Recurrence Relations of the Multi-Indexed Orthogonal Polynomials IV : closure relations and creation/annihilation operators
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Recurrence Relations ofthe Multi-Indexed Orthogonal Polynomials IV :closure relations and creation/annihilation operators
Satoru Odake
Faculty of Science, Shinshu University,Matsumoto 390-8621, Japan
Abstract
We consider the exactly solvable quantum mechanical systems whose eigenfunctionsare described by the multi-indexed orthogonal polynomials of Laguerre, Jacobi, Wilsonand Askey-Wilson types. Corresponding to the recurrence relations with constantcoefficients for the M -indexed orthogonal polynomials, it is expected that the systemssatisfy the generalized closure relations. In fact we can verify this statement for small M examples. The generalized closure relation gives the exact Heisenberg operatorsolution of a certain operator, from which the creation and annihilation operators ofthe system are obtained. The exactly solvable quantum mechanical systems described by the classical orthogonal poly-nomials in the Askey scheme have two properties, shape invariance and closure relation [1, 2].These two properties are sufficient conditions for the exact solvability. The former leads toexact solvability in the Schr¨odinger picture and the latter to that in the Heisenberg picture.The closure relation gives the exact Heisenberg operator solution of the sinusoidal coordinateand its negative/positive frequency parts provide the creation/annihilation operators [2].After the pioneer works [3, 4], new type of orthogonal polynomials, exceptional/multi-indexed orthogonal polynomials, have been studied intensively [5]–[29] (and references therein).The exceptional orthogonal polynomials (in the wide sense) {P n ( η ) | n ∈ Z ≥ } satisfy sec-ond order differential or difference equations and form a complete set, but there are miss-ing degrees, by which the constraints of Bochner’s theorem and its generalizations [30,1] are avoided. We distinguish the following two cases; the set of missing degrees I = Z ≥ \{ deg P n | n ∈ Z ≥ } is case (1): I = { , , . . . , ℓ − } , or case (2) I 6 = { , , . . . , ℓ − } ,where ℓ is a positive integer. The situation of case (1) is called stable in [9]. By using themulti-step Darboux transformations with appropriate seed solutions [32, 33], many exactlysolvable deformed quantum mechanical systems and various exceptional orthogonal polyno-mials with multi-indices can be obtained. When the virtual state wavefunctions are usedas seed solutions, we obtain case (1) and call them multi-indexed orthogonal polynomials[11, 19, 18]. When the eigenstate or pseudo virtual state wavefunctions are used as seedsolutions, we obtain case (2) [13, 20].The shape invariance of the original system is inherited by the deformed systems forcase (1), but lost for case (2) (some remnant remains [12, 13]). On the other hand, theclosure relation does not hold in the deformed systems. This is because the closure relationis intimately related to the three term recurrence relations. Roughly speaking, three termsin the r.h.s of the closure relation (2.3) correspond to three term recurrence relations. Thethree term recurrence relations characterize the ordinary orthogonal polynomial (Favard’stheorem [31]). Since the exceptional orthogonal polynomials are not ordinary orthogonalpolynomials, they do not satisfy the three term recurrence relations. They satisfy recurrencerelations with more terms [8][23]–[29]. For example, the simplest exceptional orthogonalpolynomials in [3] satisfy five term recurrence relations [8]. In our previous papers [27,29] we discussed the recurrence relations with constant coefficients for the multi-indexedorthogonal polynomials of Laguerre, Jacobi, Wilson and Askey-Wilson types. Correspondingto these recurrence relations with constant coefficients, we expect that the deformed systemssatisfy the generalized closure relations (3.2), which have more than three terms in the r.h.s.The generalized closure relation gives the exact Heisenberg operator solution of a certainpolynomial in the sinusoidal coordinate, whose negative/positive frequency parts provide thecreation/annihilation operators. The purpose of this paper is to expound these interestingsubjects.This paper is organized as follows. In section 2 we recapitulate the closure relation.The closure relation is a commutator relation between the Hamiltonian and the sinusoidalcoordinate η . It gives the exact Heisenberg operator solution of η and its negative/positivefrequency parts provide the creation/annihilation operators. In section 3 we generalize theclosure relation. If some function X satisfies the generalized closure relation, we can calcu-2ate the exact Heisenberg operator solution of X and its negative/positive frequency partsprovide the creation/annihilation operators. The discussion in this section is valid for gen-eral quantum mechanical systems but the existence of the operator X is assumed. In section4 we discuss that such operators X exist in the deformed systems described by the multi-indexed orthogonal polynomials of Laguerre, Jacobi, Wilson and Askey-Wilson types. Theoperator X is a polynomial in η giving the recurrence relations with constant coefficients forthe multi-indexed orthogonal polynomials. The exact Heisenberg operator solution of X andthe creation/annihilation operators are obtained. Explicit examples are given in section 5.The final section is for a summary and comments. In Appendix A we present some formulasfor the diagonalization of a certain matrix. In Appendix B we present more examples. In this section we recapitulate the closure relation and creation/annihilation operators.In many exactly solvable quantum mechanical models such as harmonic oscillator, radialoscillator, Darboux-P¨oschl-Teller potential and their discrete quantum mechanical counter-parts, the eigenfunction φ n ( x ) has the following factorized form H φ n ( x ) = E n φ n ( x ) , E < E < E < · · · , (2.1) φ n ( x ) = φ ( x ) ˇ P n ( x ) ( n = 0 , , , . . . ) , ˇ P n ( x ) def = P n (cid:0) η ( x ) (cid:1) , (2.2)where the function η = η ( x ) is called the sinusoidal coordinate and P n ( η )’s are classicalorthogonal polynomials in η : Hermite, Laguerre, Jacobi, Wilson, Askey-Wilson, ( q -)Racah,etc. For these systems, the Hamiltonian H and the sinusoidal coordinate η satisfy therelation, (cid:2) H , [ H , η ] (cid:3) = ηR ( H ) + [ H , η ] R ( H ) + R − ( H ) , (2.3)where R i ( z )’s are polynomials in z , R ( z ) = r (2)0 z + r (1)0 z + r (0)0 , R ( z ) = r (1)1 z + r (0)1 , R − ( z ) = r (2) − z + r (1) − z + r (0) − , (2.4)and r ( j ) i ’s are real constants ( r (2)0 = r (1)1 = r (2) − = 0 for the ordinary QM, H = p + U ( x )). Thisrelation (2.3) is called the closure relation; The double commutator (ad H ) η = [ H , [ H , η ]] isexpressed as a linear combination of (ad H ) η = η , (ad H ) η = [ H , η ] and 1 with H -dependentcoefficients R i ( H ), which are multiplied from the right. The closure relation implies that3ultiple commutators (ad H ) n η = [ H , [ H , · · · , [ H , η ] · · · ]] ( n times) are also expressed as alinear combination of η , [ H , η ] and 1 with H -dependent coefficients,(ad H ) n η = ηR ( H ) α + ( H ) n − − α − ( H ) n − α + ( H ) − α − ( H ) + [ H , η ] α + ( H ) n − α − ( H ) n α + ( H ) − α − ( H )+ R − ( H ) α + ( H ) n − − α − ( H ) n − α + ( H ) − α − ( H ) − R − ( H ) R ( H ) − δ n , (2.5)where α ± ( z ) are defined by α ± ( z ) def = (cid:0) R ( z ) ± p R ( z ) + 4 R ( z ) (cid:1) , (2.6) R ( z ) = α + ( z ) + α − ( z ) , R ( z ) = − α + ( z ) α − ( z ) . (2.7)From this, the Heisenberg operator solution of the sinusoidal coordinate η = η ( x ) can beobtained explicitly e i H t ηe − i H t = ∞ X n =0 ( it ) n n ! (ad H ) n η = a (+) e iα + ( H ) t + a ( − ) e iα − ( H ) t − R − ( H ) R ( H ) − , (2.8)where a ( ± ) = a ( ± ) ( H , η ) are a ( ± ) def = ± (cid:16) [ H , η ] − (cid:0) η + R − ( H ) R ( H ) − (cid:1) α ∓ ( H ) (cid:17)(cid:0) α + ( H ) − α − ( H ) (cid:1) − . (2.9)Although the operators a ( ± ) contain H in square roots, they do not cause any problem whenacting on the eigenfunction φ n ( x ) of H , because the operator H is replaced by the eigenvalue E n . For each model, we can check R ( E n ) >
0, namely α + ( E n ) > > α − ( E n ) . (2.10)So e iα + ( H ) t and e iα − ( H ) t in (2.8) are interpreted as the negative and positive frequency partsrespectively. As for the harmonic oscillator, the coefficients of the negative and positivefrequency parts, a (+) and a ( − ) , provide the creation and annihilation operators respectively.Since P n ( η ) are orthogonal polynomials, they satisfy the three term recurrence relations(we set P n ( η ) = 0 for n < ηP n ( η ) = A n P n +1 ( η ) + B n P n ( η ) + C n P n − ( η ) ( n ≥ . (2.11)This and (2.2) imply ηφ n ( x ) = A n φ n +1 ( x ) + B n φ n ( x ) + C n φ n − ( x ) ( n ≥ . (2.12)4ction of (2.8) on φ n ( x ) is e i H t ηe − i H t φ n ( x ) = e iα + ( E n ) t a (+) φ n ( x ) + e iα − ( E n ) t a ( − ) φ n ( x ) − R − ( E n ) R ( E n ) − φ n ( x ) . On the other hand the l.h.s. turns out to be e i H t ηe − i H t φ n ( x ) = e i H t ηe − i E n t φ n ( x )= e − i E n t e i H t (cid:0) A n φ n +1 ( x ) + B n φ n ( x ) + C n φ n − ( x ) (cid:1) = e i ( E n +1 −E n ) t A n φ n +1 ( x ) + B n φ n ( x ) + e i ( E n − −E n ) t C n φ n − ( x ) . Comparing these t -dependence (see (2.10)), we obtain α ± ( E n ) = E n ± − E n , (2.13) a (+) φ n ( x ) = A n φ n +1 ( x ) , a ( − ) φ n ( x ) = C n φ n − ( x ) , (2.14) − R − ( E n ) R ( E n ) − = B n . (2.15)Therefore a (+) and a ( − ) are creation and annihilation operators, respectively. (The normal-izability of a (+) φ n ( x ) depends on the model and n .)The creation and annihilation operators for eigenpolynomials are obtained by the simi-larity transformation in terms of the groundstate wavefunction φ ( x ). The similarity trans-formed Hamiltonian is e H def = φ ( x ) − ◦ H ◦ φ ( x ) , e H ˇ P n ( x ) = E n ˇ P n ( x ) . (2.16)Corresponding to (2.3), this also satisfies the closure relation (cid:2) e H , [ e H , η ] (cid:3) = ηR ( e H ) + [ e H , η ] R ( e H ) + R − ( e H ) . (2.17)From the creation and annihilation operators a ( ± ) = a ( ± ) ( H , η ), we obtain the creation andannihilation operators for eigenpolynomials,˜ a ( ± ) def = φ ( x ) − ◦ a ( ± ) ( H , η ) ◦ φ ( x ) = a ( ± ) ( e H , η ) , (2.18)˜ a (+) ˇ P n ( x ) = A n ˇ P n +1 ( x ) , ˜ a ( − ) ˇ P n ( x ) = C n ˇ P n − ( x ) . (2.19)The normalization constants of the orthogonality relations, h n ,( φ n , φ m ) = h n δ nm , ( f, g ) def = Z x x dx f ∗ ( x ) g ( x ) dx, (2.20)5re related to the coefficients of the three term recurrence relations. By calculating ( φ n , ηφ m ) =( ηφ n , φ m ) as h n ( A m δ n,m +1 + B m δ n,m + C m δ n,m − ) = h m ( A n δ n +1 ,m + B n δ n,m + C n δ n − ,m ) , we obtain A n h n +1 = C n +1 h n ( n ≥ . (2.21) In this section we generalize the closure relation. We consider general quantum mechanicalsystems with the Hamiltonian H and the coordinate x .Let us consider an operator X , which is a function of η = η ( x ), X = X ( η ) = X (cid:0) η ( x ) (cid:1) def = ˇ X ( x ) . (3.1)We assume that the K -times commutator of H and X has the following form:(ad H ) K X = K − X i =0 (ad H ) i X · R i ( H ) + R − ( H ) . (3.2)Here R i ( z ) = R Xi ( z ) is a polynomial in z and their coefficients are real numbers depending onthe choice of X . We call this the closure relation of order K . The closure relation reviewedin § K = 2.The closure relation of order K (3.2) implies that the multiple commutation relation of H and X , (ad H ) n X , can be expressed as a linear combination of (ad H ) i X (0 ≤ i ≤ K − H -dependent coefficients, which are multiplied from the right,(ad H ) n X = K − X i =0 (ad H ) i X · R [ n ] i ( H ) + R [ n ] − ( H ) ( n ≥ . (3.3)The initial conditions of R [ n ] i ( z ) are R [ n ] i ( z ) = (cid:26) δ ni (0 ≤ n ≤ K − − ≤ i ≤ K − R i ( z ) ( n = K ; − ≤ i ≤ K − . (3.4)By applying ad H to (3.3),(ad H ) n +1 X = (ad H )(ad H ) n X H ) (cid:16) (ad