aa r X i v : . [ m a t h . R A ] A p r HEUN ALGEBRAS OF LIE TYPE
NICOLAS CRAMP´E † , ∗ , LUC VINET ∗ , AND ALEXEI ZHEDANOV ‡ Abstract.
We introduce Heun algebras of Lie type. They are obtained from bispectralpairs associated to simple or solvable Lie algebras of dimension three or four. For su (2),this leads to the Heun-Krawtchouk algebra. The corresponding Heun-Krawtchouk operatoris identified as the Hamiltonian of the quantum analogue of the Zhukovski-Voltera gyrostat.For su (1 , ho leads to the Heun-Charlier algebra.The connections to orthogonal polynomials are achieved through realizations of the under-lying Lie algebras in terms of difference and differential operators. In the su (1 ,
1) cases, it isobserved that the Heun operator can be transformed into the Hahn, Continuous Hahn andConfluent Heun operators respectively. Introduction
This paper elaborates on Heun algebras. This subject is associated to the notion ofalgebraic Heun operator recently introduced [12]. Let us recall the main points. It is knownthat the properties of the orthogonal polynomials of the Askey scheme can be encodedalgebraically. Indeed the recurrence operator X and Y, the one of which the polynomialsare eigenfunctions, form a bispectral pair that generate a quadratic algebra. For the Askey-Wilson (AW) polynomials sitting at the top of the q-tableau, the defining relations of theAW algebra read [25]:[ X, [ X, Y ]] = ρXY X + a X + a { X, Y } + a X + a Y + a (1.1) [ Y, [ Y, X ]] = ρY XY + a { X, Y } + a Y + a Y + a X + a (1.2)where ρ = q + q − − a i , i = 1 , ... q = 1 and hence ρ = 0,the cubic terms in (1.1) and (1.2) drop and the reduced relations define the Racah algebra[9], [7]. To all families of hypergeometric polynomials belonging to the Askey classification,there corresponds an algebra of that type which is a special case or a limit of the AW algebra.The algebraic Heun operator is the generic bilinear element constructed from the generators X, Y (1.3) W = r [ X, Y ] + r { X, Y } + r X + r Y + r where r i , i = 1 , , . . . , X, W ) or (
Y, W ). The kernel of W can be viewed as solutions ofan ordinary eigenvalue problem if the parameters r i , i = 1 , . . . , r is theeigenvalue, or of a generalized problem if W is regarded as a multiparameter linear pencil.When X is multiplication by the variable and Y the hypergeometric operator, the algebrathat is realized is the Jacobi one where ρ = a = a = a = 0. It has been shown [11]that in this case W coincides with the differential Heun operator and the equation W ψ = 0amounts to the standard Heun equation with four regular Fuchsian singularities. The name algebraic Heun operator has been coined as a result of this observation; as already explainedthis concept applies to all bispectral problems and associates an operator of Heun type to thepolynomials of the Askey scheme. One would then refer to the Heun operator say of Racahtype and correspondingly to the Heun-Racah algebra for example. The exploration of thesestructures has been initiated recently with a focus on the “higher” polynomials. Attentionwas first paid to the Hahn polynomials [23]; this led naturally to a finite difference version ofthe Heun equation on the uniform lattice. Examining the little and big q-Jacobi polynomialsfrom this angle allowed [2] to give context to q-Heun operators that had been identified [21]in connection with integrable models. Last, the Heun-Askey-Wilson algebra was thoroughlyexamined [3]. We here wish to look in a similar way at the “lower” polynomials and study theHeun structures attached to bispectral pairs that generate a Lie algebra. This will lead to adescription of the Heun algebras associated to the Krawtchouk, Meixner, Meixner-Pollaczek,Laguerre and Charlier polynomials while establishing and clarifying the general foundationsof the subject.The presentation will unfold as follows. We shall begin in Section 2 with a definition of theHeun algebra of Lie type. It will be shown to be generically isomorphic to the Hahn algebra[8, 7, 6] which is obtained from the AW algebra by setting ρ = 0 as well as a = 0 in (1.1)and (1.2). We shall pursue by introducing the Lie algebra for a bispectral pair X and Y withthe so-called “ladder” property, this will amount to dropping the nonlinear terms in the AWrelations. The corresponding W will then be seen to generate together with X (or Y ), theHeun algebra of Lie type. The definitions of the Lie algebras su (2), su (1 ,
1) and ho - theharmonic oscillator one, will be recorded at the end of the section. Section 3 will focus on thecase where the Lie algebra for the bispectral pair is su (2). It will specify the associated Heunalgebra and will indicate that it cannot be mapped to the Hahn algebra over R . It will explainin addition that the Heun operator can be viewed as as the Hamiltonian of the quantumanalog of the Zhukovski-Voltera gyrostat. The case of su (1 ,
