( cq,dq,aq − ,bq − ) is the formal adjoint of µ ( a,b,c,d ) ; this is done in [3]. Let us focus on thefact that in view of (3.1) and (3.3), the Askey-Wilson polynomials are eigenfunctions of µ ( cq,dq,aq − ,bq − ) µ ( a,b,c,d ) , namely, h µ ( cq,dq,aq − ,bq − ) µ ( a,b,c,d ) i p n ( x ; a, b, c, d | q ) = ¯ ρ n p n ( x ; a, b, c, d | q ) , (3.4)8ith ¯ ρ n = q − n (1 − cdq n )(1 − abq n − ) . (3.5)Not surprisingly the factorization of the Askey-Wilson operator that this eigenvalue equationentails will coincide with the one described in the preceding section. This is readily establishedby recognizing that µ ( a,b,c,d ) = ( aq − + bq − ) Y − abq − U + Vµ ( cq,dq,aq − ,bq − ) = ( c + d ) Y − cdU + V = M (¯ α, ¯ β, , = M (¯ δ, ¯ ǫ, . (3.6)These contiguity operators are thus found to belong to the realization (2.5) of the Sklyaninalgebra ska and we see that µ ( cq,dq,aq − ,bq − ) µ ( a,b,c,d ) = M (¯ δ, ¯ ǫ, M (¯ α, ¯ β, , (3.7)with the connection with the Askey-Wilson operator provided by (2.40); we note moreoverthat the eigenvalue ¯ ρ n in (3.5) coincides with the expression obtained from (2.37) under theexchange ( a, b ) ↔ ( c, d ).Kalnins and Miller consider in addition the lowering operator τ ( a,b,c,d ) : τ ( a,b,c,d ) = 1 z − z − ( T + − T − ) = Y, (3.8)which is nothing else than our operator Y (or C ). They proceed to find its adjoint τ ( a,b,c,d ) ∗ which reads: τ ( a,b,c,d ) ∗ = q − z − z − (cid:20) (1 − az )(1 − bz )(1 − cz )(1 − dz ) z T + − (1 − az − )(1 − bz − )(1 − cz − )(1 − dz − ) z − T − (cid:21) . (3.9)These operators act as follows on the Askey-Wilson polynomials: τ ( a,b,c,d ) p n ( x ; a, b, c, d | q ) = q n (1 − q − n )(1 − abcdq n − ) p n − ( x ; aq, bq, cq, dq | q ) ,τ ( a,b,c,d ) ∗ p n − ( x ; aq, bq, cq, dq | q ) = − q − n p n ( x ; a, b, c, d | q ) . (3.10)The key point is that τ ( a,b,c,d ) ∗ can be expressed as a linear combination of the genera-tors A , B , C and D of the degenerate Sklyanin algebra ska . Let e = ( a + b + c + d ), e = ( ab + ac + ad + bc + bd + cd ), e = abc + abd + acd + bcd and e = abcd be the elementarysymmetric functions in the parameters ( a, b, c, d ), one finds indeed that τ ( a,b,c,d ) ∗ = q − h − e ( e ) − A − q − q − )( e ) B + ( q − q − )2 [ e − ( q + q − )( e ) ] C + e ( e ) D i (3.11)with q − ν = ( abcd ) . (3.12)We thus observe that the contiguity and raising operators µ ( a,b,c,d ) , τ ( a,b,c,d ) and τ ( a,b,c,d ) ∗ belongto the realization of the degenerate Sklyanin algebra which is hence represented on the Askey-Wilson polynomials. In general ν as given by the relation (3.12) above will not be an integer orhalf integer and the corresponding representation extends the finite-dimensional one discussedin Section 3 to an infinite-dimensional one. 9 roposition 3.1 The operator µ ( a,b,c,d ) , its adjoint µ ( cq,dq,aq − ,bq − ) , τ ( a,b,c,d ) and τ ( a,b,c,d ) ∗ form a basis equivalent to the set { A, B, C, D } as a representation of the degenerate Sklyaninalgebra ska . Their action on the Askey-Wilson polynomials p n ( x ; a, b, c, d | q ) is provided by (3.1) , (3.3) and (3.10) respectively. The connection formula [22], [23] of Askey and Wilson canbe used to express these formulae as combinations of polynomials p k ( x ; a, b, c, d | q ) , k = 0 , , ... ,with parameters a, b, c, d fixed, that span the representation space. Imposing that the representation be finite-dimensional amounts to enforcing the non-conventional truncation condition( q ) − N +1 = abcd, N = 2 ν + 1 (3.13)for the Askey-Wilson polynomials with base q . Quite strikingly this leads to polynomialscalled q -para Racah polynomials that have been recently characterized [8] and which are inparticular orthogonal on a bilattice composed of two Askey-Wilson grids. We wish to stressthis result. Proposition 3.2