H ) K − X · R [ n ] K − ( H ) + K − X i =0 (ad H ) i X · R [ n ] i ( H ) + R [ n ] − ( H ) (cid:17) = (cid:16) K − X i =0 (ad H ) i X · R i ( H ) + R − ( H ) (cid:17) · R [ n ] K − ( H ) + K − X i =1 (ad H ) i X · R [ n ] i − ( H ) , the recurrence relations for R [ n ] i ( z ) are obtained: R [ n +1] i ( z ) = R i ( z ) R [ n ] K − ( z ) + θ (1 ≤ i ≤ K − R [ n ] i − ( z ) ( n ≥ − ≤ i ≤ K − , (3.5)where θ (Prop) is a step function for a proposition, θ (True) = 1 and θ (False) = 0. Byintroducing a matrix A and a vector ~R [ n ] A def = R O R R . . . . . . ... O R K − R K − , ~R [ n ] def = R [ n ]0 R [ n ]1 R [ n ]2 ... R [ n ] K − , (3.6)this recurrence relations (3.5) with the initial conditions (3.4) can be rewritten as ~R [ n +1] = A ~R [ n ] ( n ≥ , ~R [0] = t (1 0 0 · · · , (3.7) R [ n +1] − = R − R [ n ] K − ( n ≥ , R [0] − = 0 . (3.8)These are easily solved as ~R [ n ] = A n ~R [0] ( n ≥ , (3.9) R [ n ] − = θ ( n ≥ R − R [ n − K − = θ ( n ≥ R − · (cid:0) A n − ~R [0] (cid:1) K ( n ≥ . (3.10)By using the properties of the matrix A given in Appendix A, let us calculate A n . Al-though the matrix elements of A depend on H , we can use the formulas in Appendix A,because the operator appearing in this calculation is H only. We assume that the matrix A has K distinct real non-vanishing eigenvalues α i = α i ( H ) and they are indexed in decreasingorder α i ( z ) = 0 , α ( z ) > α ( z ) > · · · > α K ( z ) ( z ≥ . (3.11)We note that R i (0 ≤ i ≤ K −
1) is expressed by α j , R i = ( − K − i − X ≤ j Since the operation ad H is a derivation, the closure relation of order K (3.2)is interpreted as a differential equation. More explicitly, it is stated as follows. Since aHeisenberg operator F H ( t ) satisfies the Heisenberg equation i ddt F H ( t ) = [ F H ( t ) , H ], the op-eration ad H for Heisenberg operators is a derivative − i ddt . Then the closure relation oforder K (3.2) means a differential equation for the Heisenberg operator X H ( t ) = e it H Xe − it H ,( − i ) K d K X H ( t ) dt K − K − P l =0 ( − i ) l d l X H ( t ) dt l R l ( H ) = R − ( H ), which is a K -th order linear differentialequation with ‘constant’ coefficients R l ( H ) (0 ≤ l ≤ K − 1) and a ‘constant’ inhomogeneousterm R − ( H ). Such a K -th order differential equation is converted into a coupled first orderlinear differential equation, which is expressed neatly in vector and matrix notation, seethe matrix A (3.6). Its general solution is a linear combination of e iα j t (which is a generalsolution of the homogeneous equation) plus a particular solution (which corresponds to aninhomogeneous term), see the last line of (3.18).Although these operators a ( j ) (3.19) contain H in a complicated way, they do not causeany problem when acting on the eigenfunction of H , because the operator H is replaced bythe eigenvalue. By noting[ H , t ~X ] = (cid:0) (ad H ) X, (ad H ) X, · · · , (ad H ) K − X, (ad H ) K X (cid:1) = (cid:0) (ad H ) X, (ad H ) X, · · · , (ad H ) K − X, K − X i =0 (ad H ) i − X · R i + R − (cid:1) = t ~XA + (0 , , · · · , , R − ) , (3.20)we obtain [ H , a ( j ) ] = [ H , t ~X ] ~p j ( P − ) j = (cid:0) t ~XA + (0 , , · · · , , R − ) (cid:1) ~p j ( P − ) j = (cid:0) t ~Xα j ~p j + R − p Kj ) (cid:1) ( P − ) j = a ( j ) α j . (3.21)9his relation implies H ψ ( x ) = E ψ ( x ) ⇒ H a ( j ) ψ ( x ) = a ( j ) (cid:0) H + α j ( H ) (cid:1) ψ ( x ) = a ( j ) (cid:0) E + α j ( E ) (cid:1) ψ ( x )= (cid:0) E + α j ( E ) (cid:1) a ( j ) ψ ( x ) . (3.22)Therefore the operator a ( j ) maps a solution of the Schr¨odinger equation with energy E tothat with energy E + α j ( E ). Namely it is a creation operator ( α j ( E ) > 0) or an annihilationoperator ( α j ( E ) < 0) (we have assumed ψ ( x ) and a ( j ) ψ ( x ) are normalizable).The discussion given in this section is very general but we have assumed the existence of X satisfying the generalized closure relation (3.2). In the next section we discuss that such X ’s really exist in the systems described by the multi-indexed orthogonal polynomials. In this section we consider exactly solvable systems described by the multi-indexed orthog-onal polynomials of Laguerre(L), Jacobi(J), Wilson(W) and Askey-Wilson(AW) types. Wediscuss the existence of the generalized closure relations together with their connection to therecurrence relations with constant coefficients for the multi-indexed orthogonal polynomials.We follow the notation in [23, 27, 29]. Isospectral deformations of the exactly solvable systems described by L, J, W and AWpolynomials are obtained by the M -step Darboux transformations with the virtual statewavefunctions as seed solutions. The deformed systems are labeled by D = { d , . . . , d M } = { d I1 , . . . , d I M I , d II1 , . . . , d II M II } ( M = M I + M II ), which are the degrees and types of the virtualstate wavefunctions, and their eigenstates have the following form, H D φ D n ( x ) = E n φ D n ( x ) , E < E < E < · · · , (4.1) φ D n ( x ) = Ψ D ( x ) ˇ P D ,n ( x ) ( n = 0 , , , . . . ) , ˇ P D ,n ( x ) def = P D ,n (cid:0) η ( x ) (cid:1) . (4.2)Here P D ,n ( η )’s are multi-indexed orthogonal polynomials. The Hamiltonian H D of the de-formed system is the second order differential or difference operator,L, J : H D = p + U D ( x ) , (4.3)W, AW : H D = q V D ( x ) V ∗D ( x − iγ ) e γp + q V ∗D ( x ) V D ( x + iγ ) e − γp − V D ( x ) − V ∗D ( x ) , (4.4)10here p = − i ddx and γ = 1 for W, γ = log q for AW. The deformed potential U D ( x )and potential function V D ( x ) are expressed in terms of the original U ( x ), V ( x ) and thedenominator polynomial Ξ D ( η ). The explicit forms of P D ,n ( η ), Ξ D ( η ), Ψ D ( x ), U D ( x ), V D ( x ),etc. can be found in [23, 27, 29].Since the multi-indexed orthogonal polynomials are not the ordinary orthogonal polyno-mials, they do not satisfy the three term recurrence relations. They satisfy the recurrencerelations with more terms; 3 + 2 M term recurrence relations with variable dependent coef-ficients [23]. It is conjectured that they also satisfy 1 + 2 L term