The multivariate Hahn polynomials and the singular oscillator
aa r X i v : . [ m a t h - ph ] J a n The multivariate Hahn polynomialsand the singular oscillator
Vincent X. Genest
E-mail: [email protected]
Centre de recherches mathématiques, Université de Montréal, Montréal, Québec,Canada, H3C 3J7
Luc Vinet
E-mail: [email protected]
Centre de recherches mathématiques, Université de Montréal, Montréal, Québec,Canada, H3C 3J7
Abstract.
Karlin and McGregor’s d -variable Hahn polynomials are shown to arisein the ( d + d discrete variables depend on d + d =
1. Introduction
The objective of this article is to show that the multidimensional Hahn polynomialsarise in the quantum singular oscillator model as the overlap coefficients betweenenergy eigenstate bases associated to the separation of variables in Cartesian andhyperspherical coordinates and to obtain their properties from this framework. Thisoffers an algebraic analysis of the multivariate Hahn polynomials which is resting ontheir interpretation as overlap coefficients and on the special properties of the functionsarising in the basis wavefunctions. For definiteness and ease of notation, the emphasisshall be put on the case where the Hahn polynomials in two variables appear as theCartesian vs. spherical interbasis expansion coefficients for the three-dimensionalsingular oscillator. It shall be indicated towards the end of the paper how these resultscan be extended directly to an arbitrary number of variables. he multivariate Hahn polynomials and the singular oscillator h n ( x ; α , β ; N ), arethe polynomials of degree n in the variable x defined by [28, 30] h n ( x ; α , β ; N ) = ( α + n ( − N ) n F ³ − n , n + α + β + − x α + − N ¯¯¯ ´ ,where p F q is the generalized hypergeometric function [2] and where ( a ) n stands for thePochhammer symbol (or shifted factorial)( a ) n = ( a )( a + ··· ( a + n − a ) ≡ N X x = ρ ( x ; α , β ; N ) h n ( x ; α , β ; N ) h m ( x ; α , β ; N ) = λ n ( α , β ; N ) δ nm ,with respect to the hypergeometric distribution [21] ρ ( x ; α , β ; N ) = à Nx ! ( α + x ( β + N − x ( α + β + N , (1)where ¡ Nx ¢ are the binomial coefficients. The weight function (1) is positive providedthat α , β > − α , β < − N . The normalization factor λ n reads λ n ( α , β ; N ) = α + β + n + α + β + N ! n !( N − n )! ( α + n ( β + n ( N + α + β + n ( α + β + n . (2)The polynomials h n ( x ; α , β ; N ) can be obtained from the generating function [28] F ³ − x α + ¯¯¯ − t ´ F ³ x − N β + ¯¯¯ t ´ = N X n = h n ( x ; α , β ; N )( α + n ( β + n t n n ! , (3)or from the dual generating function [24]( − N ) n n ! (1 + t ) N P ( α , β ) n µ − t + t ¶ = N X x = à Nx ! h n ( x ; α , β ; N ) t x , (4)where P ( α , β ) n ( z ) stands for the classical Jacobi polynomials [28]. In mathematicalphysics, the Hahn polynomials are mostly known for their appearance in the Clebsch-Gordan coefficients of the su (2) or su (1,1) algebras (see for example [36]). However,these polynomials have also been used in the designing of spin chains allowing perfectquantum state transfer [1, 3, 37] and moreover, they occur as exact solutions of certaindiscrete Markov processes [21].The multivariable extension of the Hahn polynomials is due to Karlinand McGregor who obtained these polynomials in [25] as exact solutions of amultidimensional genetics model. This family of multidimensional polynomials is amember of the multivariate analogue of the discrete Askey scheme proposed by Tratnikin [35] and generalized to the basic ( q -deformed) case by Gasper and Rahman in[9]. One of the key features of the polynomials in this multivariate scheme is theirbispectrality (in the sense of Duistermaat and Grünbaum [6]), which was established he multivariate Hahn polynomials and the singular oscillator q -deformed case. Since theirintroduction, the multivariate Hahn polynomials have been studied from differentpoints of view by a number of authors [20, 31, 40, 38] and used in particular forapplications in probability [18, 26]. Of particular relevance to the present article arethe papers of Dunkl [7], Scarabotti [34] and Rosengren [33], where the multivariateHahn polynomials occur in an algebraic framework.Here we give a physical interpretation of the multivariate Hahn polynomials byestablishing that they occur in the overlap coefficients between wavefunctions of thesingular oscillator model separated in Cartesian and hyperspherical coordinates. Itwill be seen that this framework provides a cogent foundation for the characterizationof these polynomials: new derivations of known formulas will be given and newidentities will come to the fore. The results presented here are in line with the physico-algebraic models that were exhibited in [13, 14], [11] and [10] where the multivariateKrawtchouk, Meixner and Charlier polynomials were identified and characterized asmatrix elements of the representations of the rotation, Lorentz and Euclidean groupson oscillator states. However the approach and techniques used in the present paperdiffer from the ones used in [13], [11] and [10] as the multivariate Hahn polynomialsdo not arise as matrix elements of Lie group representations.The outline of the paper is the following. In section 2, the singular oscillatormodel in three-dimensions is reviewed. The wavefunctions separated in Cartesian andspherical coordinates are explicitly written and the corresponding constants of motionare given. In section 3, it is shown that the expansion coefficients are expressed in termsof orthogonal polynomials in two discrete variables that are orthogonal with respect toa two-variable generalization of the hypergeometric distribution. This is accomplishedby bringing intertwining operators that raise/lower the appropriate quantum numbers.In section 4, a generating function is derived by examining the asymptotic behavior ofthe wavefunctions and this generating function is identified with the one derived byKarlin and McGregor for the multivariate Hahn polynomials. Backward and forwardstructure relations are obtained in section 5 and are seen to provide a factorizationof the pair of recurrence relations satisfied by the bivariate Hahn polynomials. Insection 6, the two difference equations are derived: one by factorization and the otherby a direct computation involving one of the symmetry operators responsible for theseparation of variable in spherical coordinates. In section 7, the explicit expression ofthe bivariate Hahn polynomials in terms of univariate Hahn polynomials is obtainedby combining the Cartesian vs. cylindrical and cylindrical vs. spherical interbasisexpansion coefficients for the singular oscillator. The connection with the recouplingof su (1,1) representations is explained in section 8. In section 9, the multivariatecase is considered. A conclusion follows with perspectives on the multivariate Racahpolynomials. A compendium of formulas for the bivariate Hahn polynomials has beenincluded in the appendix.
2. The three-dimensional singular oscillator
In this section, the 3-dimensional singular oscillator model is reviewed. The two basesfor the energy eigenstates associated to the separation of variable in Cartesian andspherical coordinates are presented in terms of Laguerre and Jacobi polynomials. Foreach basis, the symmetry operators that are diagonalized and their eigenvalues aregiven. The main object of the paper, the interbasis expansion coefficients between thesetwo bases, is defined and shown to exhibit an exchange symmetry. he multivariate Hahn polynomials and the singular oscillator The singular oscillator model in three dimensions is governed by the Hamiltonian H = X i = Ã − ∂ x i + x i + α i − x i ! , (5)where α i > − H , labeled by the non-negative integer N , have the form E N = N + α /2 + α /2 + α /2 + ( N + N + -fold degeneracy. The Schrödinger equation associated tothe Hamiltonian (5) can be exactly solved by separation of variables in Cartesianand spherical coordinates (separation also occurs in other coordinate systems), thusproviding two distinct bases to describe the states of the system. Let i and k be non-negative integers such that i + k ≤ N . We shall denote by | α , α , α ; i , k ; N 〉 C the basis vectors for the E N -energy eigenspace associated to theseparation of variables in Cartesian coordinates. The corresponding wavefunctionsread 〈 x , x , x | α , α , α ; i , k ; N 〉 C = Ψ ( α , α , α ) i , k ; N ( x , x , x ) = ξ ( α ) i ξ ( α ) k ξ ( α ) N − i − k G ( α , α , α ) L ( α ) i ( x ) L ( α ) k ( x ) L ( α ) N − i − k ( x ), (6)where L ( α ) n ( x ) are the standard Laguerre polynomials [28] and where the gauge factor G ( α , α , α ) has the form G ( α , α , α ) = e − ( x + x + x )/2 3 Y j = x α j + j .The normalization factors ξ ( α ) n = s n ! Γ ( n + α +
1) , (7)where Γ ( x ) is the gamma function [2], ensure that the wavefunctions satisfy theorthogonality relation C 〈 α , α , α ; i , k ; N | α , α , α ; i ′ , k ′ ; N ′ 〉 C = Z R + d x d x d x h Ψ ( α , α , α ) i , k ; N i ∗ Ψ ( α , α , α ) i ′ , k ′ ; N ′ = δ ii ′ δ kk ′ δ NN ′ ,where R + stands for the non-negative real line and where z ∗ stands for complexconjugation. The Cartesian basis states are completely determined by the set ofeigenvalue equations K (1)0 | α , α , α ; i , k ; N 〉 C = ( i + α /2 + | α , α , α ; i , k ; N 〉 C , K (2)0 | α , α , α ; i , k ; N 〉 C = ( k + α /2 + | α , α , α ; i , k ; N 〉 C , H | α , α , α ; i , k ; N 〉 C = E N | α , α , α ; i , k ; N 〉 C , he multivariate Hahn polynomials and the singular oscillator K ( i )0 , i = H , K ( i )0 ] =