1) will be treated in Section 4.Three situations will be distinguished depending on whether the generator X is of elliptic,hyperbolic or parabolic type. The Heun algebras will again be characterized and seen in theseinstances to be isomorphic to the Hahn algebra. Section 5 will be dedicated to the harmonicoscillator algebra ho ; the associated Heun algebra will be determined and observed not tobe equivalent to the Hahn algebra. By recalling in Section 6 certain realizations of su (2), su (1 ,
1) and ho in terms of difference and differential operators, we shall recognize in eachcase that X and Y become the bispectral operators of the Krawtchouk, Meixner, Meixner-Pollaczek, Laguerre and Charlier polynomials confirming that the Heun algebras of the Lietype identified are to be associated to each of these families of orthogonal polynomials.In view of the fact that the Heun-Meixner, Heun-Meixner-Pollaczek and Heun-Laguerrealgebras are isomorphic to the Hahn algebra, it is further shown that the correspondingHeun operators can be transformed into the Hahn, Continuous Hahn and Confluent Heunoperators respectively. The paper will end with concluding remarks in Section 7.2. Heun algebra of the Lie type
Definition 2.1.
The Heun algebra H of the Lie type is generated by X and W with thefollowing defining relations [[ X, W ] , X ] = x + x X + x X + x W , (2.1) [ W, [ X, W ]] = y + y X + y X + y X + x W + x { X, W } , (2.2) EUN ALGEBRAS OF LIE TYPE 3 where x i and y i ( i = 0 , , , ) are free parameters. In the Heun algebra H , the following element(2.3) Ω = z X + z W + z { X, W } + z XW X + z X + z W + z ([ X, W ]) + z X + z X is central if the parameters z i are given by z = 2 y − x y / , z = − x x + 2 x , z = x , (2.4) z = 2 x , z = y − x y / , z = x , (2.5) z = 1 , z = 2 y / , z = y / . (2.6) Proposition 2.1.
Generically, the Heun algebra of Lie type is isomorphic to the Hahnalgebra.
To show that, we introduce the following invertible map between the pairs ( X , W ) and( X , W ) where(2.7) W = W + µX + νX with µ and ν solutions of(2.8) 2 ν x − νx + y = 0 and y − µx − νx + 3 µνx = 0 . It follows that X and W satisfy the Hahn algebra[[ X, W ] , X ] = x + x X + x X + x W (2.9) [ W , [ X, W ]] = y + y X + x W + x { X, W } (2.10)where x = x , x = x − µx , x = x − νx , x = x (2.11) y = y − µx , y = y − µx − νx + µ x (2.12) Remark 1.
The word generically is used in the statement of Proposition 2.1 to indicatethat it assumes freeness of the parameters. We shall observe in the specific cases that willbe discussed in the following that there are instances where the parameters need to belong to C or where there are no solutions if the parameters are not constrained. In such situations,the algebraic equivalence would not prevail. Remark 2.
It is interesting to point out that the truncated reflection algebra attached to theYangian of sl(2) has been shown in [4] to be isomorphic to the Hahn algebra defined above.
We bring at this point the generic Lie algebra A for a bispectral pair X and Y . Start withthe linear relations: [ X, Y ] = Z (2.13) [ Z, X ] = aX + c Y + d (2.14) [ Y, Z ] = bY + c X + d , (2.15)where a , b , c , c , d and d are real parameters. With the presence of the constants d and d in the right-hand sides of (2.14) and (2.15), we are de facto assuming in general thepresence of an additional central element I which we will omit writing. It is readily seenthat one must have a = b for the Jacobi identity [ X, [ Y, Z ]] + [ Y, [ Z, X ]] + [ Z, [ X, Y ]] = 0 tobe verified. Now assume that X has a nondegenerate discrete real spectrum { λ p , p ∈ Z } ina representation space with u p the corresponding eigenvectors. It is easy to see that for Y N. CRAMP´E, L. VINET, AND A. ZHEDANOV to act tridiagonally on that space:
Y u p = f p +1 u p +1 + g p u p + f p u p − , in view of (2.14), onemust have(2.16) − ( λ p +1 − λ p ) = c . This implies the condition c <
0. To have (Leonard) duality requires that X and Y beon the same footing. To that end, one would thus ask that there be reciprocally a basis inwhich Y has also a real discrete spectrum and X is tridiagonal. This would necessitate inturn that c <
0. In the following, we shall want however to cover more general bispectralsituations where the spectrum of Y could be pure imaginary or continuous. We shall hencenot impose restrictions on c . As a consequence, the Heun algebras to be defined will relateto bispectral problems that are not only the purely discrete ones. Definition 2.2.