recurrence relations withconstant coefficients [27, 29] of the form, X ( η ) P D ,n ( η ) = L X k = − L r X, D n,k P D ,n + k ( η ) ( ∀ n ≥ , (4.5)in which X ( η ) is a degree L polynomial in η (with real number coefficients) and r X, D n,k ’s areconstants. We have set P D ,n ( η ) = 0 for n < 0. The polynomial X ( η ) (4.5) depends on thedenominator polynomial Ξ D ( η ) [27]: X ( η ) = ( R η Ξ D ( y ) Y ( y ) dy : L, J I [Ξ D Y ]( η ) : W, AW , deg X ( η ) = L = ℓ D + deg Y ( η ) + 1 , (4.6)where Y ( η ) is an arbitrary polynomial in η and the map I [ · ] is given in [27] and ℓ D is ℓ D = M X j =1 d j − M ( M − 1) + 2 M I M II . (4.7)So long as the two polynomials in η , Ξ D ( η ) = Ξ d ...d M ( η ) and Ξ d ...d M − ( η ) (with some mod-ification for W and AW), do not have common roots, the above form exhausts all possible X ( η ) giving rise to the recurrence relations with constant coefficients (4.5). This conjectureis proved for L and J in [29].The normalization constants of the orthogonality relations, h D n , are related to those ofthe original system (2.20),( φ D n , φ D m ) = h D n δ nm , h D n = h n M Y j =1 ( E n − ˜ E d j ) , (4.8)where ˜ E d j ’s are energies of the virtual states, see (5.2), (5.13), (5.21) and (5.26). As in (2.21),the normalization constants and the coefficients of the recurrence relations are related. Since11he recurrence relations (4.5) give X ( η ) φ D n ( x ) = L X k = − L r X, D n,k φ D n + k ( x ) ( n ≥ , (4.9)we have ( φ D n , Xφ D n − l ) = θ ( − L ≤ l ≤ L ) r X, D n − l,l h D n = ( Xφ D n , φ D n − l ) = θ ( − L ≤ l ≤ L ) r X, D n, − l h D n − l , which means r X, D n − l,l h D n = r X, D n, − l h D n − l ( − L ≤ l ≤ L ) . (4.10)Hence we obtain r X, D n, − l = h D n h D n − l r X, D n − l,l (1 ≤ l ≤ L ) . (4.11)This result is obtained for an appropriate parameter range (with which the inner product iswell-defined) but the algebraic relations (4.11) themselves are valid for any parameter range.If r X, D m,k (0 ≤ m ≤ n − − L ≤ k ≤ L ) are known ( h D n are given in (4.8)), we obtain r X, D n,k ( − L ≤ k ≤ − 1) by this relation (we can set r X, D ,k = 0 ( − L ≤ k ≤ − r X, D n,k , it is sufficient to find r X, D n,k ( n ≥ 0, 0 ≤ k ≤ L ). The topcoefficient r X, D n,L is easily obtained by comparing the highest degree terms, r X, D n,L = c X c P D ,n c P D ,n + L , (4.12)where c X and c P D ,n are X ( η ) = c X η L + (lower order terms) ,P D ,n ( η ) = c P D ,n η ℓ D + n + (lower order terms) . The explicit forms of c P D ,n are found in [29] (L, J) and [19] (W, AW). The recurrence relations with constant coefficients give rise to the generalized closure rela-tions, for which we have the following: 12 onjecture 1 For any polynomial Y ( η ) , we take X ( η ) as (4.6) . Then we have the closurerelation of order K = 2 L (3.2) and the eigenvalues of the matrix A (3.6) satisfy α ( z ) > α ( z ) > · · · > α L ( z ) > > α L +1 ( z ) > α L +2 ( z ) > · · · > α L ( z ) ( z ≥ . (4.13) Remark 1 From the form of the Hamiltonian (4.3)–(4.4), the polynomials R i ( z ) = R Xi ( z )with coefficients r ( j ) i = r X ( j ) i areL, J : R i ( z ) = [ ( K − i )] X j =0 r ( j ) i z j (0 ≤ i ≤ K − , R − ( z ) = [ K ] X j =0 r ( j ) − z j , (4.14)W, AW : R i ( z ) = K − i X j =0 r ( j ) i z j (0 ≤ i ≤ K − , R − ( z ) = K X j =0 r ( j ) − z j , (4.15)where [ a ] denotes the greatest integer not exceeding a .At present we do not have a proof of Conjecture 1 but we can verify it for small values of M , d j and deg Y ( η ) by direct calculation. Such explicit examples will be given in the nextsection and Appendix B.In the rest of this section we assume that Conjecture 1 holds. Then we have the exactHeisenberg operator solution of X (3.18) and the creation/annihilation operators a ( j ) (3.19).Action of (3.18) on φ D n ( x ) is e it H Xe − it H φ D n ( x ) = L X j =1 e iα j ( E n ) t a ( j ) φ D n ( x ) − R − ( E n ) R ( E n ) − φ D n ( x ) . On the other hand the l.h.s. turns out to be e i H t Xe − i H t φ D n ( x ) = e i H t Xe − i E n t φ D n ( x ) = e − i E n t e i H t L X k = − L r X, D n,k φ D n + k ( x )= L X k = − L e i ( E n + k −E n ) t r X, D n,k φ D n + k ( x ) . Comparing these t -dependence (see (4.13)) and using (3.15), we obtain α j ( E n ) = (cid:26) E n + L +1 − j − E n > ≤ j ≤ L ) E n − ( j − L ) − E n < L + 1 ≤ j ≤ L ) , (4.16) a ( j ) φ D n ( x ) = ( r X, D n,L +1 − j φ D n + L +1 − j ( x ) (1 ≤ j ≤ L ) r X, D n, − ( j − L ) φ D n − ( j − L ) ( x ) ( L + 1 ≤ j ≤ L ) , (4.17)13 R − ( E n ) R ( E n ) − = r X, D n, . (4.18)Therefore a ( j ) (1 ≤ j ≤ L ) and a ( j ) ( L + 1 ≤ j ≤ L ) are creation and annihilation operators,respectively. Among them a ( L ) and a ( L +1) are fundamental, a ( L ) φ D ,n ( x ) ∝ φ D n +1 ( x ) and a ( L +1) φ D ,n ( x ) ∝ φ D n − ( x ).The creation and annihilation operators for eigenpolynomials are obtained by the simi-larity transformation. The similarity transformed Hamiltonian is e H D def = Ψ D ( x ) − ◦ H D ◦ Ψ D ( x ) , e H D ˇ P D ,n ( x ) = E n ˇ P D ,n ( x ) . (4.19)Their explicit forms areL, J : e H D ( λ ) = − (cid:18) c ( η ) d dη + (cid:16) c ( η, λ [ M I ,M II ] ) − c ( η ) ∂ η Ξ D ( η ; λ )Ξ D ( η ; λ ) (cid:17) ddη + c ( η ) ∂ η Ξ D ( η ; λ )Ξ D ( η ; λ ) − c ( η, λ [ M I ,M II ] − δ ) ∂ η Ξ D ( η ; λ )Ξ D ( η ; λ ) (cid:19) , (4.20)W, AW : e H D ( λ ) = V ( x ; λ [ M I ,M II ] ) ˇΞ D ( x + i γ ; λ )ˇΞ D ( x − i γ ; λ ) (cid:18) e γp − ˇΞ D ( x − iγ ; λ + δ )ˇΞ D ( x ; λ + δ ) (cid:19) + V ∗ ( x ; λ [ M I ,M II ] ) ˇΞ D ( x − i γ ; λ )ˇΞ D ( x + i γ ; λ ) (cid:18) e − γp − ˇΞ D ( x + iγ ; λ + δ )ˇΞ D ( x ; λ + δ ) (cid:19) , (4.21)where we have written the parameter ( λ ) dependence explicitly (see [23, 27, 29] for notation).Corresponding to (3.2), this also satisfies the closure relation(ad e H ) K X = K − X i =0 (ad e H ) i X · R i ( e H ) + R − ( e H ) . (4.22)From the creation/annihilation operators a ( j ) = a ( j ) ( H , X ), we obtain the creation/annihilationoperators for eigenpolynomials,˜ a ( j ) def = Ψ D ( x ) − ◦ a ( j ) ( H , X ) ◦ Ψ D ( x ) = a ( j ) ( e H , X ) , (4.23)˜ a ( j ) ˇ P D ,n ( x ) = ( r X, D n,L +1 − j ˇ P D ,n + L +1 − j ( x ) (1 ≤ j ≤ L ) r X, D n, − ( j − L ) ˇ P D ,n − ( j − L ) ( x ) ( L + 1 ≤ j ≤ L ) . (4.24) In this section we present examples of the generalized closure relations for the deformedsystems described by the multi-indexed orthogonal polynomials of Laguerre, Jacobi, Wilsonand Askey-Wilson types. 