0) associated to theseparation of variables in Cartesian coordinates. These (Hermitian) operators havethe expression K ( i )0 = Ã − ∂ x i + x i + α i − x i ! , (8)and correspond to the one-dimensional singular oscillator Hamiltonian. For notationalconvenience, the Cartesian basis states | α , α , α ; i , k ; N 〉 C shall sometimes be writtensimply as | i , k ; N 〉 C when the explicit dependence on the parameters α i is not needed. Let m and n be non-negative integers such that m + n ≤ N . We shall denote by | α , α , α ; m , n ; N 〉 S the basis vectors for the E N -energy eigenspace associated to theseparation of variables in spherical coordinates x = r sin θ cos φ , x = r sin θ sin φ , x = r cos θ .In this case the corresponding wavefunctions are given by 〈 r , θ , φ | α , α , α ; m , n ; N 〉 S = Ξ ( α , α , α ) m , n ; N ( r , θ , φ ) = η ( α , α ) m η (2 m + α + α ) n ξ (2 m + n + α + N − m − n G ( α , α , α ) × P ( α , α ) m ( − cos2 φ )(sin θ ) m P (2 m + α + α ) n (cos2 θ )( r ) m + n L (2 m + n + α + N − m − n ( r ), (9)where P ( α , β ) n ( z ) are the Jacobi polynomials and where we have introduced the notation α i j = α i + α j , α i jk = α i + α j + α k .The normalization factors η ( α , β ) n = s n + α + β + n ! Γ ( n + α + β + Γ ( n + α + Γ ( n + β +
1) , (10)ensure that the wavefunctions Ξ ( α , α , α ) m , n ; N satisfy the orthogonality relation S 〈 α , α , α ; m , n ; N | α , α , α ; m ′ , n ′ , N ′ 〉 S = Z ∞ Z π Z π r sin θ d r d θ d φ h Ξ ( α , α , α ) m , n ; N i ∗ Ξ ( α , α , α ) m ′ , n ′ ; N ′ = δ mm ′ δ nn ′ δ NN ′ .The spherical basis states | α , α , α ; m , n ; N 〉 S are completely determined by the set ofeigenvalue equations Q (12) | α , α , α ; m , n ; N 〉 S = λ (12) m | α , α , α ; m , n ; N 〉 S , Q (123) | α , α , α ; m , n ; N 〉 S = λ (123) m , n | α , α , α ; m , n ; N 〉 S , H | α , α , α ; m , n ; N 〉 S = E N | α , α , α ; m , n ; N 〉 S , (11)where the eigenvalues λ (12) m and λ (123) m , n are given by λ (12) m = ( m + α /2 + m + α /2), λ (123) m , n = ( m + n + α /2 + m + n + α /2 + he multivariate Hahn polynomials and the singular oscillator Q (12) and Q (123) , which can be seen to commute with theHamiltonian, have the expressions Q (12) = n − ∂ φ + α − φ + α − φ − o , (12a) Q (123) = n − ∂ θ − ctg θ ∂ θ + α − θ + θ à − ∂ φ + α − φ + α − φ ! − o . (12b) In Cartesian coordinates, the spherical basis wavefunctions read 〈 x , x , x | α , α , α ; m , n ; N 〉 S = Ξ ( α , α , α ) m , n ; N ( x , x , x ) = η ( α , α ) m η (2 m + α + α ) n ξ (2 m + n + α + N − m − n G ( α , α , α ) ( x + x ) m P ( α , α ) m à x − x x + x ! × ( x + x + x ) n P (2 m + α + α ) n à x − x − x x + x + x ! L (2 m + n + α + N − m − n ( x + x + x ), (13)and the operators Q (12) , Q (123) have the form Q (12) = n J + ( x + x ) à α − x + α − x ! − o , Q (123) = n J + J + J + ( x + x + x ) à α − x + α − x + α − x ! − o , (14) where the J j are the familiar angular momentum operators J = i ( x ∂ x − x ∂ x ), J = i ( x ∂ x − x ∂ x ), J = i ( x ∂ x − x ∂ x ).For notational convenience, the spherical basis vectors | α , α , α ; m , n ; N 〉 S shallsometimes be written simply as | m , n ; N 〉 S when the explicit dependence on theparameters α i is not needed. In this paper, we shall be concerned with the overlap coefficients between the Cartesianand spherical bases. These coefficients are given by the integral C 〈 i , k ; N | m , n ; N 〉 S = Z R + d x d x d x [ Ψ ( α , α , α ) i , k ; N ( x , x , x )] ∗ Ξ ( α , α , α ) m , n ; N ( x , x , x ). (15)Since the wavefunctions are real one has C 〈 i , k ; N | m , n ; N 〉 S = S 〈 m , n ; N | i , k ; N 〉 C .One can write the expansion formulas | i , k ; N 〉 C = X m , nm + n ≤ N S 〈 m , n ; N | i , k ; N 〉 C | m , n ; N 〉 S , (16a) | m , n ; N 〉 S = X i , ki + k ≤ N C 〈 i , k ; N | m , n ; N 〉 S | i , k ; N 〉 C , (16b) he multivariate Hahn polynomials and the singular oscillator X i + k ≤ N S 〈 m , n ; N | i , k ; N 〉 C C 〈 i , k ; N | m ′ , n ′ ; N 〉 S = δ mm ′ δ nn ′ , (17a) X m + n ≤ N C 〈 i , k ; N | m , n ; N 〉 S S 〈 m , n ; N | i ′ , k ′ ; N 〉 C = δ ii ′ δ kk ′ . (17b)Upon using the explicit expressions (6) and (13) of the wavefunctions in Cartesiancoordinates and the property P ( α , β ) n ( − z ) = ( − n P ( β , α ) n ( z ) satisfied by the Jacobipolynomials, it is directly seen from (15) that the expansion coefficients obey thesymmetry relation C 〈 α , α , α ; i , k ; N | α , α , α ; m , n ; N 〉 S = ( − mC 〈 α , α , α ; k , i ; N | α , α , α ; m , n ; N 〉 S , (18)which allows one to interchange the pairs ( i , α ) and ( k , α ). This symmetry shall proveuseful in what follows.
3. The expansion coefficients as orthogonal polynomials in two variables
In this section, it is shown that the overlap coefficients between the Cartesian andspherical basis states defined in the previous section are expressed in terms oforthogonal polynomials in the two discrete variables i , k .The expansion coefficients (15) can be cast in the form C 〈 α , α , α ; i , k ; N | α , α , α ; m , n ; N 〉 S = W ( α , α , α ) i , k ; N Q ( α , α , α ) m , n ( i , k ; N ), (19)where Q ( α , α , α )0,0 ( i , k ; N ) ≡ W ( α , α , α ) i , k ; N = C 〈 α , α , α ; i , k ; N | α , α , α ;0,0; N 〉 S . ( α , α , α ) i , k ; N The coefficient W ( α , α , α ) i , k ; N in (19) can be evaluated explicitly using the definition (15) ofthe overlap coefficients. Indeed, upon taking m = n = W ( α , α , α ) i , k ; N = ξ ( α ) i ξ ( α ) k ξ ( α ) N − i − k η ( α , α )0 η ( α + α )0 ξ ( α + N Z ∞ Z ∞ Z ∞ d x d x d x e − ( x + x + x ) 3 Y j = ( x j ) α j + L ( α ) i ( x ) L ( α ) k ( x ) L ( α ) N − i − k ( x ) L ( α + N ( x + x + x ). (20)Upon using twice the addition formula for the Laguerre polynomials [2] L ( α + β + n ( x + y ) = X ℓ + k ≤ n L ( α ) ℓ ( x ) L ( β ) k ( y ),one obtains the relation L ( α + N ( x + x + x ) = X i ′ + k ′ ≤ N L ( α ) i ′ ( x ) L ( α ) k ′ ( x ) L ( α ) N − i ′ − k ′ ( x ). he multivariate Hahn polynomials and the singular oscillator W ( α , α , α ) i , k ; N = η ( α , α )0 η ( α + α )0 ξ ( α + N ξ ( α ) i ξ ( α ) k ξ ( α ) N − i − k .With the help of the identity ( a ) n = Γ ( a + n ) Γ ( a ) for the Pochhammer symbol, the aboveexpression is easily cast in the form W ( α , α , α ) i , k ; N = s N ! x ! y !( N − x − y )! ( α + i ( α + k ( α + N − i − k ( α + α + α + N . (21) We shall now show that the functions Q ( α , α , α ) m , n ( i , k ; N ) appearing in (19) arepolynomials of degree m + n in the variable i , k by obtaining their raising relations. Consider the operator C ( α , α ) + having the followingexpression in spherical coordinates C ( α , α ) + = · − ∂ φ + tg φ ( α + − ( α + φ ¸ . (22)Using the structure relation (B.4) for the Jacobi polynomials, it can be directly checkedthat one has C ( α , α ) + Ξ ( α + α + α ) m , n ; N = p ( m + m + α + Ξ ( α , α , α ) m + n ; N + . (23)Consider the matrix element C 〈 α , α , α ; i , k ; N | C ( α , α ) + | α + α + α ; m , n ; N − 〉 S .On the one hand, the action (23) and the definition (19) give C 〈 α , α , α ; i , k ; N | C ( α , α ) + | α + α + α ; m , n ; N − 〉 S = p ( m + m + α + W ( α , α , α ) i , k ; N Q ( α , α , α ) m + n ( i , k ; N ). (24)To obtain a raising relation, one needs to compute C 〈 α , α , α ; i , k ; N | C ( α , α ) + orequivalently (recalling that the wavefunctions are real) ( C ( α , α ) + ) † | α , α , α ; i , k ; N 〉 C .This computation can be performed in a straightforward fashion by writing (22) inCartesian coordinates, acting on the wavefunctions (6) and using identities of theLaguerre polynomials (see appendix of [12] for the details of a similar computation).One finds as a result( C ( α , α ) + ) † Ψ ( α , α , α ) i , k ; N = p i ( k + α + Ψ ( α + α + α ) i − k ; N − − p ( i + α + k Ψ ( α + α + α ) i , k − N − . (25)Upon combining (24) and (25) and using (21), one arrives at the following contiguityrelation for the functions Q ( α , α , α ) m , n ( i , k ; N ): c ( α , α , α ) m , n ; N Q ( α , α , α ) m + n ( i , k ; N ) = i ( k + α + Q ( α + α + α ) m , n ( i − k ; N − − k ( i + α + Q ( α + α + α ) m , n ( i , k − N − he multivariate Hahn polynomials and the singular oscillator c ( α , α , α ) m , n ; N are the coefficients given by the expression c ( α , α , α ) m , n ; N = q N ( α + α + N + α + m + m + α + α + α + . Consider the operator D ( α , α , α ) + defined as follows inspherical coordinates: D ( α , α , α ) + = ½ Q (123) − Q (12) + α + · tg θ ∂ θ − α − θ + α + ¸¾ , (27)where Q (12) and Q (123) are given by (12a) and (12b), respectively. Using the structurerelation (B.5) for the Jacobi polynomials as well as the eigenvalue equations (11), onefinds that the action of the operator D ( α , α , α ) + on the spherical basis states is D ( α , α , α ) + Ξ ( α , α , α + m , n ; N = p ( n + n + α + n + m + α + n + m + α + Ξ ( α , α , α ) m , n + N + (28)Consider the matrix element C 〈 α , α , α ; i , k ; N | D ( α , α , α ) + | α , α , α + m , n ; N − 〉 S .Upon using the action (28) and the definition (15) of the overlap coefficients, one findson the one hand C 〈 α , α , α ; i , k ; N | D ( α , α , α ) + | α , α , α + m , n ; N − 〉 S = p ( n + n + α + × p ( n + m + α + n + m + α + W ( α , α , α ) i , k ; N Q ( α , α , α ) m , n + ( i , k ; N ) (29)On the other hand, a direct computation shows that( D ( α , α , α ) + ) † Ψ ( α , α , α ) i , k ; N = p ( i + i + α + N − i − k )( N − i − k − Ψ ( α , α , α + i + k ; N − + p ( k + k + α + N − i − k )( N − i − k − Ψ ( α , α , α + i , k + N − + p i ( i + α )( N − i − k + α + N − i − k + α + Ψ ( α , α , α + i − k ; N − + p k ( k + α )( N − i − k + α + N − i − k + α + Ψ ( α , α , α + i , k − N − − (2 i + k + α + p ( N − i − k )( N − i − k + α + Ψ ( α , α , α + i , k ; N − . (30)Upon combining (29) and (30) and using (21), one obtains another contiguity relation ofthe form d ( α , α , α ) m , n ; N Q ( α , α , α ) m , n + ( i , k ; N ) = ( i + α + N − i − k )( N − i − k − Q ( α , α , α + m , n ( i + k ; N − + ( k + α + N − i − k )( N − i − k − Q ( α , α , α + m , n ( i , k + N − + i ( N − i − k + α + N − i − k + α + Q ( α , α , α + m , n ( i − k ; N − + k ( N − i − k + α + N − i − k + α + Q ( α , α , α + m , n ( i , k − N − − ( N − i − k )( N − i − k + α + i + k + α + Q ( α , α , α + m , n ( i , k ; N − he multivariate Hahn polynomials and the singular oscillator d ( α , α , α ) m , n ; N are the coefficients given by the expression d ( α , α , α ) m , n ; N = q N ( N + α + α + α + n + n + α + n + m + α + n + m + α + α + α + .Since by definition Q ( α , α , α )0,0 ( i , k ; N ) =
