An algebraic treatment of the Askey biorthogonal polynomials on the unit circle
aa r X i v : . [ m a t h . R T ] F e b An algebraic treatment of the Askey biorthogonalpolynomials on the unit circle
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. School of Mathematics, Renmin University of China, Beijing, 100872, China
February 4, 2021
Abstract
A joint algebraic interpretation of the biorthogonal Askey polynomials on the unit circleand of the orthogonal Jacobi polynomials is offered. It ties their bispectral properties toan algebra called the meta-Jacobi algebra m J . In a commentary included in his edition of Szeg˝o’s collected works, Askey [1] introduced setsof biorthogonal polynomials on the unit circle. These polynomials are defined as follows interms of the standard Gauss hypergeometric series: P n ( z ; α, β ) = ( β ) n ( α + 1) n F (cid:18) − n, α + 11 − β − n ; z (cid:19) , (1.1) Q n ( z ; α, β ) = P n ( z ; β, α ) , (1.2)where ( a ) k = a ( a + 1) . . . ( a + k − , k = 1 , , . . . and ( a ) = 1 are the Pochhammer symbols.The normalization is chosen so that P n ( z ) and Q n ( z ) are monic. The biorthogonality of thesepolynomials was proven in [2] using slightly different conventions; it here reads: − πi I | z | =1 dz ( − z ) − − β (1 − z ) α + β P m ( z, α, β ) Q n ( 1 z , α, β ) = m ! Γ( m + α + β + 1)Γ( α + 1)Γ( β + 1) δ mn , (1.3)where Γ( x ) is the standard gamma function. Remember that ( a ) n = Γ( n + α )Γ( α ) . The branch of( − z ) − − β is chosen such that ( − z ) − − β = | z | − − β if arg z = π [3], this is reflected in [1], [2] bymaking the polar variable run from − π to π . For the connection with the spherical harmonicsof the Heisenberg group, see [4], Sect 1.2. Let us also record that special cases of the Askeypolynomials were obtained in [5] as Fourier transforms of Laguerre polynomials (with weightsattached). We refer to [6] for historical remarks regarding these polynomials (see also [7]). ∗ E-mail: [email protected] † E-mail: [email protected]
1n his comments Askey expressed the opinion that the P n ( z ; α, β ) are the natural analoguesof the Jacobi polynomials on the unit circle. We here reinforce this viewpoint by offering aunified algebraic description of these Askey polynomials on S and of the Jacobi polynomials.This will involve the introduction of an algebra to be called meta-Jacobi that will be seen toaccount for the bispectrality of both classes of functions.Let us register for reference the definition and key properties of the monic Jacobi polyno-mials ˆ P ( α,β ) n ( x ) defined on the interval [0 , P ( α,β ) n ( x ) = ( − n (1 − β ) n (1 + α + n ) n F (cid:18) − n, n + α + 11 − β ; x (cid:19) . (1.4)Please note that for convenience an unconventional choice has been made for the parameters.These polynomials possess the following orthogonality property Z ˆ P ( α,β ) m ( x ) ˆ P ( α,β ) n ( x ) x − β (1 − x ) α + β dx = h n δ mn α + β > − , β < , (1.5)with the normalization factor h n given by h n = n ! Γ( n − β + 1)Γ( n + α + 1)Γ( n + α + β + 1)Γ(2 n + α + 1)Γ(2 n + α + 2) . (1.6)As is well known, in addition to satisfying a three-term recurrence relation, the polynomialsˆ P ( α,β ) n ( x ) are eigenfunctions of the hypergeometric operator M = x ( x − ∂ x + [( α + 2) x + β − ∂ x (1.7)with eigenvalue n ( n + α + 1). These bispectral properties are encoded in the Jacobi algebra J defined [8] in terms of three generators K , K and K verifying the relations[ K , K ] = K (1.8)[ K , K ] = aK + bK , (1.9)[ K , K ] = a { K , K } + bK + cK + d, (1.10)where [ A, B ] = AB − BA , { A, B } = AB + BA and a, b, c, d are structure constants. Indeed J is realized by taking K = −M K = x. (1.11)In this model where the generators K and K are the bispectral operators K = 2 x ( x − ∂ x + [( α + 2) x + β − ∂ x , (1.12)the parameters a, b, c, d are a = 2 , b = − , c = − α ( α + 2) , d = α (1 − β ) . (1.13)Headway in the algebraic description of bispectral biorthogonal functions was achievedrecently by studying polynomial and rational functions of Hahn type [9], [10]. (Related Hahnrational functions also appear in [11] and [12].) In broad strokes the general picture thatemerges is as follows. Recall that generalized eigenvalue problems (GEVP) of the form M d n = λ n Ld n where M and L are two operators and λ is the eigenvalue, naturally lead to biorthogonalfunctions which are rational (or polynomial) when M and L act tridiagonally in associated2ases [13]. Assume this to be the case. In the context mentioned above, it proved possible toadjoin a third operator X to M and L such that the biorthogonal special functions are theoverlaps between the relevant GEVP basis { d n } and the eigenbasis { e ∗ z } of the adjoint X T of X . As for the biorthogonal partner they are given reciprocally in terms of the bases forthe corresponding adjoint problems. This offers a picture which is parallel to the descriptionof hypergeometric (finite) polynomials using Leonard pairs [14]. The differential/differenceequation of the biorthogonal functions follows readily from the fact that M − λ n L whichannihilates d n acts tridiagonally in the basis { e ∗ z } . The second spectral equation stems fromthe observation that the operator R T = L T X T is such that R T e ∗ z − zL T e ∗ z = 0 and that R = XL acts tridiagonally on the basis { d n } . The algebra generated by the triplet of operators ( M, L, R )which we have called the rational
Hahn algebra ( r h ) in the particular case treated in [9], [10]thus accounts for the two GEVPs that embody the bispectrality of the biorthogonal functions.Since R factorizes as XL , the algebra generated by ( M, L, R ) can be embedded in the meta -algebra generated by (
M, L, X ). The associated family of orthogonal polynomials also arisesin this context as the overlaps between the eigenfunctions of the linear pencil W = M + µL with the vectors { e ∗ z } (or equivalently as the scalar product of the eigenbases of the adjointproblems). The bispectrality of these polynomial functions is accounted for by the algebragenerated by ( X, W ). In our paradigm study, they are the Hahn polynomials, with W and X seen to generate the known Hahn algebra h . In summary, for functions of the Hahn type,we observed that the meta-algebra m h subsumes both r h and h and thus provides a unifieddescription of both the biorthogonal and orthogonal families of functions. The two dimensionalsubalgebra of m h generated by M and L is on its own remarkable since its three-diagonalrepresentations lead alone to the corresponding orthogonal polynomials, that is the Hahn onesin this instance. The adjunction of X to form the three-generated algebra has in fact the effectof constraining the representations of M and L to be three-diagonal in the eigenbasis of X .We contend that this approach which allows the simultaneous description of hypergeometricorthogonal polynomials and associated families of biorthogonal functions extends beyond theHahn functions case from which it is drawn. We shall add support to this suggestion byshowing that the biorthogonal Askey polynomials on the unit circle together with the Jacobipolynomials are amenable to a unified treatment that follows the lines sketched above. Inso doing we will provide an algebraic interpretation of the bispectral properties of the Askeypolynomials which is of interest in its own right. We might point out that it had been shownin [15] that the recurrence relation of these polynomials can be obtained from a linear pencilin su (1 ,
1) without providing a full account of the bispectrality however.The rest of the paper is organized as follows. The meta-Jacobi m J is introduced anddiscussed in the next section. It is shown to be isomorphic to su (1 , m J module andthe appropriate overlaps are shown to yield the special functions of interest. The orthogonalityrelations are seen to follow from the completeness and orthogonality of the GEVP and EVPbases. The algebraic set-up is employed in Section 4 to derive and interpret various propertiesof the Askey polynomials P n ( z ; α, β ) and in particular their bispectrality. A differential modelof m J is obtained and used to obtain the differential equation and recurrence relation of thepolynomials P n ( z ; α, β ) as well as some contiguity formulas. Perspectives are offered in the lastsection to conclude. Computational details are included in three appendices for completenessand the convenience of the reader. 3 The meta-Jacobi algebra m J The fundamental algebraic structure upon which the subsequent analysis hinges is introducednext.