The q -para Racah polynomials with base q support a representation of thedegenerate Sklyanin algebra of dimension N = 2 ν + 1 with ν integer or half-integer. Remark 3.1
Remarkably the operator τ ( a,b,c,d ) ∗ also features centrally in Koornwinder’s study[5] of the structure relations of the Askey-Wilson polynomials. These relations amount to rais-ing and lowering relations where in contradistinction with the shift relations that we consideredabove (following Kalnins and Miller), the parameters are not affected. It is shown in [5]that such a structure relation is obtained when τ ( a,b,c,d ) ∗ (denoted by L in [5] with the factor q − omitted) acts upon the Askey-Wilson polynomial p n ( x ; a, b, c, d | q ) with base q . Note thatthe shift relation in (3.10) acts on polynomials with base q . It is also indicated in [5] that ( q − τ ( a,b,c,d ) ∗ = [ L ( a,b,c,d ) q , x ] . Remark 3.2
It is further recognized in [5] on the basis of results of Rains [24] and Rosengren[25] that the operator τ ( a,b,c,d ) ∗ generates a representation of the degenerate Sklyanin algebra ska . This is ascertained from the relation τ ( a,b,ce,de − ) ∗ τ ( qa,qb,q − c,q − d ) ∗ = τ ( a,b,c,d ) ∗ τ ( qa,qb,q − ce,q − de − ) ∗ (3.14) given in [5] and easily checked from (3.9) . As observed by Koornwinder [5], it is the trigono-metric specialization of a formula in [24] giving the defining relations of the Sklyanin algebra.We show below how (3.14) encapsulates the relations (2.43) of ska . Consider the expression (3.11) for τ ( a,b,c,d ) ∗ as a linear combination of the operators A, B, C, D .10ubstituting (3.11) in (3.14) and multiplying by ( e ) e (1 − e ) − ( ce − d ) − , one arrives at0 = ( e ) ( q − q − )( a + b ) (cid:2) A − D − ( q − q − )( BC − CB ) (cid:3) − e ) ( q − q − ) ab ( AB − qBA ) − e )( q − q − ) q − ( BD − qDB )+ ( e ) ( a + b )( qab − q − cd ) (cid:2) ( AD − DA ) − ( q − q − ) C (cid:3) − ( e ) ( q − q − )2 (cid:16) (( a + b ) q − ab − cd ) q − + ( e ) (1 + q − ) (cid:17) CD + ( e ) ( q − q − )2 (cid:16) (( a + b ) q − abq − cd ) q − + ( e ) (1 + q − ) q (cid:17) DC − ( q − q − )2 (cid:16) ab ( q + q − )( e ) + [ e (2 − q − ) + ( b cd + a cd − a b q )] (cid:17) AC − ( q − q − )2 (cid:16) − abq ( q + q − )( e ) − q − [ e (2 − q ) + ( b cd + a cd − a b q )] (cid:17) CA. (3.15)We shall illustrate how the defining relations of ska can be obtained from (3.15). First choose b = − a and c = 0. The equality (3.15) implies CA = qAC. (3.16)Substituting this back in (3.15) and multiplying by ( e ) − yields0 = ( e ) ( q − q − )( a + b ) (cid:2) A − D − ( q − q − )( BC − CB ) (cid:3) − e ) ( q − q − ) ab ( AB − qBA )+ ( a + b )( qab − q − cd ) (cid:2) ( AD − DA ) − ( q − q − ) C (cid:3) − e ) ( q − q − ) q − ( BD − qDB ) − ( e ) ( q − q − )2 (cid:16) (( a + b ) q − ab − cd ) q − + ( e ) (1 + q − ) (cid:17) CD + ( e ) ( q − q − )2 (cid:16) (( a + b ) q − abq − cd ) q − + ( e ) (1 + q − ) q (cid:17) DC + ( q − q − )( q − q − )2 (( e ) ab − q − ( e ) ) CA. (3.17)Once again, choose c = 0 for instance. The equality (3.17) implies AD − DA = ( q − q − ) C . (3.18)Repeating the same kind of argument, one obtains the other relations (2.43) that define ska .Through the realization that we have considered here, we have observed so far that thedegenerate Sklyanin algebra ska is a basic structure underneath the theory of Askey-Wilsonpolynomials. Much like a supersymmetric Hamiltonian is the “square” of supercharges, theAskey-Wilson operator is quadratic in generators realizing ska . We also saw that this isintimately connected to the application of Darboux transformations or of the factorizationmethod [3], [4] to this operator. This approach as we know is based on the identification ofraising operators. It has been realized recently that raising properties can provide a unifyingprinciple in the theory of Heun operators. We next take this angle to revisit the Heun-Askey-Wilson operator [6] and sort out the place occupied by the degenerate Sklyanin algebra in thisHeun operator picture. 