1, the relations (26) and (31) allow to constructany Q ( α , α , α ) m , n ( i , k ; N ) iteratively. Writing up the first few cases, one observes that the Q ( α , α , α ) m , n ( i , k ; N ) are polynomials of total degree m + n in the variables i , k . It is easy to see that the orthogonality relation (17a) satisfied by the transitioncoefficients C 〈 i , k ; N | m , n ; N 〉 S implies that the polynomials Q ( α , α , α ) m , n ( i , k ; N ) areorthogonal. Indeed, upon inserting (19) in the relation (17a), one finds that thepolynomials Q ( α , α , α ) m , n ( i , k ; N ) are orthonormal X i + k ≤ N w ( α , α , α ) i , k ; N Q ( α , α , α ) m , n ( i , k ; N ) Q ( α , α , α ) m ′ , n ′ ( i , k ; N ) = δ mm ′ δ nn ′ ,with respect to the discrete weight function w ( α , α , α ) i , k ; N = h W ( α , α , α ) i , k ; N i = à Ni , k ! ( α + i ( α + k ( α + N − i − k ( α + α + α + N , (32)where ¡ Nx , y ¢ are the trinomial coefficients. It is clear that the weight (32) is a bivariateextension of the Hahn weight function (1). It is also possible to obtain lowering relations for the polynomials Q ( α , α , α ) m , n ( i , k ; N )using the operators that are conjugate to C ( α , α ) + and D ( α , α , α ) + . Let us first examine the operator C ( α , α ) − = · ∂ φ + tg φ ( α + − ( α + φ ¸ . (33)It is obvious from the definitions (22) and (33) that ( C ( α , α ) ± ) † = C ( α , α ) ∓ . Furthermore,it is directly verified with the help of (B.2) that (33) has the action C ( α , α ) − Ξ ( α , α , α ) m , n ; N = p m ( m + α + Ξ ( α + α + α ) m − n ; N − . (34)Consider the matrix element C 〈 α + α + α ; i , k : N | C ( α , α ) − | α , α , α ; m , n ; N + 〉 S .Upon using (34) and the definition (19), one finds on the one hand C 〈 α + α + α ; i , k : N | C ( α , α ) − | α , α , α ; m , n ; N + 〉 S = p m ( m + α + W ( α + α + α ) i , k ; N Q ( α + α + α ) m − n ( i , k ; N ). (35) he multivariate Hahn polynomials and the singular oscillator C ( α , α ) − ) † in Cartesian coordinates and acting on the wavefunctions (6),one finds on the other hand( C ( α , α ) − ) † Ψ ( α + α + α ) i , k ; N = p ( i + k + α + Ψ ( α , α , α ) i + k ; N + − p ( i + α + k + Ψ ( α , α , α ) i , k + N + . (36)Combining (35) and (36) using (19) and (21), the following lowering relation for thepolynomials Q ( α , α , α ) m , n ( i , k ; N ) is obtained e ( α , α , α ) m , n ; N Q ( α + α + α ) m − n ( i , k ; N ) = Q ( α , α , α ) m , n ( i + k ; N + − Q ( α , α , α ) m , n ( i , k + N + e ( α , α , α ) m , n ; N = q m ( m + α + α + α + α + α + N + N + α + . Let us now consider the operator D ( α , α , α ) − = ½ Q (123) − Q (12) − α + · tg θ ∂ θ + α + θ − α ¸¾ . (37)Taking into account that ( Q (123) ) † = Q (123) and ( Q (12) ) † = Q (12) , it can be seen from thedefinitions (27) and (37) that ( D ( α , α , α ) ± ) † = D ( α , α , α ) ∓ . In view of the relation (B.3) andusing the eigenvalue equations (11), it follows that the action of the operator D ( α , α , α ) − is given by D ( α , α , α ) − Ξ ( α , α , α ) m , n ; N = p n ( n + α + n + m + α + n + m + α + Ξ ( α , α , α + m , n − N − . (38)Consider the matrix element C 〈 α , α , α + i , k ; N | D ( α , α , α ) − | α , α , α ; m , n ; N + 〉 S .Using (38) and (15), one can write C 〈 α , α , α + i , k ; N | D ( α , α , α ) − | α , α , α ; m , n ; N + 〉 S = p n ( n + α + × p ( n + m + α + n + m + α + W ( α , α , α + i , k ; N Q ( α , α , α + m , n − ( i , k ; N ). (39)The action of ( D ( α , α , α ) − ) † on the Cartesian basis wavefunctions (6) can be computedwith the result( D ( α , α , α ) − ) † Ψ ( α , α , α + i , k ; N = p ( i + i + α + N − i − k + α + N − i − k + α + Ψ ( α , α , α ) i + k ; N + + p ( k + k + α + N − i − k + α + N − i − k + α + Ψ ( α , α , α ) i , k + N + + p i ( i + α )( N − i − k + N − i − k + Ψ ( α , α , α ) i − k ; N + + p k ( k + α )( N − i − k + N − i − k + Ψ ( α , α , α ) i , k − N + − (2 i + k + α + p ( N − i − k + N − i − k + α + Ψ ( α , α , α ) i , k ; N + . (40) he multivariate Hahn polynomials and the singular oscillator f ( α , α , α ) m , n ; N Q ( α , α , α + m , n − ( i , k ; N ) = ( i + α + Q ( α , α , α ) m , n ( i + k ; N + + ( k + α + Q ( α , α , α ) m , n ( i , k + N + + i Q ( α , α , α ) m , n ( i − N + + k Q ( α , α , α ) m , n ( i , k − N + − (2 i + k + α + Q ( α , α , α ) m , n ( i , k ; N + f ( α , α , α ) m , n ; N = q n ( n + α + n + m + α + n + m + α + α + α + α + α + N + N + α + .
4. Generating function
In this section, a generating function for the bivariate polynomials Q ( α , α , α ) m , n ( i , k ; N )is derived by examining the asymptotic behavior of the wavefunctions. Thegenerating function is then seen to coincide with that of the Hahn polynomials,thus establishing that the polynomials Q ( α , α , α ) m , n are precisely the bivariate Hahnpolynomials introduced by Karlin and McGregor in [25].Consider the interbasis expansion formula (16b). Using spherical coordinates, it iseasily seen from (6), (9) and (19) that this formula can be cast in the form η ( α , α ) m η (2 m + α + α ) n ξ (2 m + n + α + N − m − n × P ( α , α ) m ( − cos2 φ )(sin θ ) m P (2 m + α + α ) n (cos2 θ )( r ) m + n L (2 m + n + α + N − m − n ( r ) = X i + k ≤ N W ( α , α , α ) i , k ; N Q ( α , α , α ) m , n ( i , k ; N ) × ξ ( α ) i ξ ( α ) k ξ ( α ) N − i − k L ( α ) i ( r sin θ cos φ ) L ( α ) k ( r sin θ sin φ ) L ( α ) N − i − k ( r cos θ ). (41)In (41), the expansion coefficients S 〈 m , n ; N | i , k ; N 〉 C = W ( α , α , α ) i , k ; N Q ( α , α , α ) m , n ( i , k ; N ),are independent of the expansion point specified by the values of the coordinates( r , θ , φ ). Let us consider the case where the value of the radial coordinate r is large.Since the asymptotic behavior of the Laguerre polynomials is of the form L ( α ) n ( x ) ∼ ( − n n ! x n + O ( x n − ).it follows that the asymptotic form of expansion formula (41) is η ( α , α ) m η (2 m + α + α ) n ξ (2 m + n + α + N − m − n P ( α , α ) m ( − cos2 φ )(sin θ ) m P (2 m + α + α ) n (cos2 θ ) = ( − n + m ( N − m − n )! X i + k ≤ N W ( α , α , α ) i , k ; N Q ( α , α , α ) m , n ( i , k ; N ) × ξ ( α ) i ξ ( α ) k ξ ( α ) N − i − k i ! k !( N − i − k )! (sin θ cos φ ) i (sin θ sin φ ) k (cos θ ) N − i − k . (42) he multivariate Hahn polynomials and the singular oscillator z = tg θ cos φ and z = tg θ sin φ , the formula (42) reads ½ ( − m + n ( N − m − n )! η ( α , α ) m η (2 m + α + α ) n ξ (2 m + n + α + N − m − n ¾ × (1 + z + z ) N − m ( z + z ) m P ( α , α ) m µ z − z z + z ¶ P (2 m + α + α ) n µ − z − z + z + z ¶ = X i + k ≤ N W ( α , α , α ) i , k ; N Q ( α , α , α ) m , n ( i , k ; N ) ( ξ ( α ) i ξ ( α ) k ξ ( α ) N − i − k i ! k !( N − i − k )! ) z i z k , (43)which has the form of a generating relation for the polynomials Q ( α , α , α ) m , n ( i , k ; N ). Let H ( α , α , α ) m , n ( i , k ; N ) denote the polynomials Q ( α , α , α ) m , n ; N ( i , k ; N ) = m ! n ! p Λ m , n ; N ( − N ) m + n H ( α , α , α ) m , n ( i , k ; N ), (44)where Λ ( α , α , α ) m , n ; N = n N ! m ! n !( N − m − n )! ( α + m ( α + m ( α + n ( α + m ( α + m ( α + N (2 m + α + n (2 m + α + n ( m + n + α + N (2 m + α + n ( m + n + α + m + n o . (45)that differ from Q ( α , α , α ) m , n ( i , k ; N ) only by a normalization factor. Performingelementary simplifications, it follows from (43) that the generating relation for thepolynomials H ( α , α , α ) m , n ( i , k ; N ) has the expression(1 + z + z ) N − m ( z + z ) m P ( α , α ) m µ z − z z + z ¶ P (2 m + α + α ) n µ − z − z + z + z ¶ = X i + k ≤ N Ã Ni , k ! H ( α , α , α ) m , n ( i , k ; N ) z i z k . (46)The generating function (46) is a bivariate generalization of the dual generatingfunction (4) for the Hahn polynomials of a single variable. Comparing the generatingfunction (46) with the one used in [25] to define the bivariate Hahn polynomials, it is nothard to see that the two generating functions coincide. Hence one may conclude thatthe polynomials H ( α , α , α ) m , n ( i , k ; N ) (and equivalently Q ( α , α , α ) m , n ( i , k ; N )) are precisely thebivariate Hahn polynomials of Karlin and McGregor. Note that on the L.H.S of (46) areessentially the Jacobi polynomials on the 2-simplex [8], as observed by Xu in [39].