Definition 2.1
The meta-Jacobi algebra m J has generators L , M and X (and the central )verifying the commutation relations [ L, M ] = L − ( α + 1) L − M (2.1)[ L, X ] = X − M, X ] = { X, L } − ( α + 1) X + β. (2.3) It is taken to be defined over the real numbers with the parameters α and β in R unless specifiedotherwise. The Casimir element is checked to be Q = { L , X } − ( α + 1) { L, X } − {
M, X } + 2 M + 2 βL. (2.4)We shall now observe that m J is isomorphic to a Lie algebra. Recall that the Lie algebra su (1 ,
1) has the commutation relations:[ J , J ± ] = ± J ± , [ J + , J − ] = − J , (2.5)and the standard Casimir operator J = J − J − J + J − . (2.6)We have: Proposition 2.1
The meta-Jacobi algebra m J is isomorphic to the Lie algebra su (1 , . This is confirmed by first observing that the commutation relations (2.5) of su (1 ,
1) are recov-ered upon using the commutation relations (2.1), (2.2), (2.3) of m J and setting J = L −
12 ( α − β + 1) (2.7) J + = X − J − = − L + ( α + 1) L + M. (2.9)That we have an isomorphism is established by noting that this map is invertible and providesthe following expressions of L , M and X in terms of the su (1 ,
1) generators: L = J + 12 ( α − β + 1) , (2.10) M = J + J − − β J −
14 ( α − β + 1)( α + β + 1) (2.11) X = J + + 1 . (2.12)The isomorphism between the two-generated subalgebras spanned by { L, M } and { J , J − } wasobserved in [16]. In light of the above formulas, the Casimir operator (2.4) of the meta-Jacobialgebra can be expressed as Q = 2 J −
12 ( α − β + 1) . (2.13)4 emark 2.1 In spite of this isomorphism it will be clear in the following that the m J presen-tation is best suited for the algebraic interpretation of the Askey polynomials. We also stick tothe terminology as it recalls the parallel with the treatment of the biorthogonal rational functionsof Hahn type. Proposition 2.2
The Jacobi algebra J defined in (1.8) , (1.9) , (1.10) admits a simple embed-ding in the meta-Jacobi algebra m J . This is seen by setting K = − M, K = X (2.14)and consequently K = −{ X, L } + ( α + 1) X − β. (2.15)Using the commutation relations (2.1), (2.2), (2.3) of m J it is straightforwardly verified that K , K , K thus defined obey those of J with the parameters given by a = 2 , b = − , c = − α ( α + 2) , d = ( α + 1) β − Q − . (2.16)Note the dependence of the parameter d on the Casimir element Q . The distinctive feature ofthe meta-Jacobi algebra lies as we see in the fact that K is resolved as a quadratic expressionin terms of the fundamental generators X and L . Remark 2.2
In the following section we shall call upon representations of su (1 , and hence of m J to interpret the Askey and Jacobi polynomials. In an irreducible representation, the Casimirelement J of su (1 , takes the form τ ( τ − . Hereafter, we shall consider representationswith τ = 12 ( α + β + 1) . (2.17) Equation (2.13) which establishes the relation between the Casimir operator Q of m J and theone of su (1 , then yields for the value of Q : Q = 2 αβ − α + β − . (2.18) Let us stress the coherence of the particular realization of the Jacobi algebra J in terms ofthe bispectral operators of the Jacobi polynomials given in the Introduction with the embeddingof J in m J given in Proposition 2.2. Indeed we see that with these choices for the Casimirelements, the parameter d of the Jacobi algebra as given in (2.16) takes the proper value: d = ( α + 1) β − Q − α (1 − β ) . In this section we shall establish the connection between the meta-Jacobi algebra m J , theAskey polynomials P n ( z ; α, β ), their biorthogonal partners Q n ( z ; α, β ) and the Jacobi polyno-mials ˆ P ( α,β ) n ( x ). To that end we shall consider a m J representation space inferred from theisomorphism of this algebra with su (1 , h | i . A T will stand forthe transpose of A : ( h u | A T ) | v i = h u | ( A | v i ).Consider the infinite-dimensional module V ( τ ) with τ ∈ R defined as follows by the actionof the generators on the basis vectors | τ, k i , k ∈ Z : J | τ, k i = ( τ + k ) | τ, k i , (3.1) J + | τ, k i = | τ, k + 1 i , (3.2) J − | τ, k i = k ( k − τ ) | τ, k − i . (3.3)(See in this connection [17].) It is readily checked that the Casimir element J = J − J − J + J − = τ ( τ −
1) on this representation space. The basis vectors are taken to be orthonormal-ized: h τ, k ′ | τ, k i = δ k ′ k . (3.4) Remark 3.1
Let us note the following.1. The representation defined above is not unitarisable [18].2. It is reducible and contains the unitary positive discrete series [19], [20], [21], [17] as anirreducible component. This submodule is spanned by the basis vectors with k ∈ Z + . Use now the formulas (2.10), (2.11), (2.12) of Proposition 2.2 that define the isomorphismbetween m J and su (1 ,
1) and take as already indicated τ = ( α + β + 1); the following actionsof L, M, X on the basis states | τ, k i are readily found: L | τ, k i = ( k + α + 1) | τ, k i , (3.5) M | τ, k i = k (cid:2) ( k + α + 1) | τ, k i + ( k + α + β ) | τ, k − i (cid:3) , (3.6) X | τ, k i = | τ, k + 1 i + | τ, k i . (3.7)The adjoint actions can be read off directly: L T | τ, k i = ( k + α + 1) | τ, k i , (3.8) M T | τ, k i = ( k + 1)( k + α + β + 1) | τ, k + 1 i + k ( k + α + 1) | τ, k i , (3.9) X T | τ, k i = | τ, k i + | τ, k − i . (3.10)Let us introduce the operator T ± on V ( τ ) such that: T ± | τ, k i = | τ, k ± i . (3.11)Consider a vector | f i = P ∞ k = −∞ f ( k ) | τ, k i in V ( τ ). We have T ± | f i = ∞ X k = −∞ f ( k ) T ± | τ, k ± i = ∞ X k = −∞ ( T ∓ f ( k )) | τ, k ± i , (3.12)where T ± stands for the shift operators acting on functions of k : T ± f ( k ) = f ( k ± emark 3.2 A realization of m J in terms of shift operators can hence be inferred from the(dual) transformations of the components of a vector | f i in the basis {| τ, k i} defined through: V | f i = P ∞ k = −∞ f ( k ) V | τ, k i = P ∞ k = −∞ (V T f ( k )) | τ, k i . Equations (3.8) , (3.9) , (3.10) thus yield: L = ( k + α + 1) , (3.13)M = ( k + 1)( k + α + β + 1) T + + k ( k + α + 1) , (3.14)X = T − + 1 . (3.15) The adjoints in this model are readily computed using T T ± = T ∓ . We are now ready to construct the bases of V ( τ ) coming in adjoint pairs, whose overlaps willprovide the algebraic interpretation we are looking for. (They will be in part the d n , d ∗ n , e z , e ∗ z of the Introduction.) The bases that will intervene are:1. The GEVP bases {| P n i} and {| Q n i} : M | P n i = ν n L | P n i M T | Q n i = ν n L T | Q n i . (3.16)It will be recalled [10], [13] that the sets {| P n i} and { L T | Q n i} form by construction twobiorthogonal ensembles of vectors: h P m | L T | Q n i = 0 , m = n. (3.17)2. The EVP bases {| z i} and { f | z i} : X | z i = z | z i , X T f | z i = z f | z i . (3.18)3. The EVP bases {| J n i} and { g | J n i} : M | J n i = µ n | J n i M T g | J n i = µ n g | J n i . (3.19) X and X T It is directly checked that the EVP (3.18) are satisfied by | z i = γ ∞ X k = −∞ ( z − − k − a | τ, k i , (3.20) f | z i = ˜ γ ∞ X k = −∞ ( z − k +˜ a | τ, k i , (3.21)with a, ˜ a ∈ R and where γ, ˜ γ ∈ C are normalization constants. That | z i and f | z ′ i are orthogonalcan be seen as follows. We have f h z ′ | z i = γ ˜ γ ∞ X k,l = −∞ ( z ′ − − k − a ( z − l +˜ a h τ, k | τ, l i . (3.22)Now let z = 1 + e iφ , z ′ = 1 + e iφ ′ , so that (3.22) becomes f h z ′ | z i = γ ˜ γ e i (˜ aφ − aφ ′ ) ∞ X k = −∞ e i ( φ − φ ′ ) k . (3.23)7e then see that upon imposing a = ˜ a + 1 , (3.24)we find f h z ′ | z i = − πiγ ˜ γδ ( z − z ′ ) (3.25)with the help of the Fourier series of Dirac’s delta function and of a standard property ofthis distribution. Since f h z ′ | z i is manifestly translation invariant, (3.25) is preserved when thevariable z lies on the unit circle centered at z = 0.We also have the completeness relation12 πiγ ˜ γ I C dz f | z ih z | = 1 (3.26)where the contour C consists in the unit circle infinitesimally deformed so that the singularityat z = 1 lies inside C . Indeed,12 πiγ ˜ γ I C dz f | z ih z | = 12 πi I C dz ( z − k − l − a +˜ a ∞ X k,l = −∞ | τ, k ih τ, l | . (3.27)Again the choice (3.24) for the integration constants a and ˜ a consistently ensures that theintegral over z becomes πi H C dz ( z − k − l − = δ kl and hence12 πiγ ˜ γ I C dz f | z ih z | = ∞ X k = −∞ | τ, k ih τ, k | = 1 . (3.28)This will play a key role in the derivation of the orthogonality relations. We shall now obtain the bases {| P n i} and {| Q n i} of V ( τ ) that satisfy the GEVP (3.16). Firstwe need to determine the set of eigenvalues ν . From the explicit two-diagonal actions (3.5),(3.6) of L and M on the basis vectors {| τ, k i} , it is readily seen that the (formal) determinantalcondition is det ( M − νL ) = ∞ Y k = −∞ [ k ( k + α + 1) − ν ( k + α + 1)] = 0 (3.29)and hence that the spectrum consists in the following values: ν n = n, n = 0 , ± , ± , . . . . (3.30) Remark 3.3
In the following, as we consider GEVPs and EVPs, we shall limit ourselves toeigenvalues corresponding to non-negative n , i.e. n ∈ Z ≥ . This will not restrain the breadthof the algebraic description since the same results would be obtained with other choices. Forcompleteness, indications on how the equations are handled for negative values of n are givenin Appendix C. | P n i = ∞ X k = −∞ d n ( k ) | τ, k i . (3.31)The generalized eigenvalue equation M | P n i = nL | P n i implies the following recurrence relationfor the expansion coefficients d n ( k ):( k + 1)( k + α + β + 1) d n ( k + 1) + ( k − n )( k + α + 1) d n ( k ) = 0 . (3.32)From (3.32), it is immediately seen that for n ≥ d n ( k ) = 0 for k > n and k ∈ Z − . (3.33)The explicit expression of the non-zero coefficients d n ( k ) reads d n ( k ) = d n (0) ( − k ( − n ) k ( α + 1) k k ! ( α + β + 1) k k = 0 , , , . . . , n. (3.34)Turn now to the adjoint GEVP M T | Q m i = m L T | Q m i which imposes on the coefficients d ∗ m ( k ) in | Q m i = ∞ X k = −∞ d ∗ m ( k ) | τ, k i (3.35)the recurrence relation k ( k + α + β ) d ∗ m ( k −
1) + ( k − m )( k + α + 1) d ∗ m ( k ) = 0 . (3.36)Assuming as previously indicated, m ≥
0, one immediately notices that (3.36) implies d ∗ m ( k ) = 0 for k < m. (3.37)In view of this fact, let k = l + m, l = 0 , , . . . , (3.38)the relation (3.36) then becomes( m + l )( l + m + α + β ) d ∗ m ( m + l −
1) + l ( l + m + α + 1) d ∗ m ( m + l ) . (3.39)It is found to have for solution d ∗ m ( m + l ) = ( − l ( m + 1) l ( m + α + β + 1) l l !( m + α + 2) l d ∗ m ( m ) l = 0 , , . . . . (3.40)Apart from the initial condition d ∗ m ( m ), equation (3.40) fully determines | Q m i = ∞ X l =0 d ∗ m ( m + l ) | τ, m + l i . (3.41)From general linear algebra considerations [13], [9], [10], we know that the vectors | P n i and L T | Q m i are biorthogonal for n = m . We have( h P n | M ) | Q m i = n ( h P n | L ) | Q m i = h P n | ( M T | Q m i ) = m h P m | ( L T | Q m i ) = m ( h P n | L ) | Q m i . (3.42)9t follows that ( n − m )( h P n | L ) | Q m i = ( n − m ) h P n | ( L T | Q m i ) = 0 (3.43)which implies the asserted biorthogonality if m = n . Since the derivation we shall provide ofthe biorthogonality of the Askey polynomials will rest on this property, we shall next directlyverify that it holds and determine the norm.From the observations made above, we see that h P n | L T | Q m i = n X k = −∞ ∞ X l =0 d n ( k ) d ∗ m ( l + m ) h τ, k | L T | τ, l + m i = n X k = −∞ ∞ X l =0 d n ( k ) d ∗ m ( l + m )( m + l + α + 1) δ k,l + m . (3.44)We readily find that h P n | L T | Q m i = 0 if m > n. (3.45)It remains to consider the situation when m ≤ n . Substituting in (3.44) the expressions (3.34)and (3.40) for d n ( k ) and d ∗ m ( m + l ), using a few properties of the Pochhammer symbols suchas x ( x + 1) l − = ( x ) l and ( x ) m + l = ( x ) m ( x + m ) l and performing one of the sums, we arrive at h P n | L T | Q m i = d n (0) d ∗ m ( m ) ( − m ( − n ) m ( α + 1) m +1 m !( α + β + 1) m n − m X l =0 ( − n + m ) l l ! . (3.46)We then recall the following formula(1 − x ) ξ = ∞ X k =0 ( − ξ ) k k ! x k (3.47)to conclude that h P n | L T | Q m i = N n δ m,n , (3.48)with N n = d n (0) d ∗ n ( n ) ( α + 1) n +1 ( α + β + 1) n . (3.49) Let us now identify some of the special functions that arise from this representation theoreticsetting. In light of the completeness relation (3.26) and the orthogonality relation (3.48), wesee that f h z | P n i and h z | L T | Q n i provide two families of biorthogonal functions on the unit circlesince 12 πiγ ˜ γ I | z | =1 dz h P n f | z ih z | L T | Q m i = h P n | L T | Q m i = N n δ m,n . (3.50)These are explicitly obtained below. 10 .3.1 The overlaps f h z | P n i From the expansions (3.21) and (3.31) of f | z i and | P n i over the orthonormal basis vectors | τ, k i we have f h z | P n i = ˜ γ n X l = −∞ ( z − l +˜ a d n ( l ) . (3.51)Upon inserting the expressions (3.34) for d n ( l ), we observe that f h z | P n i is the F polynomial f h z | P n i = ˜ γd n (0)( z − ˜ a F (cid:18) − n, α + 1 α + β + 1 ; 1 − z (cid:19) . (3.52)The Askey polynomials are then recognized with the help of the following relation [22] F (cid:18) − n, bc ; z (cid:19) = ( c − b ) n ( c ) n F (cid:18) − n, b − n + b + 1 − c ; 1 − z (cid:19) . (3.53)We find f h z | P n i = ˜ γd n (0) ( α + 1) n ( α + β + 1) n ( z − ˜ a P n ( z ; α, β ) , (3.54)where the polynomials P n ( z ; α, β ) are as defined in (1.1). Proposition 3.1
The Askey polynomials P n ( z ; α, β ) have a natural interpretation in the rep-resentation theory of the meta-Jacobi algebra. They occur according to (3.54) as the overlapsbetween two bases of the module V ( τ = ( α + β + 1)) satisfying respectively equations definedin terms of the generators X, L, M of m J . The first basis consists in the eigenvectors of X T (the transpose of X ) and the second is formed by the vectors solving the GEVP defined by L and M . h z | L T | Q m i The biorthogonal partners to the Askey polynomials are obtained in a similar fashion. From(3.8), (3.41) and (3.40) we have L T | Q m i = d ∗ m ( m )( m + α + 1) ∞ X l =0 ( − l ( m + 1) l ( m + α + β + 1) l l !( m + α + 1) l | τ, l + m i . (3.55)Combining with (3.20) and using the orthonormality of the basis vectors | τ, k i , we find h z | L T | Q m i = γd ∗ m ( m )( m + α + 1)( z − − m − a F (cid:18) m + 1 , m + α + β + 1 m + α + 1 ; 11 − z (cid:19) . (3.56)We may now use the fact that any three solutions of the hypergeometric equation are related bya linear relations and call upon transformation formulas of F series under homographic trans-formations to make the biorthogonal partners of the Askey polynomials appear in this overlap.Indeed following the steps described in Appendix A, we arrive at the following expression: h z | L T | Q m i = γd ∗ m ( m )( m + α + 1)( z − − a " Γ( m + α + 1)Γ( β + 1)Γ( m + β + 2)Γ( α ) F (cid:18) m + 1 , − αm + β + 2 ; z (cid:19) Γ( m + α + 1)Γ( m + β + 1) m ! Γ( m + α + β + 1) ( − z ) − − β (1 − z ) α + β Q m ( 1 z , α, β ) (3.57)where Q m ( z ) is defined as in (1.2). Remark 3.4