14e have verified Conjecture 1 for small M , d j and deg Y by direct calculation. Suchcalculation suggests that the polynomials R i ( z ) (0 ≤ i ≤ K − 1) depend on L (or K = 2 L )only and the dependence of d j and Y ( η ) enters in R − ( z ). We arrive at the following: Conjecture 2 The eigenvalues of the matrix A (3.6) with K = 2 L are given by (5.3) , (5.14) , (5.22) and (5.27) for L, J, W and AW, respectively. Remark 1 We can show that (5.3), (5.14), (5.22) and (5.27) satisfy (4.13) and (4.16) forappropriate parameter ranges. These α j ( z )’s depend on d j and Y ( η ) only through the degreeof X ( η ), L = ℓ D + deg Y ( η ) + 1. Remark 2 The polynomials R i ( z ) (0 ≤ i ≤ L − 1) are given by R i ( z ) = ( − i +1 X ≤ j The above expressions (5.1) with (5.3), (5.14), (5.22) and (5.27) are valid for L = 1 case, namely the original system ( D = {} , ℓ D = 0, Ξ D ( η ) = 1, X ( η ) = X min ( η ) = η ),and the generalized closure relation reduces to the original closure relation.If Conjecture 2 holds, the unknown quantity of the generalized closure relation is R − ( z )only. The data for the Laguerre polynomial are E n = 4 n, ˜ E Iv = − g + v + ) , ˜ E IIv = − g − v − ) , h n = n ! Γ( n + g + ) . (5.2)The eigenvalues of the matrix A (3.6) with K = 2 L are conjectured as α j ( z ) = (cid:26) L + 1 − j ) (1 ≤ j ≤ L ) − j − L ) ( L + 1 ≤ j ≤ L ) , (5.3)which are constants. It is trivial that these α j satisfy (4.13) and (4.16). We have R i ( z ) = 0( i : odd > 0) due to α j = − α L +1 − j . Note that R i ( z ) ( i : even ≥ 0) are also constants.15 x.1 We explain D = { I } (type I) in detail as an illustration. First we consider the lowestdegree case X ( η ) = X min ( η ), which corresponds to Y ( η ) = 1. The degree of X ( η ) is L = 2and the data of the closure relation of order K = 4 are X ( η ) = X min ( η ) = η ( η + 2 g + 1) ,R ( z ) = − , R ( z ) = 0 , R ( z ) = 80 , R ( z ) = 0 , (5.4) R − ( z ) = 64 (cid:0) z + 2(10 g + 11) z + 2(2 g + 1)(6 g + 13) (cid:1) , and ( α , α , α , α ) = (8 , , − , − a (1) and a (2) , and the annihila-tion operators, a (3) and a (4) , have the following forms, a (1) = (cid:0) − X − H ) X + 8(ad H ) X + (ad H ) X + R − ( H ) (cid:1) ,a (2) = − (cid:0) − X − H ) X + 4(ad H ) X + (ad H ) X + R − ( H ) (cid:1) ,a (3) = (cid:0) X − H ) X − H ) X + (ad H ) X − R − ( H ) (cid:1) , (5.5) a (4) = − (cid:0) X − H ) X − H ) X + (ad H ) X − R − ( H ) (cid:1) . Here differential operators H , (ad H ) X etc. are H D = − d dx + U D ( x ) , X = X (cid:0) η ( x ) (cid:1) = ˇ X ( x ) , f ′ = dfdx , f ( k ) = d k fdx k (ad H ) X = − X ′ ddx − ˇ X ′′ , (ad H ) X = 4 ˇ X ′′ d dx + 4 ˇ X ′′′ ddx + ˇ X (4) + 2 ˇ X ′ U ′D ( x ) , (ad H ) X = − X ′′′ d dx − 12 ˇ X (4) d dx − (cid:0) X (5) + 6 ˇ X ′′ U ′D ( x ) + 2 ˇ X ′ U ′′D ( x ) (cid:1) ddx − ˇ X (6) − X ′′′ U ′D ( x ) − X ′′ U ′′D ( x ) − X ′ U ′′′D ( x ) , (5.6)and the potential U D ( x ) in this case is U D ( x ) = x + g ( g + 1) x − g − x + g + − g + 1)( x + g + ) . (5.7)The eigenfunctions of H D are φ D n ( x ) = 2 e − x x g +1 x + g + P D ,n ( x ) ,P D ,n ( η ) = ( g + + η ) ∂ η L ( g − ) n ( η ) − ( g + + η ) L ( g − ) n ( η ) . (5.8)The coefficients of the 5-term recurrence relations for X ( η ) = X min ( η ) are [25, 26, 27] r X, D n, = ( n + 1)( n + 2) , r X, D n, − = (2 g + 2 n − g + 2 n + 3) , X, D n, = − ( n + 1)(2 g + 2 n + 3) , r X, D n, − = − (2 g + 2 n − g + 2 n + 3) , (5.9) r X, D n, = (cid:0) n + 4(10 g + 11) n + (2 g + 1)(6 g + 13) (cid:1) . We can check (4.16)–(4.18), (4.24), (4.11), etc.Next we take Y ( η ) = η . Then we have L = 3 and the data of the closure relation of order K = 6 are X ( η ) = η (cid:0) η + (2 g + 1) (cid:1) ,R ( z ) = 147456 , R ( z ) = − , R ( z ) = 224 , R ( z ) = R ( z ) = R ( z ) = 0 , (5.10) R − ( z ) = − (cid:0) z + 3(26 g + 33) z + 2(84 g + 240 g + 139) z + 2(2 g + 1)(2 g + 5)(10 g + 27) (cid:1) , and ( α , α , α , α , α , α ) = (12 , , , − , − , − Y ( η ) = η , we have L = 4 and the data of the closure relation of order K = 8 are X ( η ) = η (cid:0) η + (2 g + 1) (cid:1) ,R ( z ) = − , R ( z ) = 3358720 , R ( z ) = − , R ( z ) = 480 ,R ( z ) = R ( z ) = R ( z ) = R ( z ) = 0 , (5.11) R − ( z ) = 24576 (cid:0) z + 20(50 g + 67) z + 60(52 g + 152 g + 107) z + 32(6 g + 5)(18 g + 77 g + 94) z + 8(2 g + 1)(2 g + 5)(2 g + 7)(14 g + 45) (cid:1) , and ( α , α , . . . , α ) = (16 , , , , − , − , − − Ex.2 For D = { II } (type II) and Y ( η ) = 1, we have L = 2 and the data of the closurerelation of order K = 4 are X ( η ) = X min ( η ) = − η ( η + 2 g − ,R ( z ) = − , R ( z ) = 0 , R ( z ) = 80 , R ( z ) = 0 , (5.12) R − ( z ) = − (cid:0) z + 2(10 g − z + 2(2 g − g + 1) (cid:1) , and ( α , α , α , α ) = (8 , , − , − .2 Multi-indexed Jacobi polynomials The data for the Jacobi polynomial are E n = 4 n ( n + g + h ) , h n = Γ( n + g + )Γ( n + h + )2 n ! (2 n + g + h )Γ( n + g + h ) , ˜ E Iv = − g + v + )( h − v − ) , ˜ E IIv = − g − v − )( h + v + ) , (5.13)and we set a = g + h and b = g − h .The eigenvalues of the matrix A (3.6) with K = 2 L are conjectured as α j ( z ) = ( L + 1 − j ) + 4( L + 1 − j ) √ z + a (1 ≤ j ≤ L )4( j − L ) − j − L ) √ z + a ( L + 1 ≤ j ≤ L ) . (5.14)Remark that α j ( E n ) is square root free, √E n + a = 2 n + a . We can show that these α j satisfy (4.13) and (4.16) for a > L − 1. Note that α j ( z ) + α L +1 − j ( z ) = (cid:26) L + 1 − j ) (1 ≤ j ≤ L )8( j − L ) ( L + 1 ≤ j ≤ L ) , (5.15) α j ( z ) α L +1 − j ( z ) = (cid:26) L + 1 − j ) (cid:0) ( L + 1 − j ) − z − a (cid:1) (1 ≤ j ≤ L )16( j − L ) (cid:0) ( j − L ) − z − a (cid:1) ( L + 1 ≤ j ≤ L ) . Ex.1 First we consider D = { I } (type I) and take X = X min ( ⇒ ℓ D = 1, L = 2, K = 4), X ( η ) = X min ( η ) = η (cid:0) ( b + 2) η + 