5. Recurrence relations
In this section, backward and forward structure relations for the bivariate Hahnpolynomials are obtained using the raising/lowering relations of the Laguerrepolynomials. These structure relations are then used to derive by factorization therecurrence relations of the polynomials Q ( α , α , α ) m , n ; N ( i , k ; N ) and H ( α , α , α ) m , n ( i , k ; N ). To obtain a forward structure relation in the variable i , consider the first order operator A ( α ) + = · ∂ x + ( α + x − x ¸ . he multivariate Hahn polynomials and the singular oscillator A ( α ) + Ψ ( α + α , α ) i , k ; N = p i + Ψ ( α , α , α ) i + k ; N + . (47)Consider the matrix element S 〈 α , α , α ; m , n ; N | A ( α ) + | α + α , α ; i , k ; N − 〉 C . Theaction (47) gives on the one hand S 〈 α , α , α ; m , n ; N | A ( α ) + | α + α , α ; i , k ; N − 〉 C = p i + W ( α , α , α ) i + k ; N Q ( α , α , α ) m , n ( i , k ; N ). (48)Upon writing A ( α ) + in spherical coordinates and acting on (9), one finds( A ( α ) + ) † Ξ ( α , α , α ) m , n ; N = α ( α , α , α ) m , n ; N Ξ ( α + α , α ) m , n ; N − + β ( α , α , α ) m , n ; N Ξ ( α + α , α ) m − n ; N − + γ ( α , α , α ) m , n ; N Ξ ( α + α , α ) m , n − N − + δ ( α , α , α ) m , n + N Ξ ( α + α , α ) m − n + N − . (49)where the coefficients α , β , γ and δ are given by α ( α , α , α ) m , n ; N = q ( m + α + m + α + n + m + α + n + m + α + N − m − n )(2 m + α + m + α + n + m + α + n + m + α + , β ( α , α , α ) m , n ; N = q m ( m + α )( n + m + α + n + m + α + N + m + n + α + m + α )(2 m + α + n + m + α + n + m + α + , γ ( α , α , α ) m , n ; N = q n ( n + α )( m + α + m + α + N + m + n + α + m + α + m + α + n + m + α + n + m + α + , δ ( α , α , α ) m , n ; N = q mn ( m + α )( n + α )( N − m − n + m + α )(2 m + α + n + m + α )(2 n + m + α + . (50)Upon combining (48) with (49) and using (21), one obtains the forward structurerelation in the variable i for the polynomials Q ( α , α , α ) m , n ( i , k ; N ): q N ( α + α + Q ( α , α , α ) m , n ( i + k ; N ) = α ( α , α , α ) m , n ; N Q ( α + α , α ) m , n ( i , k ; N − + β ( α , α , α ) m , n ; N Q ( α + α , α ) m − n ( i , k ; N − + γ ( α , α , α ) m , n ; N Q ( α + α , α ) m , n − ( i , k ; N − + δ ( α , α , α ) m , n + N Q ( α + α , α ) m − n + ( i , k ; N − To obtain the backward structure relation in i , one considers the operator A ( α ) − = · − ∂ x + ( α + x − x ¸ . (52)It is clear that ( A ( α ) ± ) † = A ( α ) ∓ . In view of (C.1), it follows that the action of A ( α ) − on theCartesian basis wavefunctions is simply A ( α ) − Ψ ( α , α , α ) i , k ; N = p i Ψ ( α + α , α ) i − k ; N − . (53)Consider the matrix element S 〈 α + α , α ; m , n ; N − | A ( α ) − | α , α , α ; i , k ; N 〉 C . Theaction (53) implies that S 〈 α + α , α ; m , n ; N − | A ( α ) − | α , α , α ; i , k ; N 〉 C = p i W ( α + α , α ) i − k ; N − Q ( α + α , α ) m , n ( i − k ; N − he multivariate Hahn polynomials and the singular oscillator A ( α ) − ) † on the states of the spherical basis is given by( A ( α ) − ) † Ξ ( α + α , α ) m , n ; N − = α ( α , α , α ) m , n ; N Ξ ( α , α , α ) m , n ; N + β ( α , α , α ) m + n ; N Ξ ( α , α , α ) m + n ; N + γ ( α , α , α ) m , n + N Ξ ( α , α , α ) m , n + N + δ ( α , α , α ) m + n ; N Ξ ( α , α , α ) m + n − , (55)where the coefficients are given by (50). Combining (54) and (55), we obtainthe following backward structure relation in the variable i for the polynomials Q ( α , α , α ) m , n ( i , k ; N ): i q ( α + N ( α + Q ( α + α , α ) m , n ( i − k ; N − = α ( α , α , α ) m , n ; N Q ( α , α , α ) m , n ( i , k ; N ) + β ( α , α , α ) m + n ; N Q ( α , α , α ) m + n ( i , k ; N ) + γ ( α , α , α ) m , n + N Q ( α , α , α ) m , n + ( i , k ; N ) + δ ( α , α , α ) m + n ; N Q ( α , α , α ) m + n − ( i , k ; N ). (56) To obtain the forward and backward structure relations analogous to (51) and (56), onecould consider the operators B ( α ) ± = · ± ∂ x + ( α + x − x ¸ ,and follow the same steps as in subsections (5.1) and (5.2). Alternatively, one caneffectively use the symmetry relation (18) to derive these relations directly from (51)and (56) without additional computations. Upon using (18) on (51), one finds theforward structure relation q N ( α + α + Q ( α , α , α ) m , n ( i , k + N ) = α ( α , α , α ) m , n ; N Q ( α , α + α ) m , n ( i , k ; N − − β ( α , α , α ) m , n ; N Q ( α , α + α ) m − n ( i , k ; N − + γ ( α , α , α ) m , n ; N Q ( α , α + α ) m , n − ( i , k ; N − − δ ( α , α , α ) m , n + N Q ( α , α + α ) m − n + ( i , k ; N − α , α ) in the coefficients α , β , γ , δ and the signdifferences. With the help of (18), one obtains from (56) the second backward structurerelation k q ( α + N ( α + Q ( α , α + α ) m , n ( i , k − N − = α ( α , α , α ) m , n ; N Q ( α , α , α ) m , n ( i , k ; N ) − β ( α , α , α ) m + n ; N Q ( α , α , α ) m + n ( i , k ; N ) + γ ( α , α , α ) m , n + N Q ( α , α , α ) m , n + ( i , k ; N ) − δ ( α , α , α ) m + n ; N Q ( α , α , α ) m + n − ( i , k ; N ). (58)The backward and forward structure relations (51), (56), (57) and (58) are of a differentkind than those found in [32], which do not involve a change in the parameters. ( α , α , α ) m , n ( i , k ; N )The operators A ( α ) ± and the symmetry relation (18) can be used to construct the recur-rence relations satisfied by the bivariate Hahn polynomials Q ( α , α , α ) m , n ( i , k ; N ). To that he multivariate Hahn polynomials and the singular oscillator S 〈 α , α , α ; m , n ; N | A ( α ) + A ( α ) − | α , α , α ; i , k ; N 〉 C . Theactions (47) and (53) give S 〈 α , α , α ; m , n ; N | A ( α ) + A ( α ) − | α , α , α ; i , k ; N 〉 C = i W ( α , α , α ) i , k ; N Q ( α , α , α ) m , n ( i , k ; N ). (59)Note that A ( α ) + A ( α ) − is essentially the Hermitian symmetry operator K (1)0 defined in (8)since K (1)0 = A ( α ) + A ( α ) − + ( α + Q ( α , α , α ) m , n ( i , k ; N ) satisfy the 9-point recurrencerelation i Q m , n ( i , k ) = ( α m , n β m + n ) Q m + n ( i , k ) + ( α m , n γ m , n + + β m , n + δ m , n + ) Q m , n + ( i , k ) + ( γ m − n + δ m , n + ) Q m − n + ( i , k ) + ( α m , n δ m + n + β m + n − γ m , n ) Q m + n − ( i , k ) + ( α m , n + β m , n + γ m , n + δ m , n + ) Q m , n ( i , k ) + ( α m − n + δ m , n + + β m , n γ m − n + ) Q m − n + ( i , k ) + ( γ m , n δ m + n − ) Q m + n − ( i , k ) + ( α m , n − γ m , n + β m , n δ m , n ) Q m , n − ( i , k ) + ( α m − n β m , n ) Q m − n ( i , k ), (60)where the coefficients are given by (50); the explicit dependence on the parameters α i and N has been dropped to facilitate the reading. The second recurrence relation in k is directly obtained using the symmetry (18). One finds kQ m , n ( i , k ) = − ( ˜ α m , n ˜ β m + n ) Q m + n ( i , k ) + ( ˜ α m , n ˜ γ m , n + + ˜ β m , n + ˜ δ m , n + ) Q m , n + ( i , k ) − ˜( γ m − n + ˜ δ m , n + ) Q m − n + ( i , k ) − ( ˜ α m , n ˜ δ m + n + ˜ β m + n − ˜ γ m , n ) Q m + n − ( i , k ) + ( ˜ α m , n + ˜ β m , n + ˜ γ m , n + ˜ δ m , n + ) Q m , n ( i , k ) − ( ˜ α m − n + ˜ δ m , n + + ˜ β m , n ˜ γ m − n + ) Q m − n + ( i , k ) − ( ˜ γ m , n ˜ δ m + n − ) Q m + n − ( i , k ) + ( ˜ α m , n − ˜ γ m , n + ˜ β m , n ˜ δ m , n ) Q m , n − ( i , k ) − ( ˜ α m − n ˜ β m , n ) Q m − n ( i , k ), (61)where the ˜ x coefficients correspond to (50) with α ↔ α . For reference purposes, it isuseful to explicitly show the coefficients appearing in the recurrence relations (60) and(61). The recurrence relation (60) can be written as i Q m , n ( i , k ) = a m + n Q m + n ( i , k ) + a m , n Q m − n ( i , k ) + b m , n + Q m , n + ( i , k ) + b m , n Q m , n − ( i , k ) + c m , n + Q m − n + ( i , k ) + c m + n Q m + n − ( i , k ) + d m + n Q m + n − ( i , k ) + d m , n + Q m − n + ( i , k ) + e m , n Q m , n ( i , k ),with a m , n and c m , n given by a m , n = q m ( m + α )( m + α )( m + α )( n + m + α ) ( n + m + α ) ( N + m + n + α + N − m − n + m + α − (2 m + α ) (2 n + m + α ) (2 n + m + α + , c m , n = q m n ( n − m + α )( m + α )( m + α )( n + α − ( N + m + n + α + N − m − n + m + α − (2 m + α ) (2 n + m + α − (2 n + m + α − ,where b m , n and d m , n have the expression b m , n = q n ( n + α )( n + m + α + n + m + α + N + m + n + α + N − m − n + m + α + (2 m + n + α ) (2 n + m + α + × n m ( m + α )2 m + α + ( m + α + m + α + m + α + o , he multivariate Hahn polynomials and the singular oscillator d m , n = q m n ( m + α )( m + α )( m + α )( n + α )( n + m + α )( n + m + α )(2 m + α − (2 m + α ) × n (2 N + α + n + m + α − n + m + α + o ,and where e m , n reads e m , n = ( m + α + m + α + n ( n + α )( N + m + n + α + m + α + (2 n + m + α + + m ( m + α )( n + n + α + N − m − n )(2 m + α ) (2 n + m + α + + m ( m + α )( n + m + α + n + m + α + N + m + n + α + m + α ) (2 m + n + α + + ( m + α + m + α + n + m + α + n + m + α + N − m − n )(2 m + α + (2 n + m + α + .As for the relation (61), it can be written as kQ m , n ( i , k ) = − ˜ a m + n Q m + n ( i , k ) − ˜ a m , n Q m − n ( i , k ) + ˜ b m , n + Q m , n + ( i , k ) + ˜ b m , n Q m , n − ( i , k ) − ˜ c m , n + Q m − n + ( i , k ) − ˜ c m + n Q m + n − ( i , k ) − ˜ d m + n Q m + n − ( i , k ) − ˜ d m , n + Q m − n + ( i , k ) + ˜ e m , n Q m , n ( i , k ),where ˜ x m , n is obtained from x m , n by the permutation α ↔ α .