Note that the first term in this expression for h z | L T | Q m i is a power series whilethe second one which contains the polynomial Q n in the variable z has the transcendental factor z − β . Summing up:
Proposition 3.2
The biorthogonal partners Q n ( z, α, β ) of the Askey polynomials P n ( z ; α, β ) arise in the representation theory of the meta-Jacobi algebra in the overlaps, see (3.57) , betweenthe eigenbasis vectors of the generatior X and the basis vectors that obey the GEVP defined bythe operators M T and L T . The interpretation of the Askey polynomials in the framework of the meta-Jacobi algebra leadsto a natural derivation of their biorthogonality. Recall (3.50). First observe that in multiplyingthe expressions of the overlaps f h z | P n i and h z | L T | Q m i , as they are given by the formulas (3.54)and (3.57), the factor ( z − − a +˜ a reduces to 1 because of (3.24). Furthermore, one henceobserves that the product of the first term in (3.57) - a power series - with the polynomial P n ( z, α, β ) will give a vanishing contribution when integrated over the circle | z | = 1. Equation(3.50) thus yields d n (0) d ∗ m ( m ) ( m + α + 1)( α + 1) n ( α + β + 1) n Γ( m + α + 1)Γ( m + β + 1) m ! Γ( m + α + β + 1) × − πi I | z | =1 dz ( − z ) − − β (1 − z ) α + β P n ( z, α, β ) Q m ( 1 z , α, β ) = N n δ m,n . (3.58)Mindful of formula (3.49) for N n , we thus recover precisely the biorthogonality relation (1.3). M and M T We now undertake to show that the Jacobi polynomials can be described within the samealgebraic framework. We already noted in Proposition 2.2 that the elements X and M generatethe Jacobi algebra J which is thus embedded in m J . We therefore expect to see the Jacobipolynomials occur in the overlaps between the eigenvectors of X , X T and M T , M respectively.We shall hence first determine the eigenbases of V ( τ ) associated to M and M T .From the two-diagonal action (3.6) of M , we see that the spectrum { µ n } of this operatoris of the form µ n = n ( n + α + 1) . (3.59)Consider the EVPs (3.19). Set, | J n i = ∞ X k = −∞ f n ( k ) | τ, k i . (3.60)12he eigenvalue equation M | J n i = n ( n + α + 1) | J n i yields the following two-term recurrencerelation for the coefficients f n ( k ):( k + 1 + α + β ) f n ( k + 1) + [ k ( k + α + 1) − n ( n + α + 1)] f n ( k ) = 0 (3.61)which can be rewritten as( k + 1)( k + 1 + α + β ) f n ( k + 1) + ( k − n )( k + n + α + 1) f n ( k ) = 0 . (3.62)Here again, we shall focus on the case n ≥
0. The above equation is then found to imply that f n ( k ) = 0 for k > n and k ∈ Z − (3.63)and is solved by f n ( k ) = f n (0)( − k ( − n ) k ( n + α + 1) k k !( α + β + 1) k , k = 0 , , . . . , n. (3.64)Similarly, let g | J n i = ∞ X k = −∞ ˜ f n ( k ) | τ, k i . (3.65)The EVP M T g | J n i = n ( n + α + 1) g | J n i is readily seen to give: k ( k + α + β ) ˜ f n ( k −
1) + ( k − n )( k + n + α + 1) ˜ f n ( k ) = 0 . (3.66)In this instance, for m ≥
0, we observe that˜ f n ( k ) = 0 when k < n. (3.67)We set k = n + l, l = 0 , , , . . . and convert (3.66) into( n + l )( n + l + α + β ) ˜ f n ( n + l −
1) + l ( l + 2 n + α + 1) ˜ f n ( n + l ) = 0 (3.68)to find ˜ f n ( n + l ) = ˜ f n ( n )( − l ( n + 1) l ( n + 1 + α + β ) l l !(2 n + α + 2) l . (3.69)We may now verify that | J n i and g | J m i are orthogonal when m = n . This proceeds in a waysimilar to the computation of h P n | L T | Q m i carried out before. Clearly, h J n g | J m i = 0 if m > n .If m ≤ n , after some algebraic simplifications, we see that h J n g | J m i = f n (0) ˜ f m ( n )( − m ( − n ) m ( n + α + 1) m m !( α + β + 1) m F (cid:18) m − n, n + m + α + 12 m + α + 2 ; 1 (cid:19) . (3.70)From the Vandermonde formula [23] F (cid:18) − n, bc ; 1 (cid:19) = ( c − b ) n ( c ) n (3.71)we may then conclude that owing to the factor ( m − n + 1) n − m that appears, h J n g | J m i = 0unless n = m , in which case h J n g | J m i = N n δ m,n , (3.72)with N n = f n (0) ˜ f n ( n ) ( n + α + 1) n ( α + β + 1) n . (3.73)13 .5 Jacobi polynomials We will now observe how the Jacobi polynomials emerge in this framework and indicate howthis allows for another derivation of their orthogonality relation.
Let us now look at the overlaps f h z | J n i and h z g | J m i . From (3.21), (3.60) and (3.64) we obtain f h z | J n i = ˜ γf n (0)( z − ˜ a F (cid:18) − n, n + α + 1 α + β + 1 ; 1 − z (cid:19) . (3.74)Using (A.1) (or equivalently (3.53)) we arrive at f h z | J n i = ˜ γ ( z − ˜ a ( − n Γ(1 + α + β )Γ( β )Γ(2 n + α + 1)Γ(1 − β )Γ( − n + β )Γ( n + α + β + 1)Γ( n + α + 1)Γ( n + 1 − β ) ˆ P ( α,β ) n ( z ) , (3.75)where ˆ P ( α,β ) n ( z ) are the Jacobi polynomials defined in (1.4) extended to the complex plane.The second overlap is recovered from (3.20), (3.65) and (3.69). We find h z g | J m i = γ ˜ f m ( m )( z − − m − a F (cid:18) m + 1 , m + α + β + 12 m + α + 2 ; 11 − z (cid:19) . (3.76)At this point by performing the transformations described in Appendix B that make useof identities involving gamma functions and solutions of the hypergeometric equation, thefollowing formula is discovered: h z g | J m i = ( − m +1 γ ˜ f m ( m )( z − − a " Γ(2 m + α + 2)Γ( − β )Γ( m + α + 1)Γ( m − β + 1) F (cid:18) m + 1 , − m − α β ; z (cid:19) +( − m Γ(2 m + α + 2)Γ( β )Γ(2 m + α + 1)Γ(1 − β ) m ! Γ( m + α + β + 1)Γ( m + α + 1)Γ( m + 1 − β ) ( − z ) − β (1 − z ) α + β ˆ P ( α,β ) m ( z ) . (3.77) Remark 3.5
As already encountered in the expression (3.57) of h z | L T | Q m i , we see that thefirst term in (3.77) is a power series while the second that involves the Jacobi polynomialcontains the transcendental term z − β . Proposition 3.3
The Jacobi polynomials ˆ P ( α,β ) m ( z ) over C also arise in the context of themeta-Jacobi algebra m J . They occur as per the equations (3.75) and (3.77) in two overlapsbetween eigenbases of the module V ( ( α + β + 1)) : on the one hand between the eigenstatesof M and X T and on the other hand between those of M T and X . This interpretation of the Jacobi polynomials in the framework of the algebra m J entails aderivation of their orthogonality. Owing to the completeness relation (3.28), we have12 πiγ ˜ γ I C | z | =1 h J n f | z ih z g | J m i = h J n g | J m i = N n δ m,n (3.78)14 y z = x + iy C | z | =1 C ǫ e i. e i. π − ǫ Figure 1: The contour Ξwhere N n is given by (3.73). When substituting the expressions (3.75) and (3.77) for f h z | J n i and h z g | J m i , we first observe anew that the resulting factor ( z − − a +˜ a = 1 since 1 − a + ˜ a = 0.Then, we note that the product of f h z | J n i with the first term of (3.77) is a power series thatwill integrate to zero over the unit circle. Taking into account the formula (3.73) for N n andafter some simplifications, equation (3.78) thus amounts to − πi (cid:16) π sin πβ (cid:17) I C | z | =1 dz ( − z ) − β (1 − z ) α + β ˆ P ( α,β ) n ( z ) ˆ P ( α,β ) m ( z ) = h n δ m,n , (3.79)with h n given in (1.6). In obtaining (3.79) we have used the identity Γ( x )Γ(1 − x ) = π sin πx andin particular Γ( − n + β )Γ( n + 1 − β ) = π sin π ( − n + β ) = ( − n π sin πβ . (3.80)Finally, the orthogonality of the Jacobi polynomials on the interval [0 ,