The generic Lie algebra A over R for the bispectral pair X and Y is definedby (2.14) and (2.15) with (2.17) a = b, c < . As already indicated this corresponds to eliminating the non-linear terms in (1.1) and (1.2) . The following element(2.18) C = 2 d X + 2 d Y + b { X, Y } + c X + c Y + Z is central in A . Remark 3.
It will be useful as we proceed to take note of the following two obvious facts.First, like the equations (1.1) and (1.2) of which they are a special case, the equations (2.14) and (2.15) preserve their form under affine transformations of X and Y . Second, it is alsoclear that the elements X = V XV − and Y = V Y V − obtained from performing on X and Y the same similarity operation will satisfy the same relations as X and Y . Proposition 2.2.
Let X and Y be the generators of A . The Heun operator (2.19) W = r [ X, Y ] + r { X, Y } + r X + r Y + r together with X satisfy the relations of the Heun algebra H of the Lie type with x = d r − c r x = 2 d r − c r + br x = 2 br x = c (2.20) y = − d r r − d r r + c r r + d r + ( d c − bd )( r − r ) − br r − r r C (2.21) y = − d r r + c r + 8 d r r + c r + ( c c − b )( r − r ) − br r − br r − r C (2.22) y = 12 d r + 6 c r r − br r y = 8 c r . (2.23) Remark 4.
The defining relations of A are invariant under the exchange of X and Y provided one performs: c ↔ c , d ↔ d . Taking into account the definition of W (2.19) ,we see that the algebra realized by Y and W is the same as the one stemming from the pair X , Y with the coefficients obtained from the ones given in Proposition 2.2 by doing also thechanges r → − r , r ↔ r . Remark 5.
The central element Ω of H with the parameters given above is (2.24) Ω = ( r + c ( r − r )) C − c r r b − c r + 2 r c r d + 2 d r r + d ( r − r ) . EUN ALGEBRAS OF LIE TYPE 5
In the next three sections, we shall identify the specific Heun algebras that are obtainedwhen the algebra A is isomorphic to su (2), su (1 ,
1) or the oscillator algebra ho . Let us recordhere the definitions of these algebras.The Lie algebras su (2) and su (1 ,
1) are generated by J , J and J subject to(2.25) [ J , J ] = ± iJ , [ J , J ] = iJ , [ J , J ] = iJ . The upper (resp. lower) sign is associated to su (2) (resp. su (1 , c = J + J ± J . The harmonic oscillator algebra ho is solvable and has four generators N , A , A † and I that satisfy the commutation relations [20]:(2.27) [ N, A ] = − A, [ N, A † ] = A † , [ A, A † ] = I, [ I, N ] = [
I, A ] = [
I, A † ] = 0 . It is familiar that k = N − A † A is central in ho .3. The case of the Lie algebra su (2)We shall focus here on su (2) and assume therefore in this section that the generators( J , J , J ) obey the commutation relations of this algebra, that is those corresponding tothe upper signs in (2.25) and (2.26).3.1. Specialization.