6. Difference equations
In this section, the difference equations satisfied by the Hahn polynomials are obtained.The first one is obtained by factorization using the intertwining operators thatraise/lower the first degree m . The second difference equation is found by a directcomputation of the matrix elements of one of the symmetry operators associated to thespherical basis. To obtain a first difference equation for the bivariate Hahn polynomials, we startfrom the matrix element C 〈 α , α , α ; i , k : N | C ( α , α ) + C ( α , α ) − | α , α , α ; m , n ; N 〉 S where C ( α , α ) ± are the operators defined by (22) and (33). In view of the actions (23) and (34),it follows that C 〈 α , α , α ; i , k : N | C ( α , α ) + C ( α , α ) − | α , α , α ; m , n ; N 〉 S = m ( m + α + W ( α , α , α ) i , k ; N Q ( α , α , α ) m , n ( i , k ; N ). (62)Note that C ( α , α ) + C ( α , α ) − is related to the operator Q (12) defined in (12a) since C ( α , α ) + C ( α , α ) − = Q (12) − α ( α + C ( α , α ) ± on the Cartesian basis wavefunctions, one finds( C ( α , α ) + C ( α , α ) − ) † Ψ ( α , α , α ) i , k ; N = [ i ( k + α + + k ( i + α + Ψ ( α , α , α ) i , k ; N − p i ( i + α )( k + k + α + Ψ ( α , α , α ) i − k + N − p k ( i + i + α + k + α ) Ψ ( α , α , α ) i + k − N . (63) he multivariate Hahn polynomials and the singular oscillator W ( α , α , α ) i , k ; N , one finds that the bivariate Hahn polynomials satisfy the differenceequation m ( m + α + Q m , n ( i , k ) = [ i ( k + α + + k ( i + α + Q m , n ( i , k ) − i ( k + α + Q m , n ( i − k + − k ( i + α + Q m , n ( i + k − α i and N was omitted to ease thenotation. Defining the operator L as L = Υ ( i , k ) T − i T + k + Υ ( i , k ) T + i T − k − [ Υ ( i , k ) + Υ ( i , k )] I , (65)with coefficients Υ ( i , k ) = i ( k + α + Υ ( i , k ) = k ( i + α + T ± i f ( i , k ) = f ( i ± k ) (and similarly for T ± k ) are the shift operators and I stands for the identity operator, the difference equation (64) can be written as theeigenvalue equation L Q ( α , α , α ) m , n ( i , k ; N ) = − m ( m + α + Q ( α , α , α ) m , n ( i , k ; N ). It is possible to derive a second difference equation for the bivariate Hahn polynomials.To that end, consider the matrix element C 〈 α , α , α ; i , k ; N | Q | α , α , α ; m , n ; N 〉 S ,where Q is defined by Q = Q (123) − ( α + α + Q (123) given by (12b). It follows from (11) that C 〈 α , α , α ; i , k ; N | Q | α , α , α ; m , n ; N 〉 S = ( n + m )( n + m + α + W ( α , α , α ) i , k ; N Q ( α , α , α ) m , n ( i , k ; N ). (66)Upon writing Q (123) in Cartesian coordinates (see (14)) and acting on the Cartesianbasis wavefunctions, a straightforward calculation yields Q Ψ ( α , α , α ) i , k ; N = ˜ κ i , k Ψ α , α , α i , k ; N − ˜ σ i , k Ψ ( α , α , α ) i − k + N − ˜ ρ i , k Ψ ( α , α , α ) i + k − N − ˜ µ i + k Ψ ( α , α , α ) i + k ; N − ˜ µ i , k Ψ ( α , α , α ) i − k ; N − ˜ ν i , k + Ψ ( α , α , α ) i , k + N − ˜ ν i , k Ψ ( α , α , α ) i , k − N , (67)where the coefficients are of the form˜ κ i , k = i α + k α + ( N − i − k ) α − i + k + i k − i N − k N − N ),˜ σ i , k = p i ( i + α )( k + k + α + ρ i , k = p ( i + i + α + k ( k + α ),˜ µ i , k = p i ( i + α )( N − i − k + N − i − k + α + ν i , k = p k ( k + α )( N − i − k + N − i − k + α + he multivariate Hahn polynomials and the singular oscillator Q ( α , α , α ) m , n ( i , k ; N ) satisfy the following difference equation: − ( n + m )( n + m + α + Q m , n ( i , k ) = − ˜ κ i , k Q m , n ( i , k ) + i ( k + α + Q m , n ( i − k + + k ( i + α + Q m , n ( i + k − + ( k + α + N − i − k ) Q m , n ( i , k + + k ( N − i − k + α + Q m , n ( i , k − + ( i + α + N − i − k ) Q m , n ( i + k ) + i ( N − i − k + α + Q m , n ( i − k ), (69)where the explicit dependence on N and α i was again dropped for convenience. Onecan present the difference equation (69) as an eigenvalue equation in the following way.We define the operator L = Ω ( i , k ) T + i + Ω ( i , k ) T + k + Ω ( i , k ) T − i + Ω ( i , k ) T − k + Ω ( i , k ) T + i T − k + Ω ( i , k ) T − i T + k − ³ X j = Ω j ( i , k ) ´ I , (70)with coefficients Ω ( i , k ) = ( i + α + N − i − k ), Ω ( i , k ) = ( k + α + N − i − k ), Ω ( i , k ) = i ( N − i − k + α + Ω ( i , k ) = k ( N − i − k + α + Ω ( i , k ) = k ( i + α + Ω ( i , k ) = i ( k + α + L Q ( α , α , α ) m , n ( i , k ; N ) = − ( n + m )( n + m + α + Q ( α , α , α ) m , n ( i , k ; N ).