1] is recovered byemploying the contour depicted in Figure 1 and computations carried out in [3]. Schematicallythe contour Ξ = C | z | =1 + [1 ,
0] + C ǫ + [0 , x = 1 to x = 0 below the branch cut, a circle of radius ǫ around z = 0 and the segment from x = 0 to x = 1 above the branch cut. Consider the integralin (3.79) with the contour C | z | =1 replaced by the contour Ξ of figure 1. Since no singularitiesare enclosed by Ξ that integral is equal to zero.If we restrict β to be smaller than 1, i.e. if we take β < ǫ → πi I C ǫ dz ( − z ) − β (1 − z ) α + β ˆ P ( α,β ) n ( z ) ˆ P ( α,β ) m ( z ) = 0 , for β < . (3.81)It follows that the integral over C | z | =1 must be the negative of the sum of the integrals overand above the real axis. Hence, recalling the choice of branch ( − z ) − β = | z | − β when arg z = π ,we have − πi (cid:16) π sin πβ (cid:17) I C | z | =1 dz ( − z ) − β (1 − z ) α + β ˆ P ( α,β ) n ( z ) ˆ P ( α,β ) m ( z ) =152 i (cid:16) πβ (cid:17) e iπβ (1 − e − πiβ ) Z x − β (1 − x ) α + β ˆ P ( α,β ) n ( x ) ˆ P ( α,β ) m ( x ) dx. (3.82)The factors before the integral sign in the last expression cancel and this gives the orthogonalityrelation (1.5) of the Jacobi polynomials in view of (3.79) We shall indicate in this section how various properties of the biorthogonal Askey polynomialson the circle naturally follow from their interpretation based on the meta-Jacobi algebra. Recallthat | P n i = d n (0) n X k =0 ( − k ( − n ) k ( α + 1) k k !( α + β + 1) k | τ, k i . (4.1)Looking at the overlap f h z | P n i given in (3.54), without loss of generality, we can set from nowon: ˜ γ = 1 , ˜ a = 0 , a = 1 . (4.2)It is moreover natural to take the initial values d n (0) of the recurrence relation (3.32) to be d n (0; α, β ) = ( α + β + 1) n ( α + 1) n (4.3)so that f h z | P n i = P n ( z ; α, β ) , (4.4)identifying f h z | P n i precisely with the Askey polynomials. This also means that | τ, n i has coef-ficient 1 in | P n i : | P n i = ( α + β + 1) n ( α + 1) n | τ, i + · · · + | τ, n i . (4.5) L and R = XL in the basis {| P n i} We shall now show that the generator L and the product XL act in a two-diagonal fashion inthe basis {| P n i , n = 0 , , . . . } . We have L | P n ( α, β ) i = ( α + β + 1) n ( α + 1) n n X k =0 ( − k ( − n ) k ( α + 1) k k !( α + β + 1) k L | τ, k i . (4.6)From eq.(3.5) which reads L | τ, k i = ( k + α + 1) | τ, k i , and the identity( − n ) k ( k + α + 1) = ( − n ) k (cid:2) n + α + 1 + ( − n + k ) (cid:3) (4.7)= ( n + α + 1)( − n ) k − n ( − n + 1) k , we see that L | P n ( α, β ) i = ( n + α + 1) | P n ( α, β ) i − n ( n + α + β )( n + α ) | P n − ( α, β ) i . (4.8)16lternatively, using( k + α + 1)( α + 1) k = ( α + 1) k +1 = (cid:0) ( α + 1) + 1 (cid:1) k ( α + 1) , (4.9)we note that L has also the effect of shifting the parameters: L | P n ( α, β ) i = ( n + α + 1) | P n ( α + 1 , β − i . (4.10)Consider now the action of the operator R = XL . Knowing that L acts diagonally as per(3.5) on the basis vectors | τ, k i and that according to (3.7), i.e. X | τ, k i = | τ, k i + | τ, k + 1 i , wehave R | P n ( α, β ) i = ( α + β + 1) n ( α + 1) n n X k =0 ( − k ( − n ) k ( α + 1) k ( k + α + 1) k !( α + β + 1) k h | τ, k i + | τ, k + 1 i i . (4.11)Collecting the factors of the vectors | τ, k i , k = 0 , . . . , n + 1, we find R | P n ( α, β ) i =( α + β + 1) n ( α + 1) n " ( α + 1) | τ, i + n X k =1 ( − k ( α + 1) k k !( α + β + 1) k h ( − n ) k ( k + α + 1) − ( − n ) k − k ( k + α + β ) i | τ, k i (4.12)+ ( α + 1) n +1 ( α + β + 1) n | τ, n + 1 i . The two relations k ( − n ) k − = ( − n ) k − ( − n − k , (4.13)( − n ) k ( − n − − ( − n − k ( n + 1 − k ) = 0 (4.14)come in handy in deriving the following identity:( − n ) k ( k + α + 1) − k ( − n ) k − ( k + α + β )= ( − n ) k ( k + α + 1) − (cid:2) ( − n ) k − ( − n − k (cid:3) ( k + α + β )= ( − n ) k (1 − β ) + ( − n − k ( k + α + β )= ( − n ) k ( − n − β ) + ( − n − k ( n + α + β + 1) . (4.15)Clearly, (4.13) has been used in getting the second line and (4.14) has been added to the thirdline to obtain the end result. Upon inserting this relation (4.15) in (4.12), we recognize easilythat R is a two-diagonal raising operator: R | P n ( α, β ) i = ( n + α + 1) | P n +1 ( α, β ) i − ( β + n ) | P n ( α, β ) i . (4.16)In the following we shall also consider the element˜ R = XM. (4.17)
Remark 4.1
Given that the vectors | P n ( α, β ) i satisfy the GEVP M | P n ( α, β ) i = nL | P n ( α, β ) i ,the eqs. (4.8) and (4.16) readily provide the actions of M and ˜ R on these vectors. .2 A differential realization A differential model of the meta-Jacobi algebra is directly obtained. With the choices (4.2) wehave f h z | τ, k i ≡ f ( z, k ) = ( z − k . (4.18)We can dually define an operator acting on the variable z as follows O z f h z | τ, k i = f h z | O | τ, k i = O k f ( z, k ) , (4.19)where O z corresponds to the operator O acting on the module V ( ( α + β + 1)) and O k as inremark 3.2, acts on the components f ( z, k ) of the vector f | z i . With O = L, M, X we find:
Proposition 4.1
The differential operators L , M and X given below provide a realization ofthe commutation relations (2.1) , (2.2) and (2.3) of the meta-Jacobi algebra. L = ( z − ∂ z + ( α + 1) I ; (4.20) M = z ( z − ∂ z + [( α + 2) z + β − ∂ z ; (4.21) X = z. (4.22)It follows that R = XL and ˜ R = XM are realized by R = z ( z − ∂ z + ( α + 1) z ; (4.23)˜ R = z ( z − ∂ z + z [( α + 2) z + β − ∂ z . (4.24) Remark 4.2
One may also take X acting on the left on f h z | and giving the eigenvalue z sothat R f h z | P n ( α, β ) i = f h z | XL | P n ( α, β ) i = z f h z | L | P n ( α, β ) i = z L f h z | P n ( α, β ) i (4.25) and similarly for ˜ R = XM . Remark 4.3
Note that L and M have the property of stabilizing spaces of polynomials ofgiven degrees while X , R , ˜ R raise the degree by . In the spirit of studies carried out in [24],[25], [26] for example, X , R , ˜ R are operators of Heun type. Remark 4.4
Observe that M precisely coincides with the hypergeometric operator (1.7) albeitin the variable z . Remark 4.5
This differential model for m J can also be retrieved by using the Barut-Ghirardello(B-G) realization of su (1 , : J = ( z − ddz + τ,J + = ( z − , (4.26) J − = ( z − d dz + 2 τ ddz , τ = 12 ( α + β + 1) , in the formulas (2.10) , (2.11) , (2.12) giving L , M , and X in terms of the su (1 , generators.Note that the variable z is here translated by with respect to the usual B-G formulas. P n ( z ; α, β ) as f h z | P n ( α, β ) i , in view of the actions (4.8), (4.16) of L and R on the vectors | P n i , of remark 4.1 and of the realizations given above ( (4.20), (4.21),(4.22) (4.23), (4.24)) of these operators, we have the following. Proposition 4.2
The biorthogonal Askey polynomials P n ( z ; α, β ) on the unit circle satisfy thefollowing differential identities: L P n ( z ; α, β ) = ( n + α + 1) P n ( z ; α, β ) − n ( α + β + n ) α + n P n − ( z ; α, β ) , (4.27) M P n ( z ; α, β ) = n ( n + α + 1) P n ( z ; α, β ) − n ( α + β + n ) α + n P n − ( z ; α, β ) , (4.28) R P n ( z ; α, β ) = ( n + α + 1) P n +1 ( z ; α, β ) − ( β + n ) P n ( z ; α, β ) , (4.29)˜ R P n ( z ; α, β ) = n ( n + α + 1) P n +1 ( z ; α, β ) − n ( β + n ) P n ( z ; α, β ) . (4.30) The bispectral equations of the Askey polynomials can now easily be identified and interpretedin terms of generalized eigenvalue problems.