Take X and Y to be(3.1) X = αJ + βJ , Y = J with the coefficients α = cos θ and β = − sin θ .Let us first explain that there is no loss of generality with this choice keeping in mindRemark 3. We can always pick Y in a preferred direction and allowing for scaling have it asabove. Assume then that X is initially of the generic form l J + l J + l J . The adjointrepresentation will transform this su (2) element while preserving l = l + l + l . Undera rotation about the 3-axis, Y does not change and the other one can be transformed into αJ + βJ with α + β = l . Scaling finally by 1 /l we arrive at the expression for X in (3.1).It is readily checked that ( X, Y ) satisfy the relations (2.14) and (2.15) of the algebra A with(3.2) d = d = 0 , c = − , b = α and c = − ( α + β ) = − . The fact that d = d = 0 indicates that there is no center and that the algebra is three-dimensional. The Casimir C of A is related to c as follows:(3.3) C = α { X, Y } − X + c Y + ([ X, Y ]) = − β c . Observe that X is obtained from Y through an automorphism:(3.4) X = U ( θ ) Y U ( θ ) − with U ( θ ) = e iθJ . Let W be given by (2.19). From the results of the previous section, we can see that thealgebras generated by the pairs ( X, W ) and (
Y, W ) will be similar We shall therefore onlyfocus on the Heun algebra generated by X and W . N. CRAMP´E, L. VINET, AND A. ZHEDANOV
Proposition 3.1.
The elements X and W associated to su (2) as per (3.1) satisfy the rela-tions of the Heun algebra of Lie type with x = − c r , x = r α − c r , x = 2 r α , x = c , (3.5) y = r ( c r − αr ) − r r C , (3.6) y = ( r − r )( c + α ) − r r α + c r − αr r − r − r C , (3.7) y = − r ( r + αr ) , y = − r , (3.8) with α = cos θ and c = − ( α + β ) = − . Let us remark that the central element Ω of H with the parameters given above is(3.9) Ω = ( c ( r − r ) + r ) C − c r (2 r α + r ) . Proposition 3.2.
The Heun algebra of su (2) type is not isomorphic to the Hahn algebraover R . This is seen by observing that relation (2.8) in Proposition 2.1 for the parameters µ and ν has necessarily complex solutions in this su (2) case, namely(3.10) ν = − r exp( ± iθ ) , µ = − r − exp( ± iθ ) r . This is indicative of the fact that the dual Hahn polynomials cannot be obtained by tridi-agonalization of the Krawtchouk difference operator (see later).3.2. su (2) Heun operators and Hamiltonians for generalized tops.
We have alreadynoted (see Remark 3) that the bispectral pair of operators
X, Y can be chosen in variousequivalent ways from the algebraic perspective. Different choices will however lead to modi-fied expressions for the corresponding algebraic Heun operator. To point out the connectionbetween Heun operators of su (2) type and quantum Hamiltonians for tops, instead of (3.1),it will be convenient to rather adopt for X and Y (3.11) X = J + βJ , Y = J − βJ , with β = 0 an arbitrary parameter. One easily convinces oneself that this choice is equivalentto the preceding one. Start from X and Y as in (3.1) and conjugate these operators by U ( − θ/
2) with U ( θ ) given in (3.4). Scaling by sec( θ/
2) then gives the X and Y of (3.11). Itis easy to see that these two operators also satisfy the relations (2.14)-(2.15) with(3.12) d = d = 0 , a = b = (1 − β ) and c = c = − (1 + β ) . One finds that the algebraic Heun operator (2.19) is in this case of the form(3.13) W = σ (cid:0) J − β J (cid:1) + m J + m J + m J + m with σ, m i , i = 1 , , ,
4, arbitrary parameters. The first term σ ( J − β J ) is equivalent(up to an affine transformation) to the Hamiltonian of the quantum Euler top (see [17, 22]).The complete operator (3.13) corresponds to the Hamiltonian of a Euler top with additional“magnetic” interactions accounted for by the linear terms in (3.13). In classical mechanicsthis is the Zhukovsky-Volterra gyrostat [1], [16]. Note moreover that a similar Hamiltonianwas exploited to describe spin systems with anisotropy [24]. We thus see that the Heunoperator pencil on the su (2) algebra is equivalent to the Hamiltonian of a generalized Eulertop (or quantum Zhukovskii-Volterra gyrostat). EUN ALGEBRAS OF LIE TYPE 7 The case of the Lie algebra su (1 , Specializations.