2( a − (cid:1) . (5.16)The closure relation of order 4 holds with the following R i : R ( z ) = − z + a − z + a − , R ( z ) = − z + a − ) ,R ( z ) = 80( z + a − ) , R ( z ) = 40 ,R − ( z ) = 128( b + 2) (cid:16) z − (cid:0) ( b + 2) + 3 a − a + 1 (cid:1) z (5.17)+ 2( a − a − (cid:0) ( b + 2) − a − a − (cid:1)(cid:17) . The eigenvalues of A (3.6) are α ( H ) = 16 + 8 √H + a , α ( H ) = 4 + 4 √H + a ,α ( H ) = 16 − √H + a , α ( H ) = 4 − √H + a , α j ( E n ) is square root free, √E n + a = 2 n + a . We can check (3.11) for a > 3. Therefore a (1) and a (2) are the creation operators, and a (3) and a (4) are the annihilation operators. Thepotential and eigenfunctions are U D ( x ) = g ( g + 1)sin x + ( h − h − x − a + 8( a − a − b + 2) cos 2 x − g + 1)(2 h − (cid:0) a − b + 2) cos 2 x (cid:1) ,φ D n ( x ) = − x ) g +1 (cos x ) h − a − b + 2) cos 2 x P D ,n (cos 2 x ) ,P D ,n ( η ) = (1 + η ) (cid:0) a − b + 2) η (cid:1) ∂ η P ( g − ,h − ) n ( η ) − ( − h ) (cid:0) a + 1 + ( b + 2) η (cid:1) P ( g − ,h − ) n ( η ) . (5.18)The coefficients of the 5-term recurrence relations are [27] r X, D n, = ( n + 1) ( b + 2)( a + n ) (2 h + 2 n − a + 2 n ) (2 h + 2 n + 1) ,r X, D n, − = ( b + 2)(2 g + 2 n − g + 2 n + 3)( h + n − ) a + 2 n − ,r X, D n, = ( n + 1)( a − a + n )(2 g + 2 n + 3)(2 h + 2 n − a + 2 n − ( a + 2 n + 3) , (5.19) r X, D n, − = ( a − g + 2 n − g + 2 n + 3)( h + n − ) ( a + 2 n − a + 2 n − ,r X, D n, = b + 24( a + 2 n − ( a + 2 n + 1) (cid:16) − b ( b + 4) (cid:0) n ( a + n ) − ( a − a − (cid:1) + ( a + 2 n − a + 2 n + 1) (cid:0) n ( a + n ) − ( a − a − (cid:1)(cid:17) . We can check (4.16)–(4.18), (4.24), (4.11), etc. Ex.2 Next we consider D = { II } (type II) and take X = X min . Since these two cases D = { I } and D = { II } are essentially same [5], we present R − ( z ) only, R − ( z ) = − b − (cid:16) z − (cid:0) ( b − + 3 a − a + 1 (cid:1) z + 2( a − a − (cid:0) ( b − − a − a − (cid:1)(cid:17) . (5.20)More examples are presented in Appendix B. The data for the Wilson polynomial are E n = n ( n + b − , b = X i =1 a i , h n = 2 πn ! ( n + b − n Q ≤ i 1. We can show thatthese α j satisfy (4.13) and (4.16) for b > L . Note that α j ( z ) + α L +1 − j ( z ) = (cid:26) L + 1 − j ) (1 ≤ j ≤ L )2( j − L ) ( L + 1 ≤ j ≤ L ) , (5.23) α j ( z ) α L +1 − j ( z ) = (cid:26) ( L + 1 − j ) (cid:0) ( L + 1 − j ) − z − ( b − (cid:1) (1 ≤ j ≤ L )( j − L ) (cid:0) ( j − L ) − z − ( b − (cid:1) ( L + 1 ≤ j ≤ L ) . Ex.1 We consider D = { I } and take X = X min ( ⇒ ℓ D = 1, L = 2, K = 4), X ( η ) = X min ( η ) = η (cid:0) σ − σ ′ − η + 4( σ σ ′ − σ σ ′ − σ σ ′ + 2 σ ′ ) + σ + 3 σ ′ − (cid:1) . (5.24)The closure relation of order 4 holds with the following R i : R i ( z ) (0 ≤ i ≤ 4) are given byConjecture 2 and (5.1), R ( z ) = 10 , R ( z ) = 5 z ′ − , z ′ def = 4 z + ( b − ,R ( z ) = − z ′ − , R ( z ) = − z ′ − z ′ − , (5.25)and R − ( z ) is presented in (B.23) because of its lengthy expression. The coefficients of the5-term recurrence relations are found in [27] and we can check (4.16)–(4.18), (4.24), (4.11),etc. The data for the Askey-Wilson polynomial are E n = ( q − n − − b q n − ) , b = Y i =1 a i , h n = 2 π ( b q n − ; q ) n ( b q n ; q ) ∞ ( q n +1 ; q ) ∞ Q ≤ i We consider D = { I } and take X = X min ( ⇒ ℓ D = 1, L = 2, K = 4), X ( η ) = X min ( η ) = η (1 + q ) σ (cid:16) q ( σ − σ ′ q ) η − (1 + q ) (cid:0) σ (1 − σ ′ ) q + σ ′ ( σ − q ) (cid:1)(cid:17) . (5.29)The closure relation of order 4 holds with the following R i : R i ( z ) (0 ≤ i ≤ 4) are given byConjecture 2 and (5.1), R ( z ) = q − (1 − q ) (1 + 3 q + q ) z ′ , z ′ def = z + 1 + q − b ,R ( z ) = − q − (1 − q ) (cid:0) (1 − q − q − q + q ) z ′ + q − (1 + q ) (1 + 3 q + q ) b (cid:1) ,R ( z ) = − q − (1 − q ) (1 + q ) z ′ (cid:0) z ′ − q − (1 + q + 4 q + q + q ) b (cid:1) , (5.30) R ( z ) = − q − (1 − q ) (1 + q ) (cid:0) z ′ − q − (1 + q ) b (cid:1)(cid:0) z ′ − q − (1 + q ) b (cid:1) , and R − ( z ) is presented in (B.26) because of its lengthy expression. The coefficients of the5-term recurrence relations are found in [27] and we can check (4.16)–(4.18), (4.24), (4.11),etc. 21 Summary and Comments Exactly solvable quantum mechanical systems described by Laguerre, Jacobi, Wilson andAskey-Wilson polynomials satisfy the shape invariance and closure relation. The shape in-variance is inherited by the deformed systems described by the multi-indexed orthogonalpolynomials of Laguerre, Jacobi, Wilson and Askey-Wilson types. The closure relation isnot inherited, because it is related to the three term recurrence relations for the ordinaryorthogonal polynomials. These multi-indexed orthogonal polynomials satisfy the 1 + 2 L ( L ≥ 2) term recurrence relations with constant coefficients (4.5), which is obtained frommultiplication by X . We generalize the closure relation (3.2) and discuss that the deformedsystems satisfy the generalized closure relations, which corresponds to the 1 + 2 L term re-currence relations with constant coefficients. This is stated as Conjecture 1. The generalizedclosure relation gives the exact Heisenberg operator solution of X , and its negative/positivefrequency parts provide the creation/annihilation operators of the system. For small L cases, we can verify the generalized closure relations. Some examples and Conjecture 2 arepresented. Although we have discussed Laguerre, Jacobi, Wilson and Askey-Wilson cases,the method is applicable to other case (1) and case (2) exceptional/multi-indexed polyno-mials. For example, the exceptional Hermite polynomials with multi-indices are worth forinvestigation.In contrast to the shape invariance which is applicable to quantum system only, the(generalized) closure relation is applicable to not only quantum system but also classicalsystem [2]. The commutator [ · , · ] in quantum mechanics becomes the Poisson bracket {· , ·} PB in classical mechanics. To obtain the classical limit, we have to recover ~ (reduced Planckconstant) dependence, because we have used the ~ = 1 unit. Then the commutator i ~ [ · , · ]and the Hamiltonian H become the Poisson