The GEVP M | P n ( α, β ) i = nL | P n ( α, β ) i translates after projection on f h z | into the second orderdifferential equation: M P n ( z ; α, β ) = n L P n ( z ; α, β ) (4.31)with eigenvalue n and where the operators M and L are respectively given by (4.28) and(4.27). The recurrence relation is obtained by considering the GEVP R P n ( z ; α, β ) = z L P n ( z ; α, β ) (4.32)which is satisfied by construction (see (4.25) in Remark 4.2). Expressing in (4.32) the two-diagonal actions (4.27) and (4.29) of L and R , one arrives at the recurrence relation P n +1 ( x ) + b n P n ( x ) = x ( P n ( x ) + g n P n − ( x )) (4.33)where b n = − β + nα + n + 1 , g n = − n ( n + α + β )( α + n )( α + n + 1) . (4.34)This recurrence relation was obtained by Hendriksen and van Rossum in [3]. It was derived in[15] by considering linear pencils in sl . It is also constructed through a gluing procedure byKim and Stanton in their recent study of R I polynomials [27]. Remark 4.6
It is manifest from this recurrence relation of R I - type [28], that z (resp. X ) isa lower Hessenberg matrix on the space of Askey polynomials (resp. in the basis {| P n ( α, β ) i} ).This feature of the representation theory of m J was also observed in the study of the meta-Hahnalgebra [10]. Proposition 4.3
The biorthogonal Askey polynomials defined on the unit circle are bispectral.They satisfy the differential equation (4.31) and the recurrence relation of R I - type (4.33) withcoefficients (4.34) . Both spectral equations are of GEVP type. .4 Contiguity relations Some contiguity relations for the Askey polynomials arise also naturally in the meta-Jacobialgebra framework. Indeed, we already observed in eq. (4.10) that the generator L has theeffect of performing the shifts α → α + 1, β → β − | P n ( α, β ) i . That M has asimilar effect follows from the fact that M = nL in the GEVP basis | P n ( α, β ) i . This translatesinto the following for the polynomials P n ( α, β ) = f h z | P n ( α, β ) i . Proposition 4.4
The Askey polynomials P n ( α, β ) verify the following contiguity equations: L P n ( z ; α, β ) = ( α + n + 1) P n ( z ; α + 1 , β − , (4.35) M P n ( z ; α, β ) = n ( α + n + 1) P n ( z ; α + 1 , β − , (4.36) where L and M are the differential operators (4.20) and (4.21) respectively. Remark 4.7
Given the explicit form (1.1) of the Askey polynomials, the above relations canbe checked directly on P n ( z ; α, β ) with the differential operators L and M . Having done this,comparing (4.35) and (4.36) offers a way to show that the Askey polynomials are solutions ofthe GEVP (4.31) . We shall finally examine how solving the GEVP M f ( z ) = n L f ( z ) and the adjoint problemcompares to the representation theoretic computations that were performed of the overlaps f h z | P n ( α, β ) i and h z | L T | Q m ( α, β ) i . A first look shows that the GEVPs in the differential modelwill be of hypergeometric nature as is confirmed by the expressions of the overlaps. Let usfocus on this more closely.Given the expressions (4.20) and (4.21) for L and M , we see that M f ( z ) = n L f ( z ) takesthe form of the hypergeometric equation [22] z (1 − z ) d fdz + (cid:2) c − ( a + b + 1) z (cid:3) dfdz − abf = 0 , (4.37)with parameters a = − n, b = α + 1 , c = 1 − n − β. (4.38)In the following we shall use Bateman’s nomenclature [22] of the 24 Kummer solutions; theseare arranged in six sets such that the four elements in each set represent the same function.The representatives u , u , ..., u of the sets are in general different although (3.53) is a casewhere u ∝ u . With the parameters given by (4.38), it is immediate to see that the solution u = F (cid:18) a, bc ; z (cid:19) (4.39)will yield directly (up to a constant) the Askey polynomials P n ( z ; α, β ).Consider now the adjoint operators: L T = (1 − z ) ∂ z + α I (4.40) M T = z ( z − ∂ z + (cid:2) (2 − α ) z − β − (cid:3) ∂ z − α I . (4.41)20he adjoint GEVP M T f ∗ ( z ) = m L f ∗ ( z ) also turns out to yield the hypergeometric equation(4.37) but this time with parameters a = m + 1 , b = − α, c = 1 + β + m. (4.42)Recall that L T f ∗ ( z ) will provide a solution orthogonal to f ( z ). Selecting u for f ∗ also willlead to functions trivially orthogonal over the unit circle. Consider instead f ∗ ( z ) = u = (1 − z ) − a F (cid:18) a, c − ba + 1 − b ; 11 − z (cid:19) . (4.43)Using ( k + m + α + 1)( m + α + 2) k = ( m + α + 1)( m + α + 1) k (4.44)it is easy to find that in that case, L T f ∗ ( z ) = ( m + α + 1)(1 − z ) − m − F (cid:18) m + 1 , m + α + β + 1 m + α + 1 ; 11 − z (cid:19) . (4.45)We thus see that choosing the solution u , yields the result (3.56) obtained algebraically for h z | L T | P m ( α, β ) i . (Recall that the constant a in this expression is here set equal to 1.) Thereader is reminded of eq. (3.57) where this function is seen to be composed of two parts: one,a power series in z and the other, the function orthogonal to the Askey polynomial dressedwith the hypergeometric weight.Let us point out that within the differential realization, it is possible to pick a solution of M T f ∗ ( z ) = m L f ∗ ( z ) that will solely give the biorthogonal partner Q m ( z , α, β ) multiplied bythe weight. Indeed take f ∗ ( z ) to be rather given by the solution u = (1 − z ) − b F (cid:18) b, c − ab + 1 − a ; 11 − z (cid:19) . (4.46)Substituting the parameters (4.42) we have in this instance f ∗ ( z ) = (1 − z ) α F (cid:18) − α, β − α − m ; 11 − z (cid:19) = ∞ X k =0 ( − α ) k ( β ) k ( − α − m ) k (1 − z ) − k + α k ! . (4.47)The action of L T is again readily computed when the argument is a function of (1 − z ): L T f ∗ ( z ) = ∞ X k =1 ( − α ) k ( β ) k ( − α − m ) k (1 − z ) − k + α ( k − ∞ X l =0 ( − α ) l +1 ( β ) l +1 ( − α − m ) l +1 (1 − z ) − l + α − l != αβ ( α + m ) (1 − z ) α − F (cid:18) − α + 1 , β + 11 − α − m ; 11 − z (cid:19) , (4.48)where we have used ( x ) l +1 = x ( x + 1) l . We thus observe that the action of L T is to effect : α → α − , β → β + 1; in view of the parameter identification (4.42), the action of L T on u yields again u with the following