The situation where su (1 ,
1) is the underlying algebra will have, notsurprisingly, close similarities with the picture in the su (2) case; the main differences willcome from the richer orbit structure. Let us indeed explain how the choices for the bispectralpair ( X, Y ) can be standardized. Throughout this section, the elements J κ with κ = 1 , , su (1 ,
1) and to have one discrete spectrum in play, we shallagain take Y to be the compact generator J . Following a reasoning similar to the one givenin the last section, we observe that a generic element l J + l J + l J can be transformedinto αJ + βJ under a rotation while not changing Y . The difference here is that the adjointaction of su (1 ,
1) preserves the non-definite form l = l + l − l . This will hence require that β − α = l . There will therefore be three distinct cases according to whether l is negative,positive or zero; elements in these classes are respectively said to be elliptic, hyperbolic andparabolic.The upshot of this is that the bispectral operators X and Y in the su (1 ,
1) case will haveexactly the same form as those of the su (2) case, namely(4.1) X = αJ + βJ , Y = J but will fall into the three categories where allowing for scaling, α and β will be parametrizedas follows:elliptic l < α = cosh θ β = − sinh θ (4.2) hyperbolic l > α = − cos φ β = 1(4.3) parabolic l = 0 α = 1 β = 1(4.4)with 0 < φ < π .Owing to the fact that X and Y have the same general expressions in both the su (2) and su (1 ,
1) cases many formulas will be almost identical to those for su (2) as we determine theHeun algebras associated to su (1 , X and Y will again obey the relations (2.14) and (2.15)of the algebra A with the only change with respect to (3.2) being(4.5) c = β − α . The Casimir operator C of A will be related exactly as in (3.3) to the Casimir element c of su (1 ,
1) given by the lower part of (2.26). We may point out that the relations (3.4) remaintrue for the elliptic case. Always keeping the definition (2.19) for W , Proposition 3.1 readilytranslates to the su (1 ,
1) case.
Proposition 4.1.
The elements X and W associated to su (1 , as per (4.1) satisfy therelations of the Heun algebra of Lie type with the coefficients x s and y s , s = 0 , , , as in (3.6) - (3.8) and with α given by (4.2) , (4.3) , (4.4) for each of the possible classes and c given by (4.5) . The central element Ω of the corresponding Heun algebras is again given by (3.9) withthe appropriate α and c .4.2. Connections to the Hahn algebra .
Having found for su (1 , Y is of elliptic, hyperbolic or parabolic type, we now observe N. CRAMP´E, L. VINET, AND A. ZHEDANOV that these algebras are isomorphic to the Hahn algebra. Indeed, the relation (2.8) for theparameters µ and ν has real solutions that read in light of (4.5):(4.6) ν = − r α ± β and µ = − r α ± β − r . For the parabolic case α = β = 1, only the solution with the upper sign is permitted. Theoperators X and W therefore satisfy the Hahn algebra with the parameters x = ( α − β ) r , x = ± βr , x = ± βr , x = β − α , (4.7) y = ∓ βr r − r r C, y = β ( r − r ) ∓ βr r − r C . (4.8) 5.
The case of the harmonic oscillator algebra
We shall focus in this section on the situation where the algebra A of Definition 2.2 isisomorphic to the harmonic oscillator algebra ho (2.27). Consider for the bispectral pair( X, Y ):(5.1) X = N + χ ( A + A † ) + χ I , Y = N , where χ is a real constant. Here again, it is straightforward to check that the relations (2.14)and (2.15) of the algebra A are satisfied with(5.2) b = 1 c = c = − d = d = χ and one readily observes that X, Y, Z and I also form a basis for ho . The Casimir C of A becomes:(5.3) C = 2 χ ( X + Y ) + { X, Y } − X − Y + Z = χ [4 k + χ − k = N − A † A . We now bring on the associated algebraic Heun operator W givenin (2.19) as the generic bilinear expression in these X and Y . Anew, it is seen that thegenerator Y is obtained from X by an automorphism:(5.4) Y = U ( χ ) XU ( χ ) − with U ( χ ) = e χ ( A − A † ) . Recalling from the argument given in Section 2 that the Heun algebras generated by thepairs (
X, W ) and (
Y, W ) are isomorphic, we shall only concentrate on the former with thefollowing result directly obtained from Proposition 2.2.
Proposition 5.1.