11 S -Heun operators and the Heun-Askey-Wilson operator The standard Heun operator that defines the ordinary second order differential equation withfour regular singularities [26] has the property of raising the degree of polynomials by one.It can also be obtained as a bilinear expression in the bispectral operators of the Jacobipolynomials, namely, multiplication by the variable and the hypergeometric operator [27].Both viewpoints have been built upon to develop a broad perspective on operators of Heuntype and the algebras they realize. The tridiagonalization method based on the hypergeometricoperator has been generalized to any bispectral situation and the concept of algebraic Heunoperator [28] has emerged in this fashion. In a nutshell this construct amounts to forming thegeneric bilinear expression in the bispectral operators. The raising property has been usedto arrive at Heun operators defined on various lattices. In summary, one looks in this casefor the most general second-order operator that raises by one the degree of polynomials onspecified grids. Applied to the Askey-Wilson lattice or polynomials, both approaches have ledequivalently to the Heun-Askey-Wilson operator [6]. (The Heun-Racah and Heun-Bannai-Itooperators have similarly been obtained [29].)Let us mention that the Heun-Askey-Wilson operator has been shown [7] to arise as adegeneration of the one-variable Ruijsenaars-van Diejen Hamiltonian [30], [31], [32]. It has alsobeen found that this operator can be diagonalized with the help of the algebraic Bethe ansatz[9]. We shall expand this by relating here the Heun-Askey-Wilson operators to our observationson Sklyanin algebras. To that end, we shall first focus on determining the most general firstorder operators acting on the Askey-Wilson grid that raise the degree of polynomials by one.We shall call them special Heun operators or S-Heun operators for short. These can be viewedas second order operators without diagonal terms. Indeed if the operator (4.5) given below ismultiplied by T + , we readily see that it takes the form of a first order operator A ( z ) T + A ( z )on a grid with base q . Looking for S-Heun operators is in fact a more basic problem thansearching for the generic second order operator with the raising property as a way of arrivingdirectly at the Heun operator of Askey-Wilson type. It is hence not surprising that there willbe factorization connotations. This undertaking will reveal that the S-Heun operators form afive-dimensional space that includes the operators ( A, B, C, D ) realizing ska . We shall furtherobserve that the Heun-Askey-Wilson operator has a quadratic expression in terms of theseS-Heun operators. S -Heun operators Before we apply the raising condition to determine the S-Heun operators that act through q -differences on the symmetric variable x = z + z − , for reference, let us first go over the mostsimple case of first order differential operators that raise by one the degree of polynomials inthe variable z . Consider the operator SS = F ( z ) ddz + G ( z ) (4.1)and demand that Sp n ( z ) = ˜ p n +1 ( z ) with p n and ˜ p n polynomials of degree n . It is readily seenthat the most general admissible