bracket {· , ·} PB and the classical Hamiltonian H cl respectively, in the ~ → F is ddt F = −{H cl , F } PB and it is solved as F ( t ) = ∞ P n =0 ( − t ) n n ! (ad PB H cl ) n F , where ad PB H cl stands for the operation (ad PB H cl ) F = {H cl , F } PB . By the similar calculation in § 3, thegeneralized closure relation gives us the time evolution of X , X ( t ), explicitly.For the ordinary orthogonal polynomials, any recurrence relations of the form X ( η ) P n ( η ) = L P k = − L r Xn,k P n + k ( η ) can be obtained from the three term recurrence relations. On the otherhand, for multi-indexed orthogonal polynomials of L, J, W and AW types, different choices22f X give different recurrence relations with constant coefficients (4.5) in general. Cor-respondingly we have infinitely many creation/annihilation operators. It is a challengingproblem to study their relations. Even for the simplest choice X = X min , there are L cre-ation operators a ( j ) (1 ≤ j ≤ L ) and L annihilation operators a ( j ) ( L + 1 ≤ j ≤ L ) and it isa good problem to study their relations (commutators etc.). In harmonic oscillator, the co-herent state is obtained as an eigenstate of the annihilation operator. It is also a challengingproblem to find the eigenstates of the annihilation operators a ( j ) ( L + 1 ≤ j ≤ L ). Acknowledgments I thank R. Sasaki for discussion and reading of the manuscript. I am supported in part byGrant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Scienceand Technology (MEXT), No.25400395. A Diagonalization of Some Matrix In this appendix we present some formulas for the diagonalization of a matrix (A.1).Let us consider a K × K matrix A , A = R O R R . . . . . . ... O R K − R K − , (A.1)where R i ’s are real numbers. Its characteristic polynomial is | xE − A | = x K − K − X i =0 R i x i . (A.2)We assume that the matrix A has K distinct real non-vanishing eigenvalues, | xE − A | = K Y j =1 ( x − α j ) , α > α > · · · > α K , α i = 0 . (A.3)Then R i is expressed by α j , R i = ( − K − i − X ≤ j In this appendix we present more examples of the generalized closure relations, which supportConjecture 1 and Conjecture 2. We take X ( η ) = X min ( η ) ( Y ( η ) = 1). The order of thegeneralized closure relation is K = 2 L = 2( ℓ D + 1). Since R i ( z ) (0 ≤ i ≤ L − 1) areobtained by Conjecture 2 and (5.1), we present R − ( z ) only. B.1 Laguerre type We present R − ( z ) for all D with K ≤ K = 2 D = {} : R − ( z ) = − z + 2 g + 1) . (B.1) K = 4 D = { I } : R − ( z ) = 64 (cid:0) g )(13 + 6 g ) + 2(11 + 10 g ) z + 3 z (cid:1) , (B.2)24 = { II } : R − ( z ) = − (cid:0) g − g ) + 2(10 g − z + 3 z (cid:1) . (B.3) K = 6 D = { I } : R − ( z ) = − (cid:0) g )(3 + 2 g )(41 + 14 g ) + 4(61 + 108 g + 36 g ) z + 24(3 + 2 g ) z + 5 z (cid:1) , (B.4) D = { II } : R − ( z ) = − (cid:0) g − g − g ) + 4(25 − g + 36 g ) z + 12(4 g − z + 5 z (cid:1) , (B.5) D = { I , I } : R − ( z ) = − (cid:0) g )(5 + 2 g )(45 + 14 g ) + 4(97 + 132 g + 36 g ) z + 12(7 + 4 g ) z + 5 z (cid:1) , (B.6) D = { II , II } : R − ( z ) = 1536 (cid:0) g − g − g − 1) + 4(61 − g + 36 g ) z + 24(2 g − z + 5 z (cid:1) . (B.7) K = 8 D = { I } : R − ( z ) = 12288(8(1 + 2 g )(3 + 2 g )(5 + 2 g )(113 + 30 g )+ 16(935 + 2072 g + 1284 g + 224 g ) z + 4(1405 + 1848 g + 492 g ) z + 20(41 + 22 g ) z + 35 z (cid:1) , (B.8) D = { II } : R − ( z ) = − (cid:0) g − g − − − g + 20 g )+ 16( − 245 + 968 g − g + 224 g ) z + 4(805 − g + 492 g ) z + 20(22 g − z + 35 z (cid:1) , (B.9) D = { I , I } : R − ( z ) = 24576 (cid:0) g )(5 + 2 g )(17 + 6 g )(39 + 10 g )+ 16(1625 + 2984 g + 1500 g + 224 g ) z + 4(2005 + 2136 g + 492 g ) z + 20(47 + 22 g ) z + 35 z (cid:1) , (B.10) D = { II , II } : R − ( z ) = 24576 (cid:0) g − g − g − g − − 647 + 1736 g − g + 224 g ) z + 4(1333 − g + 492 g ) z + 20(22 g − z + 35 z (cid:1) , (B.11)25 = { I , I , I } : R − ( z ) = 12288 (cid:0) g )(7 + 2 g )(251 + 144 g + 20 g )+ 16(2411 + 3944 g + 1716 g + 224 g ) z + 4(2629 + 2424 g + 492 g ) z + 20(53 + 22 g ) z + 35 z (cid:1) , (B.12) D = { II , II , II } : R − ( z ) = 12288 (cid:0) g − g − g − g − − g − g + 224 g ) z + 4(1885 − g + 492 g ) z + 20(22 g − z + 35 z (cid:1) , (B.13) D = { I , II } : R − ( z ) = 73728 (cid:0) g − g )(5 + 6 g )(19 + 10 g )+ 16( − − g + 156 g + 224 g ) z + 4( − 299 + 168 g + 492 g ) z + 20(3 + 22 g ) z + 35 z (cid:1) . (B.14)We have also calculated for K = 10 and K = 12 cases, K = 10 : D = { I } , { II } , { I , I } , { II , II } , { I , I } , { II , II } , { I , I , I } , { II , II , II } , { I , I , I , I } , { II , II , II , II } , { I , II } , { I , II } ,K = 12 : D = { I } , { II } , { I , I } , { II , II } , { I , I } , { II , II } , { I , I , I } , { II , II , II } , { I , I , I } , { II , II , II } , { I , I , I , I } , { II , II , II , II } , { I , I , I , I , I } , { II , II , II , II , II } , { I , II } , { I , II } , { I , I , II } , { I , II , II } , { I , II } , and checked Conjecture 1 and Conjecture 2. B.2 Jacobi type We present R − ( z ) for all D with K ≤ K = 2 D = {} : R − ( z ) = 16 b ( a − . (B.15) K = 4 D = { I } : R − ( z ) = 128(2 + b ) (cid:16) a − a − (cid:0) (2 + b ) − − a − a (cid:1) − (cid:0) (2 + b ) + 1 − a + 3 a (cid:1) z + z (cid:17) , (B.16)26 = { II } : R − ( z ) = (cid:0) R − ( z ) in (B.16) (cid:1)(cid:12)(cid:12) b →− b . (B.17) K = 6 D = { I } : R − ( z ) = 3072(1 − a ) (cid:16) a − a − (cid:0) a − a (2 + a )(3 + a ) − a )(5 + a + a )(3 + b ) − (14 − a + 3 a )(3 + b ) + (10 + 3 a + 3 a )(3 + b ) + 2(3 + b ) − (3 + b ) (cid:1) + 3 (cid:0) a − a ( − 13 + 2 a ) + (46 − a − a + 12 a − a )(3 + b ) − − a − a )(3 + b ) − a )(3 + b ) − b ) + (3 + b ) (cid:1) z + 6 (cid:0) ( a − a − (4 − a )(3 + b ) + 2(3 + b ) − (3 + b ) (cid:1) z + 3(3 + b ) z (cid:17) , (B.18) D = { II } : R − ( z ) = − (cid:0) R − ( z ) in (B.18) (cid:1)(cid:12)(cid:12) b →− b , (B.19) D = { I , I } : R − ( z ) = − − a )(4 + b ) (cid:16) a − a − (cid:0) a − a (2 + a )(3 + a )+ 3(2 + a )(5 + a + a )(3 + b ) − (14 − a + 3 a )(3 + b ) − (10 + 3 a + 3 a )(3 + b ) + 2(3 + b ) + (3 + b ) (cid:1) + 3 (cid:0) a − a ( − 13 + 