The elements X and W associated to ho as per (5.1) and (2.19) satisfythe relations of the Heun algebra of Lie type with x = χ r + r , x = 2 χ r + r + r , x = 2 r , x = − , (5.5) y = χ (2 r − r − r − r r + 2 r r ) − ( r + r ) r − r r C , (5.6) y = 4 χ r (2 r − r ) − ( r + r ) − r r − r C , (5.7) y = 12 χ r − r ( r + r ) , y = − r . (5.8)With these parameters, the central element Ω of H is(5.9) Ω = ( r + r − r ) C + χ ( r − r ) − χ r ( r − r ) + 2 r r + r . In this case, looking at the relations (2.8) for µ and ν one finds that there are no solutions.The Heun algebra associated to the oscillator algebra is therefore not isomorphic to the Hahnalgebra. EUN ALGEBRAS OF LIE TYPE 9 Realizations in terms of difference operators
It is well known that various families of orthogonal polynomials are related to the Liealgebras that we have considered so far [10, 5]. We shall review these results in this sectionby considering different models for su (2), su (1 ,
1) and ho and by showing that the associatedoperators Y are realized within the corresponding representations as the operators of whichthe polynomials are eigenfunctions. Since X will always be multiplication by the variable,this will allow us to say that the Heun algebras of Lie type that have been identified are theHeun algebras of Krawtchouk, Meixner, Meixner-Pollaczek, Laguerre and Charlier type.6.1. su (2) . There is a model of su (2) in terms of the difference operators T ± (6.1) T ± f ( x ) = f ( x ± . It has the generators given as follows: J = sin ( θ/
2) ( x − N ) T + + cos( θ ) ( x − N/ − cos ( θ/ xT − , (6.2) J − = sin θ (cid:0) ( x − N ) T + − x + N + xT − (cid:1) , (6.3) J + = sin θ (cid:0) tan ( θ/ N − x ) T + − x + N − cot ( θ/ xT − (cid:1) , (6.4)with J = ( J + + J − ) and J = − i ( J + − J − ). As per Section 3.1, we have the bispectraloperators X and Y given by X = cos( θ ) J − sin( θ ) J = x − N Y = J . (6.5)One readily recognizes in view of (6.2), that Y becomes the difference operator of theKrawtchouk polynomials K n ( x ; sin ( θ/ , N ) [15]:(6.6) Y K n ( x ; sin ( θ/ , N ) = (cid:18) n − N (cid:19) K n ( x ; sin ( θ/ , N ) . The operators J ± = J ± iJ are moreover lowering and raising operators for these polyno-mials: J − K n ( x ; sin ( θ/ , N ) = n cot( θ/ K n − ( x ; sin ( θ/ , N ) , (6.7) J + K n ( x ; sin ( θ/ , N ) = ( N − n ) tan( θ/ K n +1 ( x ; sin ( θ/ , N ) . (6.8)The associated operator WW = sin ( θ N − x )( N r + r − r − r − r x ) T + + 2 r cos( θ ) x + ρ x + ρ + cos ( θ x ( N r − r + r − r − r x ) T − (6.9)is hence the Heun-Krawtchouk operator. The parameters ρ and ρ are given by ρ = cos( θ )( r − N r ) + r , ρ = 12 cos( θ ) N ( N r − r ) − N r r . (6.10)The algebra given in Proposition 3.1 can thus appropriately be called the Heun-Krawtchoukalgebra since its generators can be realized using the bispectral operators of the Krawtchoukpolynomials. su (1 , . Three models of the positive discrete series representation will be presentedin correspondance with the situations where Y is of elliptic, hyperbolic and parabolic type.The orthogonal polynomials of Meixner, Meixner-Pollaczek and Laguerre will arise corre-spondingly [18, 14]. Furthermore, owing to the fact that the associated Heun algebras areisomorphic to the Hahn one, it will be recorded that the respective Heun operators W canbe transformed into the Hahn, Continuous Hahn and Jacobi operators.6.2.1. Elliptic case.