functions F ( z ) and G ( z ) are F ( z ) = α + α z + α z , G ( z ) = β + β z. (4.2) S therefore belongs to a 5-dimensional vector space with the following natural basis L = ddz , M = 1 , M = z ddz , R = z, R = z ddz , (4.3)12btained by setting all coefficients α i and β i equal to zero except for one.These operators can be combined to form the usual finite-dimensional differential realizationof dimension 2 j + 1 on monomials z n , n = 0 , , . . . of the Lie algebra sl , i.e.: J = z ddz − j = M − jM , J + = z ddz − jz = R − jR , J − = ddz = L. (4.4)This corresponds to the q → U q ( su ) given in Section 2.Consider now the q -difference operator S = A ( z ) T + + A ( z ) T − (4.5)where A , ( z ) are functions of z . Note that these S-Heun operators can be viewed as “squareroots” of the general (second order) Heun operators used in [6]. Impose again a raising conditionon polynomials P n ( x ( z )) of degree n in x ( z ) = z + z − : SP n ( x ( z )) = ˜ P n +1 ( x ( z )) (4.6)for all n = 0 , , , . . . .It is sufficient to check property (4.6) for the elementary Askey-Wilson monomials χ n ( z ) = z n + z − n , (4.7)that is to verify that Sχ n ( z ) = n +1 X k =0 a nk χ k ( z ) (4.8)for some coefficients a nk . Let us look at the action of S on the two Askey-Wilson monomials χ n ( x ) of lowest degrees. For n = 0, the raising condition reads A ( z ) + A ( z ) = a + a χ ( z ) (4.9)and similarly, for n = 1 we have A ( z )( zq + z − q − ) + A ( z )( zq − + z − q ) = a + a χ ( z ) + a χ ( z ) (4.10)where a , a , a , a , a are arbitrary parameters. Evaluating the action of S on the higherdegree Askey-Wilson monomials does not give rise to new parameters: the higher coefficients a nk with n ≥ a k and a k . Hence these 5 parametersaccount for all the degrees of freedom that the most general S-Heun operator defined on theAskey-Wilson grid possesses.Combining (4.9) and (4.10), we find for A ( z ) A ( z ) = π ( z ) z (1 − z )(1 − q ) , (4.11)where π ( z ) is a polynomial of degree four: π ( z ) = ( a q − a ) z + ( qa − a ) z − ((1 + q ) a − qa ) z + q ( a − qa ) z + q ( a − qa ) . (4.12)From the observation that both the lhs and rhs of the system (4.9)–(4.10) are invariant under z → z − , it follows that A ( z ) = A ( z − ) . (4.13)This leads to the following proposition. 13 roposition 4.1 The most general S-Heun operators on the Askey-Wilson grid which arerequired by definition to be of the form (4.5) and to raise by one the degrees of polynomials in x = z + z − are specified by the functions A , ( z ) given in (4.11) – (4.13) . As the operator S depends on 5 free parameters, it gives rise as in the differential case to a5-dimensional linear space of S-Heun operators. A natural basis for this space is formed bythree sets which correspond respectively to lowering, stabilizing and raising operators:( i ) Taking a = 1 as the only non-zero parameter in (4.12) leads to the operator denoted L which decreases the degree of any polynomial in x ( z ) by 1 and changes its parity.( ii ) Taking either a = 1 or a = 1 as the only non-zero parameter, one obtains stabilizingoperators, denoted either M or M . Both preserve the degree as well as the parity of anypolynomial in x ( z ).( iii ) The choice a = 1 and all other parameters equal to 0 leads to the raising operator R , while the choice a = q , a = 1 and all other parameters 0 yields the operator R . Bothincrease by one the degree of any polynomial in x ( z ) and change parity.For the sake of completeness, we give below the full expressions of these 5 operators L = 1 q − q − z − z − ( T + − T − ) ,M = 1 q − q − z − z − (cid:0) ( qz + q − z − ) T − − ( q − z + qz − ) T + (cid:1) ,M = 1 q − q − z − z − ( z + z − )( T + − T − ) ,R = 1 q − q − z − z − ( z + z − ) (cid:0) ( qz + q − z − ) T − − ( q − z + qz − ) T + (cid:1) ,R = 1 q − q − (cid:18) q z − z − ( z + z − )( zT − − z − T + ) − ( zT − + z − T + ) (cid:19) . (4.14) Proposition 4.2