2 a ) − (46 − a − a + 12 a − a )(3 + b )+ 2(1 − a − a )(3 + b ) + 2(1 + 4 a )(3 + b ) − b ) − (3 + b ) (cid:1) z + 6 (cid:0) ( a − a + (4 − a )(3 + b ) + 2(3 + b ) + (3 + b ) (cid:1) z − b ) z (cid:17) , (B.20) D = { II , II } : R − ( z ) = − (cid:0) R − ( z ) in (B.20) (cid:1)(cid:12)(cid:12) b →− b . (B.21) B.3 Wilson type We present R − ( z ) for all D with K ≤ K = 2 D = {} : R − ( z ) = − z + ( b − b ) z − ( b − b . (B.22) K = 4 D = { I } : R − ( z )= 18 ( b − b − (cid:16) b + b ( σ − σ ′ ) − b ( σ + 7 σ ′ ) + 16( σ σ ′ + 4 σ ′ σ − σ ′ σ ′ )27 8 (cid:0) σ σ ′ + σ σ ′ b + σ σ ′ ( σ + 15 σ ′ ) + σ ′ (21 σ − σ ′ ) (cid:1) − (cid:0) σ σ ′ + σ σ ′ (7 σ + 2 σ ′ )+ σ σ ′ (14 σ − σ ′ ) − σ ′ σ − σ σ ′ + 4 σ ′ (2 σ − σ σ ′ + σ ′ ) (cid:1) − (cid:0) σ σ ′ ( σ + 3 σ ′ )( σ σ ′ + σ ′ σ ) − σ σ ′ − σ σ ′ (5 σ − σ σ ′ + 7 σ ′ ) + σ ′ σ ( σ − σ ′ ) (cid:1) − σ σ ′ + σ ′ σ ) (cid:0) σ σ ′ − σ σ ′ ( σ − σ ′ ) − σ ′ σ (cid:1)(cid:17) + 12 (cid:16) − σ + 5 σ ′ ) + 8( σ + 2 σ σ ′ + 16 σ ′ − σ + 6 σ ′ ) − σ + 53 σ σ ′ + 143 σ σ ′ − σ ′ + 52 σ σ ′ + 220 σ ′ σ ) + 8 (cid:0) σ + 7 σ σ ′ + 15 σ σ ′ − σ σ ′ − σ ′ + σ (9 σ + 11 σ ′ )+ 2 σ σ ′ (37 σ − σ ′ ) − σ ′ (11 σ + 13 σ ′ ) + 12 σ ( σ − σ ′ ) (cid:1) − (cid:0) σ σ ′ ( σ − σ σ ′ − σ σ ′ − σ ′ ) + σ (3 σ + 5 σ ′ ) + σ σ ′ (23 σ − σ ′ ) − σ σ ′ (35 σ + 11 σ ′ ) − σ ′ (17 σ − σ ′ )+ 2 σ (6 σ − σ σ ′ + 11 σ ′ ) + 2 σ ′ ( σ + 4 σ σ ′ + 6 σ ′ ) (cid:1) − (cid:0) σ σ ′ b − σ σ ′ − σ σ ′ ( σ − σ ′ ) + σ σ ′ (27 σ − σ ′ ) + σ σ ′ (19 σ − σ ′ ) − σ ′ σ − σ (3 σ − σ σ ′ + 19 σ ′ )+ 4 σ σ ′ (3 σ − σ ′ )( σ + σ ′ ) + σ ′ ( σ − σ ′ )(9 σ + 7 σ ′ ) (cid:1) + 8 (cid:0) σ σ ′ σ ′ + σ σ ′ (5 σ − σ ′ )+ σ σ ′ (4 σ − σ ′ ) − σ σ ′ σ − σ σ ′ + σ σ ′ (5 σ − σ ′ )( σ + σ ′ )+ σ σ ′ (7 σ − σ ′ )( σ + σ ′ ) + 4 σ ′ σ (cid:1)(cid:17) z − (cid:16) 14 + 13 σ + 67 σ ′ − σ + 11 σ σ ′ − σ ′ + 10 σ − σ ′ ) + 2 (cid:0) σ + 3 σ σ ′ − σ σ ′ − σ ′ + σ (21 σ − σ ′ ) − σ ′ (31 σ + 9 σ ′ ) (cid:1) + 2 (cid:0) σ σ ′ ( σ + 12 σ σ ′ − σ ′ ) − σ ( σ − σ ′ )+ 6 σ σ ′ (7 σ − σ ′ ) + σ ′ (9 σ − σ ′ ) + 2(3 σ − σ σ ′ − σ ′ ) (cid:1) − (cid:0) σ σ ′ ( σ − σ ′ )+ σ σ ′ − σ ′ σ + σ σ ′ (11 σ − σ ′ ) + σ σ ′ (6 σ − σ ′ ) + σ (3 σ − σ σ ′ − σ ′ )+ σ ′ (5 σ + 6 σ σ ′ − σ ′ ) (cid:1)(cid:17) z + 8 (cid:0) − σ − σ ′ ) − (13 σ − σ ′ ) σ ′ − σ + 2 σ ′ + 3( σ − σ ′ ) σ σ ′ + 3( σ σ − σ ′ σ ′ ) − σ σ ′ + σ ′ σ (cid:1) z + 12( − σ − σ ′ ) z , (B.23) D = { II } : R − ( z ) = (cid:0) R − ( z ) in (B.23) (cid:1)(cid:12)(cid:12) ( a ,a ) ↔ ( a ,a ) . (B.24) B.4 Askey-Wilson type We present R − ( z ) for all D with K ≤ K = 2 D = {} : R − ( z ) = − q − ( q − (cid:0) ( b + b q − ) z + (1 − b q − )( b − b ) (cid:1) . (B.25)28 = 4 D = { I } : R − ( z ) × q σ (1 − q ) (1 + q )=( q − σ σ ′ )( q − σ σ ′ ) (cid:16) − q (1 + q )( σ − q σ ′ ) − (1 − q )(1 − q − q )( σ − q σ ′ )+ σ (cid:0) σ + q (1 + q + q ) σ ′ (cid:1) + q (1 + q ) σ σ ′ ( σ − q σ ′ ) − qσ ′ (cid:0) (1 + q + q ) σ + q σ ′ (cid:1) − (1 − q )( σ − q σ ′ ) − (1 + q ) (cid:0) (1 − q ) σ σ ′ ( σ + q σ ′ ) + σ σ ′ ( σ − q σ ′ ) − q (1 − q ) σ ′ σ ( σ + qσ ′ ) − − q ) σ σ ′ ( σ − q σ ′ ) (cid:1) − q σ σ ′ (cid:0) (1 + q + q ) σ + q σ ′ (cid:1) + (1 + q ) σ σ ′ σ σ ′ ( σ − q σ ′ ) + qσ ′ σ (cid:0) σ + q (1 + q + q ) σ ′ (cid:1) + (1 − q ) σ σ ′ ( σ − q σ ′ )+ σ σ ′ (cid:0) q (1 + q )( q σ σ ′ − σ ′ σ ) − (1 − q )(1 + q − q ) σ σ ′ ( σ − q σ ′ ) (cid:1)(cid:17) + q (cid:16) q (cid:0) − q (1 + q ) σ + 3 q (1 + q ) σ ′ − (3 − q + 3 q )( σ − q σ ′ ) (cid:1) + q (cid:16) qσ (cid:0) σ + q (1 + q + q ) σ ′ (cid:1) − (1 + q )(1 + q + q − q ) σ σ ′ ( σ − q σ ′ ) − q σ ′ (cid:0) (1 + q + q ) σ + q σ ′ (cid:1) − (1 + 3 q − q )( σ − q σ ′ ) (cid:17) + q (cid:16) q (1 + q ) σ σ ′ (cid:0) (1 + 2 q )(2 − q + q ) σ − q (1 + q )(1 − q + 3 q − q ) σ ′ (cid:1) + q (1 + q )(1 + q − q ) σ σ ′ ( σ − q σ ′ )+ q (1 + q ) σ ′ σ (cid:0) (1 + q )(1 − q + 3 q − q ) σ − q (1 + 2 q )(2 − q + q ) σ ′ (cid:1) + (2 − q − q + 13 q − q − q + 3 q ) σ σ ′ ( σ − q σ ′ ) (cid:17) + q (cid:16) − σ σ ′ (cid:0) (1 + 2 q + q ) σ + 2 q (1 − q ) σ σ ′ − q (1 + 2 q + q ) σ ′ (cid:1) + (1 + q )(1 − q − q − q + q ) σ σ ′ σ σ ′ × ( σ − q σ ′ ) − σ ′ σ (cid:0) (1 + 2 q + q ) σ − q (1 − q ) σ σ ′ − q (1 + 2 q + q ) σ ′ (cid:1) + (1 + q )(3 − q + 3 q ) σ σ ′ ( σ − q σ ′ ) (cid:17) + σ σ ′ (cid:16) − (1 + q ) σ σ ′ (cid:0) (1 + q )(1 − q + 2 q − q ) σ + q (2 + q )(1 − q + 2 q ) σ ′ (cid:1) − (1 + q )(1 − q − q ) σ σ ′ ( σ − q σ ′ ) + q (1 + q ) σ ′ σ × (cid:0) (2 + q )(1 − q + 2 q ) σ + q (1 + q )(1 − q + 2 q − q ) σ ′ (cid:1) + (3 − q − q + 13 q − q − q + 2 q ) σ σ ′ ( σ − q σ ′ ) (cid:17) + σ σ ′ (cid:16) − q σ σ ′ (cid:0) (1 + q + q ) σ + q σ ′ (cid:1) + (1 + q )(2 − q − q − q ) σ σ ′ σ σ ′ ( σ − q σ ′ ) + 2 qσ ′ σ (cid:0) σ + q (1 + q + q ) σ ′ (cid:1) + (2 − q − q ) σ σ ′ ( σ − q σ ′ ) (cid:17) + σ σ ′ (cid:0) q (1 + q )( q σ σ ′ − σ ′ σ ) − (3 − q + 3 q ) σ σ ′ ( σ − q σ ′ ) (cid:1)(cid:17) z + q (cid:16) − q (cid:0) q (1 + q )( σ − q σ ′ ) + (1 − q + q )( σ − q σ ′ ) (cid:1) + q (cid:0) qσ σ + q (1 + q + q ) σ σ ′ − (1 + q )(2 + 2 q + 2 q − q ) σ σ ′ ( σ − q σ ′ ) − q σ ′ σ ′ − q (1 + q + q ) σ ′ σ − (2 + 3 q − q )( σ − q σ ′ ) (cid:1) + q (1 + q ) σ σ ′ (cid:0) (1 + q − q + q ) σ − q (1 − q + q + q ) σ ′ (cid:1) + q (1 + q ) σ σ ′ ( σ − q σ ′ ) + q (1 + q ) σ ′ σ (cid:0) (1 − q + q + q ) σ − q (1 + q − q + q ) σ ′ (cid:1) 29 (1 − q + q )(1 − q + q ) σ σ ′ ( σ − q σ ′ ) − q σ σ ′ (cid:0) (1 + q + q ) σ + q σ ′ (cid:1) + (1 + q ) (cid:0) − q (1 + q + q ) (cid:1) σ σ ′ σ σ ′ ( σ − q σ ′ ) + qσ ′ σ (cid:0) σ + q (1 + q + q ) σ ′ (cid:1) + (1 − q − q ) σ σ ′ ( σ − q σ ′ ) + 3 σ σ ′ (cid:0) q (1 + q ) σ σ ′ − q (1 + q ) σ ′ σ − (1 − q + q ) σ σ ′ ( σ − q σ ′ ) (cid:1)(cid:17) z + q (cid:16) − q (cid:0) q (1 + q )( σ − q σ ′ ) + (1 − q + q )( σ − q σ ′ ) (cid:1) − q (1 + q )( σ − q σ ′ ) × (cid:0) (1 + q + q ) σ σ ′ + σ + q σ ′ (cid:1) + q (1 + q )( q σ σ ′ − σ ′ σ ) − (1 − q + q ) σ σ ′ ( σ − q σ ′ ) (cid:17) z + 2 q ( σ − q σ ′ ) z , (B.26) D = { II } : R − ( z ) = (cid:0) R − ( z ) in (B.26) (cid:1)(cid:12)(cid:12) ( a ,a ) ↔ ( a ,a ) . 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