parameters: a = m + 1 , b = − α + 1 , c = 2 + β + m. (4.49)21ow another expression for u is u = ( − z ) a − c (1 − z ) c − a − b F (cid:18) − a, c − ab + 1 − a ; 1 z (cid:19) . (4.50)Using the parameters (4.49), we then find that choosing u as solution of the hypergeometricequation stemming from the adjoint GEVP M T f ∗ ( z ) = m L f ∗ ( z ) leads to L T f ∗ ( z ) ∝ ( − z ) − − β (1 − z ) α + β Q m ( 1 z , α, β ) . (4.51)That is, we obtain as unique term, up to a factor, the orthogonal partner of P n ( z ; α, β ) multi-plied by the weight. It is now time to wrap up and offer perspectives. We have presented a unified algebraic inter-pretation of the biorthogonal Askey polynomials on the circle and of the Jacobi polynomials onthe interval [0 , L, M, X verifying quadraticrelations that we have called the meta-Jacobi algebra and denoted by m J . The Askey poly-nomials P n ( z ; α, β ) arise as overlaps between the basis elements that are on the one hand thesolutions, on an infinite dimensional module, of the generalized eigenvalue problem defined bythe generators L and M and on the other hand, the eigenvectors of the adjoint of X . Thebiorthogonal partners Q n ( z ; α, β ) are obtained similarly from the reciprocal adjoints. The sameframework was seen to provide an algebraic picture for the Jacobi polynomials as overlaps be-tween the eigenbases of M and of X T (or of M T and X ). Proofs of the orthogonality relationswere found to follow. With the introduction of a differential model for the meta-Jacobi algebra,the bispectrality of the Askey polynomials P n ( z ; α, β ) was accounted for in particular; theirdifferential equation and the recurrence relation were explicitly obtained and found to be ofGEVP form.The meta-Jacobi algebra is actually isomorphic to the Lie algebra su (1 , R I . To be sure, it is with enthusiasm that we plan to pursue the investigations of meta-algebrasand their relations to special functions. A The computation of h z | L T | Q m i We provide in this Appendix the details on how the result of Proposition 3.2 is obtained.22iven the expression (3.56) for h z | L T | Q m i , we use the following linear relation betweenKummer solutions of the hypergeometric equation [22](Sect 2.9, eq. (35)): F (cid:18) a, bc ; z (cid:19) = Γ( a + 1 − c )Γ( b + 1 − c )Γ( a + b + 1 − c )Γ(1 − c ) F (cid:18) a, ba + b + 1 − c ; 1 − z (cid:19) (A.1) − Γ( a + 1 − c )Γ( b + 1 − c )Γ( c − a )Γ( b )Γ(1 − c ) z − c (1 − z ) c − a − b F (cid:18) − a, − b − c ; z (cid:19) . This yields F (cid:18) m + 1 , m + α + β + 1 m + α + 1 ; 11 − z (cid:19) =Γ(1 − α )Γ( β + 1)Γ( m + β + 2)Γ( − m − α ) F (cid:18) m + 1 , m + α + β + 1 m + β + 2 ; zz − (cid:19) − (A.2)Γ(1 − α )Γ( β + 1)Γ( m + α ) m ! Γ( m + α + β + 1)Γ( − m − α ) ( − z ) − m − β − (1 − z ) m + α + β +1 2 F (cid:18) − m, − m − α − β − m − α ; 11 − z (cid:19) . Now use [22] (Sect. 2.9 eqs. (1) & (4) and (9) & (11)): F (cid:18) a, c − bc ; zz − (cid:19) = (1 − z ) a F (cid:18) a, bc ; z (cid:19) (A.3)and F (cid:18) a, c − ba + 1 − b ; 11 − z (cid:19) = (1 − z ) a ( − z ) − a F (cid:18) a, a + 1 − ca + 1 − b ; 1 z (cid:19) (A.4)to reexpress the two F ’s on the right hand side of (A.2) as functions of z and z respec-tively. Recalling then the definition (1.2) of the biorthogonal partner Q m ( z, α, β ) of the Askeypolynomials, one arrives at (3.57) with the help of the relationΓ( − m − α )Γ( m + α + 1) = ( − m +1 Γ( α )Γ(1 − α ) (A.5)which is a consequence of the identityΓ( x )Γ(1 − x ) = π sin πx . (A.6) B The determination of h z g | J n i Details on how formula (3.77) for h z g | J m i is obtained are given here. We need to transformthe F that occurs in the expression (3.76) of this overlap. First we use the following relationbetween three solutions of the hypergeometric equation [22] (Sect 2.9, eq. (34)): F (cid:18) a, bc ; z (cid:19) = Γ( c )Γ( b − a )Γ( c − a )Γ( b ) ( − z ) − a F (cid:18) a, a + 1 − ca + 1 − b ; 1 z (cid:19) (B.1)23 Γ( c )Γ( a − b )Γ( c − b )Γ( a ) ( − z ) a − c (1 − z ) c − a − b F (cid:18) − a, c − ab + 1 − a ; 1 z (cid:19) . From this identity we find: F (cid:18) m + 1 , m + α + β + 12 m + α + 2 ; 11 − z (cid:19) = ( z − m +1 × " Γ(2 m + α + 2)Γ( α + β )Γ( m + α + 1)Γ( m + α + β + 1) F (cid:18) m + 1 , − m − α − α − β ; 1 − z (cid:19) + (B.2)Γ(2 m + α + 2)Γ( − α − β ) m ! Γ( m + 1 − β ) z − β ( z − α + β F (cid:18) − m, m + α + 1 α + β + 1 ; 1 − z (cid:19) . We now apply the relation (A.1) to convert each of the two F s on the right hand side of(B.2) that are functions of (1 − z ) into combinations of F s that are functions of z . This leadsto F (cid:18) m + 1 , m + α + β + 12 m + α + 2 ; 11 − z (cid:19) =Γ(2 m + α + 2)Γ( α + β )Γ(1 − α − β )Γ( − β )Γ( m + α + 1)Γ( m + α + β + 1)Γ(1 + m − β )Γ( − m − α − β ) ( z − m +1 2 F (cid:18) m + 1 , − m − α β ; z (cid:19) + " ( − ( α + β ) Γ(2 m + α + 2)Γ( α + β )Γ(1 − α − β )Γ( β ) m ! Γ( m + α + 1)Γ( − m − α )Γ( m + α + β + 1) (B.3)+ Γ(2 m + α + 2)Γ( − α − β )Γ( α + β + 1)Γ( β ) m ! Γ( m + 1 − β )Γ( − m + β )Γ( m + α + β + 1) z − β ( z − m +1+ α + β F (cid:18) − m, m + α + 11 − β ; z (cid:19) . Simplifications are carried out through repeated use of the identity (A.6) and various implica-tions such as (A.5) and by observing that1sin π ( α + β ) (cid:16) ( − α + β sin πα + sin πβ (cid:17) = e iπα sin π ( α + β ) (cid:16) e iπβ sin πα + e − iπα sin πβ (cid:17) = e iπα = ( − α . (B.4)One finally obtains F (cid:18) m + 1 , m + α + β + 12 m + α + 2 ; 11 − z (cid:19) = (1 − z ) m +1 × " Γ(2 m + α + 2)Γ( − β )Γ( m + α + 1)Γ( m − β + 1) F (cid:18) m + 1 , − m − α β ; z (cid:19) + (B.5)24(2 m + α + 2)Γ( β ) m ! Γ( m + α + β + 1) ( − z ) − β (1 − z ) α + β F (cid:18) − m, m + α + 11 − β ; z (cid:19) which readily gives (3.77). C Negative indices
In the main part of the paper, it sufficed for the purpose of interpreting the Askey polynomialsand their biorthogonal partners to focus on GEVP and EVP solutions with non-negative(integer) eigenvalues. For completeness we briefly indicate in this appendix how the situationswith negative integers can be treated and seen to lead to redundant information.