The operators L , M , M , R , R are linearly independent. They form abasis for the linear space of S-Heun operators. Note that the 3 operators L , M , M span the 3-dimensional subspace of all “stabilizing”S-Heun operators. This means that any operator S = α L + α M + α M preserves thedegree of any polynomial in x ( z ), if at least one of α , α is nonzero. Comparing (2.5) and(4.14), it is immediate to see that Y = ( q − q − ) L, U = M + qM , V = M + q − M , (4.15)and that the operators ( L, M , M ) equivalently realize ska . We can thus rephrase as followsthe observations of Subsection 2.3 according to which the Askey-Wilson operator is given as aquadratic expression in the operators ( Y, U, V ) representing ska : Proposition 4.3
The Askey-Wilson operator can be given as the most general quadratic com-bination of the S-Heun operators L , M , M that stabilize the degree of polynomials in x ( z ) . We know that the operators
A, C, D in the realization (2.45) of ska are proportional to U, Y, V respectively. It is not difficult to see that B in that same realization can be given as the followingcombination of L, R , R : B = ( q + q − )[( q ν − q − ν ) − ( q − q − )]2( q − q − ) L + q − ν R + ( q ν − − q − ν )2( q − q − ) R . (4.16)We thus have: 14 roposition 4.4 The realization (2.45) of ska is obtained from linear combinations of S-Heun operators on the Askey-Wilson grid. In addition, the operator x can be constructed as a quadratic polynomial in the the elementaryS-Heun operators; we have indeed: x = 1 q − q − (cid:2) (1 + q − )( qM R − R M ) + 2 q − ( qM R − R M ) (cid:3) . (4.17)It follows that the Askey-Wilson algebra can be realized by combining quadratically the fivebasic S-Heun operators. We shall now obtain a formula for the Heun-Askey-Wilson operator in terms of S-Heun oper-ators.Consider the most general quadratic combination of the operators L , M , M , R , R thatraises the degree of polynomials in x ( z ) by at most one. There should hence be no terms in R and R . Using the relations in the Appendix A, one can show that this combination maybe written as follows Q HAW = α L + α LM + α M + α M M + α M L + α M + β M R + β R M + β R M (4.18)where the γ i ’s and δ i ’s are arbitrary parameters. On functions f ( z ) this operator takes theform: Q HAW f ( z ) = [ A ( z ) T + A ( z − ) T − + A ( z ) I ] f ( z ) , (4.19)with A ( z ) = Q ( z ) z (1 − z )(1 − q z ) , A ( z ) = − ( A ( z ) + A ( z − )) + p ( x ) , (4.20)where Q ( z ) is a generic polynomial of degree 6 in z and p ( x ) is a generic polynomial of degree1 in the variable x = z + z − . The exact parameters are expressible in terms of those of (4.18): p ( x ) = β x + α , Q ( z ) = 1 q ( q − q − ) X k =0 r k z k , (4.21) r = β q − β q + β q + β q , r = α q − α q + α q ,r = − β q + β q + 2 β q + ( α + β ) q + ( α + β − β ) q + β q,r = − α q + ( α + α ) q + α q + ( α + α ) q − α q,r = − β q + β q + ( α + β − β ) q + ( α + β ) q + 2 β q + β q,r = α q − α q + α q , r = β q + β q . (4.22)This operator is recognized as the Askey-Wilson Heun operator which has been identified andcharacterized in [6]. (See also [7] and [13].) It is immediately seen that the Askey-Wilsonoperator is recovered upon taking the β i ’s equal to zero, which is equivalent to removing theterms involving raising S-Heun operators from Q HAW .15his formula giving Q HAW as the most general quadratic combination in the S-Heun