The generators of su (1 ,
1) can be realized as follows: J = − ( x + κ ) sinh ( θ/ T + + ( x + κ/
2) cosh( θ ) − cosh ( θ/ xT − (6.11) J − = − sinh θ (cid:0) ( x + κ ) T + − x − κ + xT − (cid:1) (6.12) J + = − sinh θ (cid:0) tanh ( θ/ x + κ ) T + − x − κ + coth ( θ/ xT − (cid:1) . (6.13)in terms of the shift operators (6.1). As per Section 4.1, we introduce the following bispectraloperators X = cosh( θ ) J − sinh( θ ) J = x + κ Y = J . (6.14) Y is then identified as the difference operator of the Meixner polynomials M n ( x ; κ, tanh ( θ/ Y M n ( x ; κ, tanh ( θ/ (cid:16) κ n (cid:17) M n ( x ; κ, tanh ( θ/ J + and J − are ladder operators for these polynomials: J − M n ( x ; κ, tanh ( θ/ n coth( θ/ M n − ( x ; κ, tanh ( θ/ J + M n ( x ; κ, tanh ( θ/ κ + n ) tanh( θ/ M n +1 ( x ; κ, tanh ( θ/ . (6.17)The associated operator WW = − ( x + κ )(2 r x + r κ − r + r + r ) sinh ( θ/ T + + 2 r cosh( θ ) x + ρ x + ρ + x ( r x + r κ + r − r + r ) cosh ( θ/ T − (6.18)is hence the Heun-Meixner operator. We have introduced ρ = cosh( θ )(2 r κ + r ) + r , ρ = cosh( θ ) κ r κ + r ) + r κ r . (6.19)The algebra given in Proposition 4.1 may thus be called the Heun-Meixner algebra sinceit can be realized using the bispectral operators of the Meixner polynomials. We knowthat this Heun-Meixner algebra is isomorphic to the Hahn one. This can be underscoredwithin the present su (1 ,
1) representation by recovering the Hahn operator from W recallingthe observations made in Section 4.2. Consider to simplify, the conjugated operator f W = x ( θ/ W tanh x ( θ/
2) which reads as follows f W = 1tanh x ( θ/ W tanh x ( θ/ − ( r e − θ + r ) X − r e − θ X = − r sinh( θ ) h x ( x + r κ + r − r + r r ) T − + ( κ + x )( x + r κ − r + r + r r ) T + − (2 x + κ )( x + r κ + r r ) i + r (6.20) EUN ALGEBRAS OF LIE TYPE 11
We readily recognize the difference operator of the Hahn polynomials in the square bracketof the second line of (6.20).6.2.2.
Hyperbolic case.
The operators J = 12 i sin φ (cid:0) e iφ ( λ − ix ) T + + 2 ix cos( φ ) − e − iφ ( λ + ix ) T − (cid:1) (6.21) J ± = e iφ i sin φ (cid:0) e ± iφ ( λ − ix ) T + + 2 ix − e ∓ iφ ( λ + ix ) T − (cid:1) (6.22)satisfy the su (1 ,
1) commutation relations with the shift operators T ± defined by T ± f ( x ) = f ( x ± i ). With X and Y given according to Section 4.1 by X = − cos( φ ) J + J = x sin( φ ) , Y = J , (6.23)we see that Y is realized in this instance as the difference operator of the Meixner-Pollaczekpolynomials P ( λ ) n ( x ; φ ) [15]:(6.24) Y P ( λ ) n ( x ; φ ) = ( n + λ ) P ( λ ) n ( x ; φ ) , while J + and J − act as raising and lowering operators: J + P ( λ ) n ( x ; φ ) = ( n + 1) P ( λ ) n +1 ( x ; φ ) , J − P ( λ ) n ( x ; φ ) = (2 λ + n − P ( λ ) n − ( x ; φ ) . (6.25)The associated operator WW = ( ix − λ ) (cid:18) ir x + r − r ir φ ) (cid:19) T + + 2 r cos( φ ) x + ρ x + r + e − iφ ( λ + ix ) (cid:18) ir x − r − r ir φ ) (cid:19) T − (6.26)is hence the Heun-Meixner-Pollaczek operator. Here ρ = sin( φ ) r +cot( φ ) r . In this case thealgebra of Proposition 4.1 is really the Heun-Meixner-Pollaczek algebra. In this realization,upon scaling and conjugating W according to f W = r e iφx W e − iφx one finds f W = ( λ − ix ) (cid:18) r − r r − ir r sin( φ ) − ix (cid:19) T + − x − xr r sin( φ ) + r r +( λ + ix ) (cid:18) r − r r + ir r sin( φ ) + ix (cid:19) T − . (6.27)This operator is the difference operator that is diagonalized by the continuous Hahn polyno-mials ( − ix p n ( x ; λ, r − r r + ir r sin( φ ) , λ, r − r r − ir r sin( φ ) ) with eigenvalues − n − (2 n + 1) λ + ( λ + n ) r + r r .6.2.3. Parabolic case.