C.1
Consider equation (3.32) and assume that n <
0. Let n = − s − , s = 0 , , . . . (C.1)In this case the recursion relation still implies d n ( k ) = 0 for k > n but no longer bounds k from below. Write k in the form k = − s − − l, l = 0 , , . . . (C.2)Upon substituting (C.1) and (C.2) and taking d n ( k ) ≡ ˜ d s ( s + l ), equation (3.32) becomes( s + l )( l + s − α − β ) ˜ d s ( s + l −
1) + l ( l + s − α ) ˜ d s ( s + l ) = 0 . (C.3)We observe that this last relation coincide with the condition (3.36) that had been obtainedfrom the adjoint GEVP with a positive eigenvalue under the substitutions m → s, α → − α − , β → − β + 1 , d ∗ m ( m + l ) → ˜ d s ( s + l ) . (C.4)Hence, d n ( k ) = d − s − ( − s − − l ) = ( − l ( s + 1) l ( s − α − β + 1) l l !( s − α + 1) l d n ( n ) , s, l = 0 , , , . . . . (C.5) C.2
Examine now equation (3.36) when m <
0. Set m = − s − , s = 0 , , . . . . In this case therecursion equation implies that d ∗ m ( k ) = 0 for k < m and also truncates at k = 0. The non-zerovalues of d ∗ n ( k ) therefore only occur for k = − l − , l = 0 , . . . s. (C.6)Incorporating the above redefinitions in (3.36) and taking d ∗ m ( k ) = d ∗− s − ( − l − ≡ ˜ d ∗ m ( l ) weget ( l + 1)( l − α − β + 1) ˜ d ∗ m ( l + 1) + ( l − s )( l − α ) ˜ d ∗ m ( l ) = 0 (C.7)and see that this equation can be retrieved from (3.32) under the substitutions n → m, α → − α − , β → − β − , d n ( k ) → ˜ d ∗ m ( l ) . (C.8)It follows that for negative md ∗ m ( k ) = d ∗− s − ( − l −
1) = ( − l ( − s ) l ( − α ) l l !( − α − β + 1) l d ∗ n ( − , l = 0 , . . . s. (C.9)25 .3 We may check the orthogonality of | P n i and L T | Q m i , m = n , for various possibilities regardingthe sign of the indices m and n . In summary, the summation ranges are as follows: • For n ≥ , m ≥ | P n i = n X k =0 d n ( k ) | τ, k i , (C.10) | Q m i = ∞ X k = m d m ( k ) | τ, k i ; (C.11) • For n < , m < | P n i = n X k = −∞ d n ( k ) | τ, k i , (C.12) | Q m i = − X k = m d m ( k ) | τ, k i . (C.13)It is manifest that the orthogonality prevails when one index is non-negative and the otheris negative. When the two indices are negative, the proof of orthogonality follows the onegiven for two non-negative indices since as we observed the change of signs basically flips thecoefficients d and d ∗ . C.4
Regarding the special functions, in light of this exchange of the expansion coefficients, the rolesof | P n i and of L T | Q m i are inverted when the indices are negative. For instance, we have | Q − s − i = s X l =0 ( − l ( − s ) l ( − α ) l l !( − α − β + 1) l | τ, − l − i . (C.14)The overlap of L T | Q − s − i with the state | z i given in (3.20) is then found to be h z | L T | Q − s − i = αd − s − ( − z − ˜ a − F (cid:18) − s, − α − α − β ; 1 − z (cid:19) . (C.15)Owing again to (3.53), we see that the Askey polynomials arise in this case in the overlap h z | L T | Q − s − i with a change of parameters. C.5
Things can be seen to proceed similarly in the treatment of the Jacobi polynomials if negativeindices are considered. 26 cknowledgments
The authors are grateful to Tom Koornwinder for correspondence and bringing some referencesto their attention. They have much appreciated Erik Koelink’s comments on the manuscriptand are thankful to Julien Gaboriaud for kind assistance. The work of LV is supported in partby a Discovery Grant from the Natural Sciences and Engineering Research Council (NSERC)of Canada. AZ who is funded by the National Foundation of China (Grant No.11771015)gratefully acknowledges the hospitality of the CRM over an extended period and the award ofa Simons CRM professorship..
References [1] R. Askey. Discussion of Szeg¨o’s paper “Beitr¨age zur Theorie der Toeplitzschen Formen”.
Gabor Szeg¨o. Collected works , 1:303–305, 1982.[2] R. Askey. Some problems about special functions and computations.
Rend. Sem. Mat.Univ. Politec. Torino , pages 1–22, 1985.[3] E. Hendriksen and H. van Rossum. Orthogonal Laurent polynomials. In
IndagationesMathematicae (Proceedings) , volume 89, pages 17–36. Elsevier, 1986.[4] G. Greiner and T. H. Koornwinder. Variations on the Heisenberg spherical harmonics.
Stichting Mathematisch Centrum. Zuivere Wiskunde , (ZW 186/83), 1983.[5] L.-C. Shen. Orthogonal polynomials on the unit circle associated with the Laguerre poly-nomials.
Proceedings of the American Mathematical Society , 129(3):873–879, 2001.[6] N. M. Temme. Uniform asymptotic expansion for a class of polynomials biorthogonal onthe unit circle.
Constructive Approximation , 2(1):369–376, 1986.[7] J. Borrego-Morell and F. R. Rafaeli. On a class of biorthogonal polynomials on the unitcircle.
Journal of Mathematical Analysis and Applications , 438(1):465–473, 2016.[8] V. Genest, M. Ismail, L. Vinet, and A. Zhedanov. Tridiagonalization of the hypergeomet-ric operator and the Racah–Wilson algebra.
Proceedings of the American MathematicalSociety , 144(10):4441–4454, 2016.[9] S. Tsujimoto, L. Vinet, and A. Zhedanov. An algebraic description of the bispectral-ity of the biorthogonal rational functions of Hahn type.
Proceedings of the AmericanMathematical Society , 149(2):715–728, 2021.[10] L. Vinet and A. Zhedanov. A unified algebraic underpinning for the Hahn polynomialsand rational functions.
Journal of Mathematical Analysis and Applications , page 124863,2020.[11] W. Koepf and M. Masjed-Jamei. Two classes of special functions using Fourier transformsof some finite classes of classical orthogonal polynomials.
Proceedings of the AmericanMathematical Society , 135(11):3599–3606, 2007.[12] M. Masjed-Jamei and W. Koepf. Two classes of special functions using Fourier transformsof generalized ultraspherical and generalized Hermite polynomials.
Proceedings of theAmerican Mathematical Society , 140(6):2053–2063, 2012.2713] A. Zhedanov. Biorthogonal rational functions and the generalized eigenvalue problem.
Journal of Approximation Theory , 101(2):303–329, 1999.[14] P. Terwilliger. Two linear transformations each tridiagonal with respect to an eigenbasisof the other.
Linear algebra and its applications , 330(1-3):149–203, 2001.[15] F. A. Gr¨unbaum, L. Vinet, and A. Zhedanov. Linear operator pencils on Lie algebrasand Laurent biorthogonal polynomials.
Journal of Physics A: Mathematical and General ,37(31):7711, 2004.[16] J. Gaddis. Two-generated algebras and standard-form congruence.
Communications inAlgebra , 43(4):1668–1686, 2015.[17] R. E. Howe and E. C. Tan.
Non-Abelian Harmonic Analysis: Applications of SL (2,R) .Springer Science & Business Media, 2012.[18] G. Tomasini and B. Ørsted. Unitary representations of the universal cover of SU(1,1) andtensor products.
Kyoto Journal of Mathematics , 54(2):311–352, 2014.[19] N. J. Vilenkin and A. U. Klimyk.
Representations of Lie groups and special functions ,volume 1. Kluwer Academic Publishers, 1991.[20] D. R. Masson and J. Repka. Spectral Theory of Jacobi Matrices in l ( Z ) and the su(1,1)Lie Algebra. SIAM journal on mathematical analysis , 22(4):1131–1146, 1991.[21] W. Groenevelt and E. Koelink. Meixner functions and polynomials related to Lie algebrarepresentations.
Journal of Physics A: Mathematical and General , 35(1):65, 2001.[22] H. Bateman.
Higher transcendental functions [volumes i-iii] , volume 1. McGraw-Hill BookCompany, 1953.[23] G. Gasper and M. Rahman.
Basic hypergeometric series , volume 96. Cambridge universitypress, 2004.[24] F. A. Gr¨unbaum, L. Vinet, and A. Zhedanov. Tridiagonalization and the Heun equation.
Journal of Mathematical Physics , 58(3):031703, 2017.[25] S. Tsujimoto, L. Vinet, and A. Zhedanov. The rational Heun operator and Wilsonbiorthogonal functions. arXiv preprint arXiv:1912.11571 , 2019.[26] J. Gaboriaud, S. Tsujimoto, L. Vinet, and A. Zhedanov. Degenerate Sklyanin algebras,Askey–Wilson polynomials and Heun operators.