op-erators on the Askey-Wilson grid provides a novel characterization of the Heun-Askey-Wilsonoperator. As pointed out at the beginning of this section, this operator was identified in [6] onthe one hand as the most general second order q -shift operator that raises by one the degreeof polynomials on the Askey-Wilson grid and on the other hand, as the tridiagonalization ofthe Askey-Wilson operator as per the algebraic Heun construct. The presentation obtainedhere with the S-Heun operators as basic building blocks has the merit of providing, typically,a factorization of Q HAW . Indeed it is seen that the Heun-Askey-Wilson operator can also bewritten generically in the form: Q HAW = ( ξ L + ξ M + ξ M )( η L + η M + η M + η R + η R ) + κ. (4.23)This formula for Q HAW should be compared with equation (2.30) that provides the factoriza-tion of the Askey-Wilson operator as the product of two ska elements. It is hence manifestfrom (4.23) that Q HAW reduces to the Askey-Wilson operator when g = h = 0. To conclude, let us first summarize our observations and second offer a brief outlook.We have considered realizations of the Sklyanin algebras ska and ska in terms of q -difference operators and we determined the first order operators of that type – the S-Heunoperators – that are the basic constituents of the most general degree raising operator in thatclass. Within these realizations, our salient observations are: • The Askey-Wilson operator factorizes as the product of two linear combinations of ele-ments in ska ; • In analogy with the dynamical enlargement of a symmetry algebra with the inclusionof ladder operators, the contiguity and shift operators of the Askey-Wilson polynomialshave been shown to generate a realization of the degenerate Sklyanin algebra ska whichformally includes a realization of the Askey-Wilson algebra. • The q -para Racah polynomials (with base q ) have been identified as forming a basis forthe finite-dimensional representations of the degenerate Sklyanin algebra ska . • The set of S-Heun operators is five-dimensional and has a subset that realizes ska . • The operator multiplication by x has a quadratic expression in terms of the S-Heunoperators. • The Heun-Askey-Wilson operator can also be written as a quadratic expression in theS-Heun operators.With respect to these last two points, let us mention the following. We recall that the algebraicHeun construct gives the Heun-Askey-Wilson operator Q HAW as a bilinear operator in x andthe Askey-Wilson operator. In view of the first and next to last points, this implies that Q HAW is quartic in the S-Heun operators, an expression that must be reducible to the quadraticformula (4.18) obtained here.This study raises a number of questions. Let us mention two: ( i ) How does the examinationof the S-Heun operators extend when the raising property is applied to rational functions as16n [7]? ( ii ) What other algebraic structures akin to the degenerate Sklyanin algebras willemerge when the S-Heun operator approach is adapted to other lattices such as for examplethe quadratic one on which the Wilson polynomials are defined? We plan on addressing theseand other related questions in the near future. Acknowledgments
The authors benefitted from discussions with Nicolas Cramp´e and Slava Spiridonov. JG holdsan Alexander-Graham-Bell scholarship from the Natural Science and Engineering ResearchCouncil (NSERC) of Canada. The work of ST is partially supported by JSPS KAKENHI(Grant Numbers 19H01792, 17K18725). The research of LV is funded in part by a DiscoveryGrant from NSERC. The work of AZ is supported by the National Science Foundation of China(Grant No.11771015).