A model of su (1 ,
1) in terms of differential operator is given by J = − x d dx − (1 + a − x ) ddx + 1 + a J − = x d dx + (1 + a ) ddx (6.29) J + = x d dx + (1 + a − x ) ddx + x − − a (6.30) where a is a free parameter associated to this realization. For the parabolic case, the bispectaloperators X and Y were taken to be X = J + J = 12 x and Y = J (6.31)in Section 4.1. It follows that Y can here be identified with the difference operator of theLaguerre polynomials L ( a ) n ( x ) [15]:(6.32) Y L ( a ) n ( x ) = ( n + ( a + 1) / L ( a ) n ( x ) . One also gets J + L ( a ) n ( x ) = − ( n + 1) L ( a ) n +1 ( x ) , J − L ( a ) n ( x ) = − ( a + n ) L ( a ) n − ( x ) . (6.33)The associated operator W (6.34) W = − x ( r x + r ) d dx + ( r x + ( r − r − r a + r ) x − r (1 + a )) ddx + ρ x + ρ is hence the Heun-Laguerre operator. We have defined ρ = 12 ( r − r + ( a + 2) r ) and ρ = r + 12 ( r − r + r )( a + 1) . (6.35)In this case, the algebra given in Proposition 4.1 should hence be referred to as the Heun-Laguerre algebra since we have a realization of it based on the bispectral operators of theLaguerre polynomials. It can be also showed that the eigenvalue equation W f ( x ) = λf ( x )becomes the confluent Heun equation [13], [19]. Let us remark that this connection betweenthe confluent Heun equation and the Heun operators associated to the Laguerre differentialequation had already been pointed out in [11]. Finally note that the conjugated operator f W = e − x/ W e x/ f W = e − x/ W e x/ − ( r / r ) X − r X = − x ( r x + r ) d dx + (( r − r a − r ) x − r (1 + a )) ddx + 12 ( r − r )( a + 1) + r (6.36)is recognized to be the Jacobi differential operator which together with X generates theHahn algebra. Indeed, f W is diagonalized by the Jacobi polynomial P ( a, − r /r ) n (cid:16) r xr + 1 (cid:17) with eigenvalues ( r − r )( n + ( a + 1)) − r n ( n + a ) + r .6.3. Harmonic oscillator algebra ho . The oscillator algebra can also be realized as followsin terms of the shift operators (6.1):(6.37) N = − xT − + x + χ − χ T + , A † = − χ + 1 χ xT − , A = χ ( T + − . As in Section 5, the bispectral pair (
X, Y ) is X = N + χ ( A + A † ) + χ = x , Y = N . (6.38)The operator Y is the difference operator of the Charlier polynomial C n ( x, χ ) [15]:(6.39) Y C n ( x, χ ) = nC n ( x, χ ) , and A † and A are their raising and lowering operators:(6.40) A † C n ( x, χ ) = − χC n +1 ( x, χ ) , AC n ( x, χ ) = − nχ C n − ( x, χ ) . EUN ALGEBRAS OF LIE TYPE 13
The associated operator W (6.41) W = x ( r − r − r − r x ) T − + χ ( r − r − r − r x ) T + +2 r x +(2 r χ + r + r ) x + r χ + r is hence the Heun-Charlier operator. The algebra given in Proposition 5.1 should thereforebe called the Heun-Charlier algebra since it is realized with X = x and W constructed from x and the Charlier operator. 7. Concluding remarks
The results presented here provide a comprehensive picture of the Heun operators andalgebras associated to orthogonal polynomials that admit a Lie theoretical interpretation.This complements the previous studies of the Jacobi and Hahn polynomials from this Heunangle which led respectively to the description of the standard Heun differential operatorand its discrete version within the framework of bispectral problems. Missing is the paralleltreatment of the Racah polynomials especially since the q → Acknowledgments:
N.C. is gratefully holding a CRM–Simons professorship. The re-search of L.V. is supported in part by a Natural Science and Engineering Council (NSERC)of Canada discovery grant and that of A.Z. by the National Science Foundation of China(Grant No. 11711015).
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Theoret. Math. Phys. 89 (1991)1146. † Institut Denis-Poisson CNRS/UMR 7013 - Universit´e de Tours - Universit´e d’Orl´eans;Parc de Grammont, 37200 Tours, France.
E-mail address : [email protected] ∗ Centre de recherches, math´ematiques, Universit´e de Montr´eal; P.O. Box 6128, Centre-ville Station; Montr´eal (Qu´ebec), H3C 3J7, Canada.
E-mail address : [email protected] ‡ Department of Mathematics, School of Information; Renmin University of China; Bei-jing 100872, China.
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