7. Expression in hypergeometric series
In this section, the explicit expression for the bivariate Hahn polynomials Q ( α , α , α ) m , n interms of the Hahn polynomials in one variable is derived. This is done by introducingan ancillary basis of states corresponding to the separation of variables in cylindricalcoordinates and by evaluating explicitly the Cartesian vs. cylindrical and cylindrical vs.spherical interbasis expansion coefficients in terms of the univariate Hahn polynomials. Let p and q be non-negative integers such that p ≤ q ≤ N . We shall denote by | α , α , α ; p , q ; N 〉 P the basis vectors for the E N -energy eigenspace associated to theseparation of variables in cylindrical-polar coordinates x = ρ cos ϕ , x = ρ sin ϕ , x = x .In these coordinates, the wavefunctions have the expression 〈 ρ , ϕ , x | α , α , α ; p , q ; N 〉 P = A ( α , α , α ) p , q ; N ( ρ , ϕ , x ) = η ( α , α ) p ξ (2 p + α + q − p ξ ( α ) N − q G ( α , α , α ) P ( α , α ) p ( − cos2 ϕ )( ρ ) p L (2 p + α + q − p ( ρ ) L ( α ) N − q ( x ), (72) he multivariate Hahn polynomials and the singular oscillator Z ∞ Z π /20 Z ∞ h A ( α , α , α ) p , q ; N ( ρ , ϕ , x ) i ∗ A ( α , α , α ) p ′ , q ′ ; N ′ ( ρ , ϕ , x ) ρ d ρ d ϕ d x = δ pp ′ δ qq ′ δ NN ′ .In Cartesian coordinates, the wavefunctions of the cylindrical basis take the form 〈 x , x , x | α , α , α ; p , q ; N 〉 P = η ( α , α ) p ξ (2 p + α + q − p ξ ( α ) N − q G ( α , α , α ) ( x + x ) p P ( α , α ) p à x − x x + x ! L (2 m + α + q − p ( x + x ) L ( α ) N − q ( x ). (73) Let us obtain the explicit expression for the expansion coefficients P 〈 p , q ; N | i , k ; N 〉 C between the states of the cylindrical-polar and Cartesian bases. These expressionsare already known (see for example [27]) but we give here a new derivation of thesecoefficients using a generating function technique [4, 12, 17].Upon comparing the formulas (6) and (73) for the Cartesian and cylindrical-polarwavefunctions, it is clear that one can write P 〈 p , q ; N | i , k ; N 〉 C = δ q , i + k P 〈 p ; q | i ; q 〉 C ,where P 〈 p ; q | i ; q 〉 C are the coefficients appearing in the expansion formula ξ ( α ) i ξ ( α ) q − i L ( α ) i ( x ) L ( α ) q − i ( x ) = q X p = P 〈 p ; q | i ; q 〉 C × η ( α , α ) p ξ (2 p + α + q − p ( x + x ) p P ( α , α ) p à x − x x + x ! L (2 p + α + q − p ( x + x ). (74)Since the coefficients P 〈 p ; q | i ; q 〉 C are independent of x , x , the expansion formula (74)holds regardless of the value taken by these coordinates, i.e. (74) is a formal expansion.Let us set x + x =
0. Upon using the formula [2]( x + y ) m P ( α , β ) m µ x − yx + y ¶ = ( α + m m ! x m F ³ − m , − m − βα + ¯¯¯ − yx ´ ,and Gauss’s summation formula [2] as well as taking x = u , one finds that theexpansion formula (74) reduces to the generating relation ξ ( α ) i ξ ( α ) q − i L ( α ) i ( u ) L ( α ) q − i ( − u ) = q X p = P 〈 p ; q | i ; q 〉 C η ( α , α ) p ξ (2 p + α + q − p n ( p + α + p (2 p + α + q − p p !( q − p )! o u p .The above relation can be written as F ³ − i α + ¯¯¯ − u ´ F ³ i − q α + ¯¯¯ u ´ = ( i !( q − i )!( α + i ( α + q − i ξ ( α i ξ ( α q − i ) × q X p = P 〈 p ; q | i ; q 〉 C η ( α , α ) p ξ (2 p + α + q − p n ( p + α + p (2 p + α + q − p p !( q − p )! o u p . (75) he multivariate Hahn polynomials and the singular oscillator P 〈 p ; q | i ; q 〉 C = s ρ ( i ; α , α ; q ) λ p ( α , α ; q ) h p ( i ; α , α ; q ),where ρ ( x ; α , β ; N ) and λ n ( α , β ; N ) are respectively given by (1) and (2). The completeexpression for the overlap coefficients P 〈 α , α , α ; p , q ; N | α , α , α ; i , k ; N 〉 C betweenthe states of the cylindrical and Cartesian bases is thus expressed in terms of the Hahnpolynomials h n ( x ; α , β ; N ) in the following way: P 〈 α , α , α ; p , q ; N | α , α , α ; i , k ; N 〉 C = δ q , i + k s ρ ( i ; α , α ; q ) λ p ( α , α ; q ) h p ( i ; α , α ; q ). (76) Upon comparing the expressions (13) and (73) giving the wavefunctions of the sphericaland cylindrical-polar bases in Cartesian coordinates, it is easy to see that the overlapcoefficients S 〈 m , n ; N | p , q ; N 〉 P between these two bases is of the form S 〈 m , n ; N | p , q ; N 〉 P = δ mp S 〈 n ; N | q ; N 〉 P ,where S 〈 n ; N | q ; N 〉 P are the coefficients arising in the expansion ξ (2 m + α + q − m ξ ( α ) N − q L (2 m + α + q − m ( x + x ) L ( α ) N − q ( x ) = N − m X n = S 〈 n ; N | q ; N 〉 P η (2 m + α + α ) n ξ (2 m + n + α + N − m − n × ( x + x + x ) n P (2 m + α + α ) n à x − x − x x + x + x ! L (2 m + n + α + N − m − n ( x + x + x ). (77)Taking x = S 〈 α , α , α ; m , n ; N | α , α , α ; p , q ; N 〉 P = δ mp s ρ ( q − m ;2 m + α + α ; N − m ) λ n (2 m + α + α ; N − m ) h n ( q − m ;2 m + α + α ; N − m ). (78) ( α , α , α ) m , n ( i , k ; N )The expansion formulas (76) and (78) can be combined to obtain the explicit expressionfor the bivariate Hahn polynomials in terms of the univariate Hahn polynomials. he multivariate Hahn polynomials and the singular oscillator W ( α , α , α ) i , k ; N Q ( α , α , α ) m , n ( i , k ; N ) = S 〈 α , α , α ; m , n ; N | α , α , α ; i , k ; N 〉 C = N X p = N X q = p S 〈 m , n ; N | p , q ; N 〉 P P 〈 p , q ; N | i , k ; N 〉 C = s ρ ( i ; α , α ; i + k ) λ m ( α , α ; i + k ) ρ ( i + k − m ;2 m + α + α ; N − m ) λ n ( i + k − m ;2 m + α + α ; N − m ) × h m ( i ; α , α ; i + k ) h n ( i + k − m ;2 m + α + α ; N − m ). (79)With the expression (21), one finds the following expression for the polynomials Q ( α , α , α ) m , n ( i , k ; N ): Q ( α , α , α ) m , n ( i , k ; N ) = ³ Λ ( α , α , α ) m , n ; N ´ − h m ( i ; α , α ; i + k ) h n ( i + k − m ;2 m + α + α ; N − m ), (80)where Λ ( α , α , α ) m , n ; N are the normalization coefficients defined in (45). The explicitexpression (80) for the bivariate Hahn polynomials Q ( α , α , α ) m , n ( i , k ; N ) corresponds toKarlin and McGregor’s [25]. From the results of this section, it is clear that thecomplete theory of the univariate Hahn polynomials could also be worked out from theirinterpretation as interbasis expansion coefficients for the two-dimensional singularoscillator.
8. Algebraic interpretation
In this section, an algebraic interpretation of the overlap coefficients between theCartesian and spherical bases is presented in terms of su (1,1) representations. It isseen that these overlap coefficients can be assimilated to generalized Clebsch-Gordancoefficients, a result that entails a connection with the work of Rosengren [33]. su (1,1)The su (1,1) algebra has for generators the elements K and K ± that satisfy thecommutation relations [16, 36][ K , K ± ] = ± K ± , [ K − , K + ] = K . (81)The Casimir operator C , which commutes with every generator, is of the form C = K − K + K − − K . (82)Let ν > V ( ν ) denote the infinite-dimensional vector spacespanned by the basis vectors e ( ν ) n , n ∈ { } . If V ( ν ) is endowed with the actions K e ( ν ) n = ( n + ν ) e ( ν ) n , K + e ( ν ) n = p ( n + n + ν ) e ( ν ) n + , K − e ( ν ) n = p n ( n + ν − e ( ν ) n − , (83) he multivariate Hahn polynomials and the singular oscillator V ( ν ) becomes an irreducible su (1,1)-module; the representation (83) belongs to thepositive discrete series [36]. On this module the Casimir operator acts as a multiple ofthe identity C e ( ν ) n = ν ( ν − e ( ν ) n ,as expected from Schur’s lemma. Consider three mutually commuting sets { K ( i )0 , K ( i ) ± } , i = su (1,1) generators. These generators can be combined as follows to producea fourth set of generators: K (123)0 = K (1)0 + K (2)0 + K (3)0 , K (123) ± = K (1) ± + K (2) ± + K (3) ± .There is a natural representation for this realization of su (1,1) on the tensor productspace V ( ν ) ⊗ V ( ν ) ⊗ V ( ν ) ; in this representation each set of generators { K ( i )0 , K ( i ) ± } actson V ( ν i ) only. A convenient basis for this module is the direct product basis spannedby the vectors e ( ν ) n ⊗ e ( ν ) n ⊗ e ( ν ) n with the actions of the generators { K ( i )0 , K ( i ) ± } on thevectors e ( ν i ) n i as prescribed by (83). In general, this representation is not irreducibleand it can be completely decomposed in a direct sum of irreducible representations V ( ν ) also belonging to the positive-discrete series. To perform this decomposition, one canproceed in two steps by first decomposing V ( ν ) ⊗ V ( ν ) in irreducible modules V ( ν ) andthen decomposing V ( ν ) ⊗ V ( ν ) in irreducible modules V ( ν ) for each occurring values of ν . A natural basis associated to this decomposition scheme, which we shall call the“coupled” basis, is provided by the vectors e ( ν , ν ) n , n ∈ { } , satisfying C (12) e ( ν , ν ) n = ν ( ν − e ( ν , ν ) n , C (123) e ( ν , ν ) n = ν ( ν − e ( ν , ν ) n , K (123)0 e ( ν , ν ) n = ( n + ν ) e ( ν , ν ) n , (84)where C (12) is the Casimir operator associated to the decomposition of V ( ν ) ⊗ V ( ν ) : C (12) = [ K (12)0 ] − K (12) + K (12) − − K (12)0 , (85)with K ( i j )0 = K ( i )0 + K ( j )0 , K ( i j ) ± = K ( i ) ± + K ( j ) ± and where C (123) is the Casimir operatorassociated to the decomposition of V ( ν ) ⊗ V ( ν ) : C (123) = [ K (123)0 ] − K (123) + K (123) − − K (123)0 . (86)It is well known (see for example [5]) that the occurring values of ν and ν are givenby ν ( m ) = m + ν + ν , ν ( m , n ) = n + m + ν + ν + ν , (87)where m , n are non-negative integers. The direct product and coupled bases span thesame representation space and the corresponding basis vectors are thus related by alinear transformation. Furthermore, since these vectors are both eigenvectors of K (123)0 the transformation is non-trivial if and only if the involved vectors correspond to thesame eigenvalue of K (123)0 . Let λ K = N + ν + ν + ν , N ∈ { N } , be the eigenvaluesof K (123)0 , then for each N one has e ( ν ) i ⊗ e ( ν ) k ⊗ e ( ν ) N − i − k = X m , nm + n ≤ N C ( ν , ν , ν ) m , n ( i , k ; N ) e ( ν ( m ), ν ( m , n )) N − m − n , (88) he multivariate Hahn polynomials and the singular oscillator i , k are positive integers such that i + k ≤ N . The coefficients C ( ν , ν , ν ) m , n ( i , k ; N ) aregeneralized Clebsch-Gordan coefficients for the positive-discrete series of irreduciblerepresentations of su (1,1); the reader is referred to [5, 36] for the standard Clebsch-Gordan problem, which involves only two representations of su (1,1). The connection between the singular oscillator model and the combination of three su (1,1) representations can be established as follows. Consider the following coordinaterealizations of the su (1,1) algebra K ( i )0 = Ã − ∂ x i + x i + α i − x i ! , (89) K ( i ) ± = Ã ( x i ∓ ∂ x i ) − α i − x i ! , (90)where i = C ( i ) takes the value ν i ( ν i −
1) with ν i = α i +
12 , i = H = K (123)0 . One can check using (6) and (89) that thestates | i , k ; N 〉 C of the Cartesian basis provide, up to an inessential phase factor, arealization of the tensor product basis in the addition of three irreducible modules V ( ν i ) of the positive-discrete series. Hence we have the identification | i , k ; N 〉 C ∼ e ( ν ) i ⊗ e ( ν ) k ⊗ e ( ν ) N − i − k , (92)with ν i given by (91). Upon computing the Casimir operators C (12) and C (123) in therealization (89) from their definitions (85) and (86) and comparing with the operators Q (12) and Q (123) given in Cartesian coordinates by (14), it is directly checked that C (12) ∼ Q (12) , C (123) ∼ Q (123) . (93)It is also checked that the eigenvalues (11) correspond to (87) and thus we havethe following identification between the spherical basis states and the coupled basisvectors: | m , n ; N 〉 S ∼ e ( ν ( m ), ν ( m , n )) N − m − n . (94)In view of (88), (92) and (94), the interbasis expansion coefficients between the sphericaland Cartesian bases S 〈 m , n ; N | i , k ; N 〉 C = W ( α , α , α ) i , k ; N Q ( α , α , α ) m , n ( i , k ; N ),given in terms of the bivariate Hahn polynomials Q ( α , α , α ) m , n ( i , k ; N ) correspond to thegeneralized Clebsch-Gordan coefficients C ( ν , ν , ν ) m , n ; N ( i , k ; N ) ≃ W (2 ν − ν − ν − i , k ; N Q (2 ν − ν − ν − m , n ( i , k ; N ),where the ≃ symbol is used to account for the possible phase factors coming from thechoices of phase factors in the basis states. he multivariate Hahn polynomials and the singular oscillator
9. Multivariate case
In this section, it shown how the results of the previous sections can be directlygeneralized so as to find the Hahn polynomials in d -variables as the interbasisexpansion coefficients between the Cartesian and hyperspherical eigenbases for thesingular oscillator model in ( d +
1) dimensions.