A Quadratic algebraic relations for L , M , M , R , R The homogeneous quadratic algebraic relations between the five S -Heun operators L , M , M , R , R are collected below:[ M , M ] = ( q + q − ) L , (A.1) M L − ( q + q − ) LM = LM , (A.2) LM + M L = 0 , (A.3) M + M + ( q + q − ) M M = 1 , (A.4) LR = 1 − M , (A.5) R L = 1 − M , (A.6) LR = − L + q − M + M M + q, (A.7) R L = − L + qM + q M M , (A.8) R M + M R = 0 , (A.9) M R + R M = 2( q + q − ) M L − ( q + q − ) LM , (A.10) qR M − M R = R M + ( q + q − ) LM , (A.11) R M − ( q + q − ) M R = M R − ( q + q − )( q − q − ) M L, (A.12) M R − ( q + q − ) R M = R M + 2( q + q − ) M L − (2 q − + 1 + q ) LM , (A.13) R − qR R + q − R R = − q + q − ) M M − ( q + q − ) M + 2[( q + q − ) − ( q − q − ) ] L (A.14)These relations are checked directly from the expressions of the operators in (4.14). Theyprovide the necessary reorderings to reexpress the the most general quadratic combination ofthe 5 operators as in (4.18). 17 eferences [1] A. S. Gorsky and A. V. Zabrodin. Degenerations of Sklyanin algebra and Askey-Wilson polynomials. Journal of Physics A: Mathematical and General , 26(15), 1993. arXiv:hep-th/9303026 .[2] P. B. Wiegmann and A. V. Zabrodin. Algebraization of difference eigenvalueequations related to U q ( sl ). Nuclear Physics, Section B , 451(3):699–724, 1995. arXiv:cond-mat/9501129 .[3] E. G. Kalnins and W. J. Miller. Symmetry techniques for q -series: Askey-Wilson polyno-mials. Rocky Mountain Journal of Mathematics , 19(1):223–230, 1989.[4] G. Bangerezako. The factorization method for the Askey–Wilson polynomials.
Journal ofComputational and Applied Mathematics , 107(2):219–232, 1999. arXiv:math/9805143 .[5] T. H. Koornwinder. The structure relation for Askey–Wilson polynomials.
Journal ofComputational and Applied Mathematics , 207:214–226, 2007. arXiv:math/0601303 .[6] P. Baseilhac, S. Tsujimoto, L. Vinet, and A. Zhedanov. The Heun-Askey-Wilson Algebraand the Heun Operator of Askey-Wilson Type.
Annales Henri Poincar´e , 20(9):3091–3112,2019. arXiv:1811.11407 .[7] S. Tsujimoto, L. Vinet, and A. Zhedanov. The rational Heun operator and Wilsonbiorthogonal functions. pages 1–14, 2019. arXiv:1912.11571 .[8] J. M. Lemay, L. Vinet, and A. Zhedanov. A q -generalization of the para-Racah poly-nomials. Journal of Mathematical Analysis and Applications , 462(1):323–336, 2018. arXiv:1708.03368 .[9] P. Baseilhac, L. Vinet, and A. Zhedanov. The q -Heun operator of big q -Jacobi type andthe q -Heun algebra. Ramanujan Journal , 2019. arXiv:1808.06695 .[10] R. Floreanini and L. Vinet. q -difference realizations of quantum algebras. Physics LettersB , 315(3-4):299–303, 1993.[11] V. K. Dobrev. q -difference intertwining operators for U q ( sl ( n )): General setting and thecase n = 3. Journal of Physics A: Mathematical and General , 27(14):4841–4857, 1994. arXiv:hep-th/9405150 .[12] E. K. Sklyanin. Some algebraic structures connected with the Yang-Baxter equation.Representations of quantum algebras.
Functional Analysis and Its Applications , 17(4):273–284, 1983.[13] P. Baseilhac, X. Martin, L. Vinet, and A. Zhedanov. Little and big q -Jacobi poly-nomials and the Askey–Wilson algebra. Ramanujan Journal , 51(3):629–648, 2020. arXiv:1806.02656 .[14] N. Iyudu and S. Shkarin. Three dimensional Sklyanin algebras and Gr¨obner bases.
Journalof Algebra , 470:379–419, 2017. arXiv:1601.00564 .[15] L. Chekhov, M. Mazzocco, and V. Rubtsov. Quantised Painlev´e monodromy manifolds,Sklyanin and Calabi-Yau algebras. pages 1–41, 2019. arXiv:1905.02772 .1816] R. Koekoek, P. A. Lesky, and R. F. Swarttouw.
Hypergeometric Orthogonal Polynomialsand Their q -Analogues . Springer Monographs in Mathematics. Springer Berlin Heidelberg,2010.[17] A. S. Zhedanov. “Hidden symmetry” of Askey–Wilson polynomials. Theoretical andMathematical Physics , 89(2):1146–1157, 1991.[18] T. H. Koornwinder. The Relationship between Zhedanov’s Algebra AW (3) and the Dou-ble Affine Hecke Algebra in the Rank One Case. SIGMA. Symmetry, Integrability andGeometry: Methods and Applications , 3:063, dec 2006. arXiv:math/0612730 .[19] Y. I. Granovskii and A. S. Zhedanov. Linear covariance algebra for SL q (2). Journal ofPhysics A: Mathematical and General , 26:L357, 1993.[20] P. Baseilhac, L. Vinet, and A. Zhedanov. The q -Onsager algebra and multivariable q -special functions. Journal of Physics A: Mathematical and Theoretical , 50(39), 2017. arXiv:1611.09250 .[21] S. P. Smith. The four-dimensional Sklyanin algebras.
K-Theory , 8(1):65–80, 1994.[22] R. Askey and J. Wilson.
Some basic hypergeometric orthogonal polynomials that generalizeJacobi polynomials . American Mathematical Society, 1985.[23] G. Gasper and M. Rahman.
Basic Hypergeometric Series . Cambridge University Press,2nd edition, 2004.[24] E. M. Rains. BC n -symmetric abelian functions. Duke Mathematical Journal , 135(1):99–180, 2006. arXiv:math/0402113 .[25] H. Rosengren. An elementary approach to 6 j -symbols (classical, quantum, ra-tional, trigonometric, and elliptic). Ramanujan Journal , 13(1-3):131–166, 2007. arXiv:math/0312310 .[26] G. Kristensson.
Second order differential equations: Special functions and their classifi-cation . 2010.[27] F. A. Gr¨unbaum, L. Vinet, and A. Zhedanov. Tridiagonalization and the Heun equation.
Journal of Mathematical Physics , 58(3), 2017. arXiv:1602.04840 .[28] F. A. Gr¨unbaum, L. Vinet, and A. Zhedanov. Algebraic Heun Operator and Band-Time Limiting.
Communications in Mathematical Physics , 364(3):1041–1068, 2018. arXiv:1711.07862 .[29] G. Bergeron, N. Cramp´e, S. Tsujimoto, L. Vinet, and A. Zhedanov. The Heun-Racah andHeun-Bannai-Ito algebras. pages 1–18, 2020. arXiv:2003.09558 .[30] J. F. Van Diejen. Integrability of difference Calogero-Moser systems.
Journal of Mathe-matical Physics , 35(6):2983–3004, 1994.[31] S. N. M. Ruijsenaars. Integrable BC N Analytic Difference Operators: Hidden ParameterSymmetries and Eigenfunctions. In
New Trends in Integrability and Partial Solvability ,pages 217–261, 2004.[32] K. Takemura. Degenerations of Ruijsenaars–van Diejen operator and q -Painlev´e equations. Journal of Integrable Systems , 2(1):1–27, 2017. arXiv:1608.07265arXiv:1608.07265