Let α = ( α ,... , α d + ) with α i > − d + H = d + X i = à − ∂ x i + x i + α i − x i ! .The energy spectrum E N of this Hamiltonian is of the form E N = N +| α | /2 + ( d + | α | = α + ··· + α d + ,and exhibits a ¡ N + dd ¢ -fold degeneracy. Let i = ( i ,... , i d + ) with i d + = N − P dj = i j andlet | α ; i 〉 C denote the states spanning the Cartesian basis. In Cartesian coordinates, thecorresponding wavefunctions have the expression 〈 x | α ; i 〉 C = Ψ ( α ) i ( x ) = G ( α ) ( x ) d + Y k = ξ ( α k ) i k L ( α k ) i k ( x k ), (95)where x = ( x ,... , x d + ) is the coordinate vector and where the gauge factor G ( α ) ( x ) is G ( α ) ( x ) = e −| x | /2 d + Y k = x α k + k ,with | x | = x + ··· x d + . With the normalization coefficients ξ ( α ) n as in (7) one has Z R d + + C 〈 α ; i ′ | x 〉〈 x | α ; i 〉 C d x = δ ii ′ .Let n = ( n ,... , n d + ) with n d + = N − P dk = n d and let | α ; n 〉 S denote the states spanningthe hyperspherical basis. In Cartesian coordinates, the corresponding wavefunctionsare given by 〈 x | α ; n 〉 S = Ξ ( α ) n ( x ) = G ( α ) ( x ) × ( d Y k = η ( a k , α k + ) n k ¡ | x k + | ¢ n k P ( a k , α k + ) n k à x k + −| x k | | x k + | !) ξ ( a d + ) n d + L ( a d + ) n d + ¡ | x | ¢ , (96)where the following notations were used: | y k | = y + ··· + y k , a k = a k ( α , n ) = | n k − |+| α k | + k −
1, (97a) | y k | = y + ··· y k , | y | =
0. (97b) he multivariate Hahn polynomials and the singular oscillator ξ ( α ) n given by (7) and η ( α , β ) m given by (10) ensure that one has Z R d + + S 〈 α ; n ′ | x 〉〈 x | α ; n 〉 S d x = δ n , n ′ .It is directly seen that the wavefunctions of the hyperspherical basis are separated inthe hyperspherical coordinates x = r cos θ sin θ ··· sin θ d , x = r sin θ sin θ ··· sin θ d ,... x k = r cos θ k − sin θ k ··· sin θ d ,... x d + = r cos θ d ,The operators that are diagonal on (96) and their eigenvalues are easily obtainedthrough the correspondence (93) with the combining of d + su (1,1); theycorrespond to the Casimir operators C (12) , C (123) , C (1234) , etc.The overlap coefficients between the Cartesian and hyperspherical bases aredenoted C 〈 α ; i | α ; n 〉 S and are defined by the integral C 〈 α ; i | α ; n 〉 S = Z R d + + h Ξ ( α ) n ( x ) i ∗ Ψ ( i ) i ( x ) d x , (98)from which one easily sees that C 〈 α ; i | α ; n 〉 S = S 〈 α ; n | α ; i 〉 C .The overlap coefficients provide the expansion formulas | α ; n 〉 S = X | i |= N C 〈 α ; i | α ; n 〉 S | α ; i 〉 C , | α ; i 〉 C = X | n |= N S 〈 α ; n | α ; i 〉 C | α ; n 〉 S ,between the hyperspherical and Cartesian bases. Since the Cartesian andhyperspherical basis vectors are orthonormal, the interbasis expansions coefficientssatisfy the discrete orthogonality relations X | i |= N S 〈 α ; n ′ | α ; i 〉 C C 〈 α ; i | α ; n 〉 S = δ nn ′ , X | n |= N C 〈 α ; i ′ | α ; n 〉 S S 〈 α ; n | α ; i 〉 C = δ ii ′ . The interbasis expansion coefficients can be cast in the form C 〈 α ; i | α ; n 〉 S = W ( α ) i Q ( α ) n ( i ), (99) he multivariate Hahn polynomials and the singular oscillator W ( α ) i is defined by W ( α ) i = C 〈 α ; i | α ; 〉 S , (100)with = (0, ··· ,0, N ). The explicit expression for (100) is easily found by repeatedlyusing the addition formula for the Laguerre polynomials on the hypersphericalwavefunctions (96) in the integral expression (98). One then finds W ( α ) i = vuutà Ni ,... , i d ! ( α + i ··· ( α d + + i d + ( | α | + d + N , (101)where ¡ Nx ,..., x d ¢ are the multinomial coefficients. The explicit formula for the completeinterbasis expansion coefficients (99) in terms of the univariate Hahn polynomials canbe obtained by introducing a sequence of “cylindrical” coordinate systems correspondingto the coordinate couplings ( x , x ), ( x , x , x ), etc.. Upon using (100), one finds in thisway that the Q ( α ) n ( i ) appearing in (99) are of the form Q ( α ) n ( i ) = h Λ ( α ) m i − d Y k = h n k ( | i k |−| n k − | ; a k ( α , n ); α k + ; | i k + |−| n k − | ), (102)where Λ ( α ) m is an easily obtained normalization factor and where the notations (97) havebeen used. It is directly seen from (102) that the functions Q ( α ) m ( i ) are polynomials oftotal degree | n | in the variables i that satisfy the orthogonality relation X | i |= N w ( α ) i Q ( α ) n ′ ( i ) Q ( α ) n ( i ) = δ nn ′ ,with respect to the multivariate hypergeometric distribution w ( α ) i = h W ( α ) i i = Q d + k = ¡ i k + α k i k ¢¡ N +| α |+ dN ¢ .The properties of the multivariate Hahn polynomials Q ( α ) n ( i ) can be derived using thesame methods as in the previous sections.
10. Conclusion
In this paper, we have shown that Karlin and McGregor’s d -variable Hahn polynomialsarise as interbasis expansion coefficients in the ( d + su (1,1) representations was also established.A natural question that arises from our considerations is whether a similarinterpretation can be given for the multivariate Racah polynomials, which have oneparameter more than the multivariate Hahn polynomials. The answer to that questionis in the positive. Indeed, the multivariate Racah polynomials can be seen to occur asinterbasis expansion coefficients in the so-called generic ( d + he multivariate Hahn polynomials and the singular oscillator d -sphere. With the usual embedding x + ··· + x d + = d -sphere in the ( d + H = X ≤ i < j ≤ d + · i ³ x i ∂ x j − x j ∂ x i ´¸ + d + X k = α k − x k ,and the ( d − d = d = d + d -sphere. Acknowledgments
The authors wish to thank Willard Miller Jr. and Sarah Post for stimulatingdiscussions. VXG benefits from an Alexander-Graham-Bell fellowship from the NaturalSciences and Engineering Research Council of Canada (NSERC). The research of LV issupported in part by NSERC.
Appendix A. A compendium of formulas for bivariate Hahn polynomials
In this appendix we give for reference a compendium of formulas for the bivariate Hahnpolynomials; some of them can be found in the literature, others not as far as we know.Recall that the univariate Hahn polynomials h n ( x ; α , β ; N ) are defined by h n ( x ; α , β ; N ) = ( α + n ( − N ) n F ³ − n , n + α + β + − x α + − N ¯¯¯ ´ ,where p F q is the generalized hypergeometric function [2]. Appendix A.1. Definition
The bivariate Hahn polynomials P ( α , α , α ) n , n ( x , x ; N ) are defined by P ( α , α , α ) n , n ( x , x ; N ) = − N ) n + n h n ( x ; α , α ; x + x ) h n ( x + x − n ;2 n + α + α + α ; N − n ).It is checked that P ( α , α , α ) n , n ( x , x ; N ) are polynomials of total degree n + n in thevariables x and x . Appendix A.2. Orthogonality
The polynomials P ( α , α , α ) n , n ( x , x ; N ) satisfy the orthogonality relation X x , x x + x ≤ N ω ( α , α , α ) x , x ; N P ( α , α , α ) n , n ( x , x ; N ) P ( α , α , α ) m , m ( x , x ; N ) = λ ( α , α , α ) n , n ; N δ m , n δ m , n . he multivariate Hahn polynomials and the singular oscillator ω ( α , α , α ) x , x ; N is given by ω ( α , α , α ) x , x ; N = ¡ x + α x ¢¡ x + α x ¢¡ N − x − x + α N − x − x ¢¡ N + α + α + α + N ¢ ,and the normalization factor λ ( α , α , α ) n , n ; N reads λ ( α , α , α ) n , n ; N = n ! n !( N − n − n )! N ! ( α + n ( α + n ( α + n ( α + α + n ( α + α + n ( α + α + α + N × (2 n + α + α + n (2 n + α + α + α + n ( n + n + α + α + α + N (2 n + α + α + α + n ( n + n + α + α + α + n + n . Appendix A.3. Recurrence relations
The bivariate Hahn polynomials P n , n ( x , x ) satisfy the recurrence relation x P n , n ( x , x ) = a n , n P n + n ( x , x ) + b n , n P n , n + ( x , x ) + c n , n P n − n + ( x , x ) + d n , n P n − n + ( x , x ) + e n , n P n , n ( x , x ) + f n , n P n + n − ( x , x ) − g n , n P n + n − ( x , x ) − h n , n P n , n − ( x , x ) − i n , n P n − n ( x , x ),where the coefficients are given by a n , n = ( n + α + α + n + n + α + α + α + n + n + α + α + α + n + n − N )(2 n + α + α + n + α + α + n + n + α + α + α + n + n + α + α + α + , b n , n = (2 n + n + α + α + α + n + n ( α + α + + ( α + α + α )]( n + n − N )(2 n + α + α )(2 n + α + α + n + n + α + α + α + n + n + α + α + α + , c n , n = n ( n + α )( n + α )( n + n − N )(2 n + α + α )(2 n + α + α + n + n + α + α + α + n + n + α + α + α + , d n , n = n ( n + α )( n + α )(2 n + n + α + α + N + α + α + α + n + α + α )(2 n + α + α + n + n + α + α + α + n + n + α + α + α + , f n , n = n ( n + α )( n + α + α + n + n + α + α + α + N + α + α + α + n + α + α + n + α + α + n + n + α + α + α + n + n + α + α + α + , g n , n = n ( n − n + α )( n + α − n + α + α + N + n + n + α + α + α + n + α + α + n + α + α + n + n + α + α + α + n + n + α + α + α + , h n , n = n ( n + α )(2 n + n ( α + α + + ( α + α + α ))(2 n + n + α + α + N + n + n + α + α + α + n + α + α )(2 n + α + α + n + n + α + α + α + n + n + α + α + α + , i n , n = n ( n + α )( n + α )(2 n + n + α + α )(2 n + n + α + α + N + n + n + α + α + α + n + α + α )(2 n + α + α + n + n + α + α + α + n + n + α + α + α + ,and by e n , n = n ( n + α + n + α + α + n + α )( N + n + n + α + α + α + n + α + α + n + α + α + n + n + α + α + α + n + n + α + α + α + + n ( n + α )( n + n + α + N − n − n )(2 n + α + α )(2 n + α + α + n + n + α + α + α + n + n + α + α + α + + n ( n + α )(2 n + n + α + α + n + n + α + α + α + N + n + n + α + α + α + m + α ) (2 m + n + α + + ( n + α + n + α + α + n + n + α + α + n + n + α + α + α + N − n − n )(2 n + α + α + n + α + α + n + n + α + α + α + n + n + α + α + α + .The bivariate Hahn polynomials also satisfy the recurrence relation x P n , n ( x , x ) = − ˜ a n , n P n + n ( x , x ) + ˜ b n , n P n , n + ( x , x ) − ˜ c n , n P n − n + ( x , x ) − ˜ d n , n P n − n + ( x , x ) + ˜ e n , n P n , n ( x , x ) − ˜ f n , n P n + n − ( x , x ) + ˜ g n , n P n + n − ( x , x ) − ˜ h n , n P n , n − ( x , x ) + ˜ i n , n P n − n ( x , x ), he multivariate Hahn polynomials and the singular oscillator y n , n are obtained from y n , n by the permutation α ↔ α Appendix A.4. Difference equations
The bivariate Hahn polynomials P ( α , α , α ) n , n ( x , x ; N ) satisfy the eigenvalues equation L P ( α , α , α ) n , n ( x , x ; N ) = − n ( n + α + α + P ( α , α , α ) n , n ( x , x ; N )where L = x ( x + α + T − x T + x + x ( x + α + T + x T − x − ( x ( x + α + + x ( x + α + I ,where T ± x i are the usual forward and backward shift operators in the variable x i . Thebivariate Hahn polynomials also satisfy L P ( α , α , α ) n , n ( x , x ; N ) = − ( n + n )( n + n + α + α + α + P ( α , α , α ) n , n ( x , x ; N ),where L is the difference operator L = ( N − x − x ) £ ( x + α + T + x + ( x + α + T + x ¤ + x ( x + α + T − x T + x + ( N − x − x + α + £ x T − x + x T − x ¤ + x ( x + α + T + x T − x − h ( N − x − x )( x + x + α + α + + ( x + x )( N − x − x + α + + x ( x + α + + x ( x + α + i I . Appendix A.5. Generating Function
The polynomials P ( α , α , α ) n , n ( x , x ; N ) have for generating function(1 + z + z ) N − n ( z + z ) n P ( α , α ) n µ z − z z + z ¶ P (2 n + α + α + α ) n µ − z − z + z + z ¶ = X x , x x + x ≤ N N ! x ! x !( N − x − x )! P ( α , α , α ) n , n ( x , x ; N ) n ! n ! z x z x . Appendix A.6. Forward shift operators
One has the forward relation − N P ( α , α , α ) n + n ( x , x ; N ) = x ( x + α + P ( α + α + α ) n , n ( x − x ; N − − x ( x + α + P ( α + α + α ) n , n ( x , x − N − he multivariate Hahn polynomials and the singular oscillator − N ( n + α + P ( α , α , α ) n , n + ( x , x ; N ) = ( x + α + N − x − x )( N − x − x − P ( α , α , α + n , n ( x + x ; N − + ( x + α + N − x − x )( N − x − x − P ( α , α , α + n , n ( x , x + N − + x ( N − x − x + α + N − x − x + α + P ( α , α , α + n , n ( x − x ; N − + x ( N − x − x + α + N − x − x + α + P ( α , α , α + n , n ( x , x − N − − ( N − x − x )( N − x − x + α + x + x + α + α + P ( α , α , α + n , n ( x , x ; N − Appendix A.7. Backward shift operators
The backward relations are given by − n ( n + α + α + N + P ( α + α + α ) n − n ( x , x ; N ) = P ( α , α , α ) n , n ( x + x ; N + − P ( α , α , α ) n , n ( x , x + N + − n (2 n + n + α + α + n + n + α + α + α + N + P ( α , α , α + n , n − ( x , x ; N ) = ( x + α + P ( α , α , α ) n , n ( x + x ; N + + x P ( α , α , α ) n , n ( x − x ; N + + ( x + α + P ( α , α , α ) n , n ( x , x + N + + x P ( α , α , α ) n , n ( x , x − N + − (2 x + x + α + α + P ( α , α , α ) n , n ( x , x ; N + Appendix A.8. Structure relations
One has N P ( α , α , α ) n , n ( x + x ; N ) = ( n + α + α + n + n + α + α + α + N − n − n )(2 n + α + α + n + n + α + α + α + P ( α + α , α ) n , n ( x , x ; N − − n ( n + α )(2 n + n + α + α + N + n + n + α + α + α + n + α + α + n + n + α + α + α + P ( α + α , α ) n − n ( x , x ; N − − n ( n + α )( n + α + α + N + n + n + α + α + α + n + α + α + n + n + α + α + α + P ( α + α , α ) n , n − ( x , x ; N − + n ( n + α )( N − n − n )(2 n + α + α + n + n + α + α + α + P ( α + α , α ) n − n + ( x , x ; N − P ( α , α , α ) n , n ( x , x ; N ) = ( − n P ( α , α , α ) n , n ( x , x ; N ), we also have N P ( α , α , α ) n , n ( x , x + N ) = ( n + α + α + n + n + α + α + α + N − n − n )(2 n + α + α + n + n + α + α + α + P ( α , α + α ) n , n ( x , x ; N − + n ( n + α )(2 n + n + α + α + N + n + n + α + α + α + n + α + α + n + n + α + α + α + P ( α , α + α ) n − n ( x , x ; N − − n ( n + α )( n + α + α + N + n + n + α + α + α + n + α + α + n + n + α + α + α + P ( α , α + α ) n , n − ( x , x ; N − − n ( n + α )( N − n − n )(2 n + α + α + n + n + α + α + α + P ( α , α + α ) n − n + ( x , x ; N − he multivariate Hahn polynomials and the singular oscillator x N P ( α + α , α ) n , n ( x − x ; N − = ( n + α + n + n + α + α + n + α + α + n + n + α + α + α + P ( α , α , α ) n , n ( x , x ; N ) − (2 n + n + α + α + α + n + α + α + n + n + α + α + α + P ( α , α , α ) n + n ( x , x ; N ) − ( n + α + n + α + α + n + n + α + α + α + P ( α , α , α ) n , n + ( x , x ; N ) + n ( n + α )(2 n + α + α + n + n + α + α + α + P ( α , α , α ) n + n − ( x , x ; N ). x N P ( α , α + α ) n , n ( x , x − N − = ( n + α + n + n + α + α + n + α + α + n + n + α + α + α + P ( α , α , α ) n , n ( x , x ; N ) + (2 n + n + α + α + α + n + α + α + n + n + α + α + α + P ( α , α , α ) n + n ( x , x ; N ) − ( n + α + n + α + α + n + n + α + α + α + P ( α , α , α ) n , n + ( x , x ; N ) − n ( n + α )(2 n + α + α + n + n + α + α + α + P ( α , α , α ) n + n − ( x , x ; N ). Appendix B. Structure relations for Jacobi polynomials
The Jacobi polynomials P ( α , β ) n ( z ) are defined by [28] P ( α , β ) n ( z ) = ( α + n n ! F ³ − n , n + α + β + α + ¯¯¯ − z ´ . (B.1)The following structure relations hold for the Jacobi polynomials [29]: ∂ z P ( α , β ) n ( z ) = ( n + α + β + P ( α + β + n − ( z ), (B.2) £ ( z − ∂ z + ( α + ∂ z ¤ P ( α , β ) n ( z ) = ( n + α )( n + α + β + P ( α , β + n − ( z ), (B.3) £ (1 − z ) ∂ z + [( β − α ) − ( α + β ) z ] ¤ P ( α , β ) n ( z ) = − n + P ( α − β − n + ( z ), (B.4) n (1 + z )( z − ∂ z + (1 + z )[1 + α − β + (1 + α + β ) z ] ∂ z + β [2 + α (1 + z ) + β ( z − o P ( α , β ) n ( z ) = n + n + β ) P ( α , β − n + ( z ). (B.5) Appendix C. Structure relations for Laguerre polynomials
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