Symmetry, Integrability and Geometry: Methods and Applications Special Functions of Hypercomplex Variable on the Lattice Based on SU(1,1)

Based on the representation of a set of canonical operators on the lattice $h\mathbb{Z}^n$, which are Clifford-vector-valued, we will introduce new families of special functions of hypercomplex variable possessing $\mathfrak{su}(1,1)$ symmetries. The Fourier decomposition of the space of Clifford-vector-valued polynomials with respect to the ${\rm SO}(n)\times \mathfrak{su}(1,1)$-module gives rise to the construction of new families of polynomial sequences as eigenfunctions of a coupled system involving forward/backward discretizations $E_h^{\pm}$ of the Euler operator $E=\sum\limits_{j=1}^nx_j \partial_{x_j}$. Moreover, the interpretation of the one-parameter representation $\mathbb{E}_h(t)=\exp(tE_h^--tE_h^+)$ of the Lie group ${\rm SU}(1,1)$ as a semigroup $\left(\mathbb{E}_h(t)\right)_{t\geq 0}$ will allows us to describe the polynomial solutions of an homogeneous Cauchy problem on $[0,\infty)\times h{\mathbb Z}^n$ involving the differencial-difference operator $\partial_t+E_h^+-E_h^-$.


Introduction
In the investigation of special functions, the representation theory through Lie groups allows to compute families of orthogonal polynomials in terms of hypergeometric series expansions (see, e.g., [17]). Recent approaches towards discrete quantum mechanics, as for instance in [14], reveals that the representation of finite difference operators as canonical generators of a certain Lie algebra provides a general scheme to construct sequences of polynomials as eigenfunctions of a discrete Hamiltonian operator. These sequences of polynomials that appear on the literature under the name of Sheffer sequences (cf. [6,12]) or Appell sequences (cf. [16]) give rise to families of Bernoulli and Euler polynomials beyond the classical rising/falling factorial polynomials (see also [1,Section 3]).
Seen the fact that Clifford algebras of signature (0, n) encode the structure of the special orthogonal group SO(n) of n × n matrices (cf. [4, Subsection I.1]), it remains natural to study multi-variable extensions of the above approaches to the Euclidean space R n in terms of hypercomplex variables. In [13,Section 2] and in [2,Section 3] the authors obtained the hypercomplex extension of Bernoulli and Euler polynomials, respectively; In [3,Section 4] the authors considered discrete versions of Fueter polynomials as an alternative hypercomplex extension for the raising/lowering Clifford-vector-valued polynomials considered in [8,Section 3]. In [9,Section 2] the authors shown that such families of Clifford-vector-valued polynomials of discrete variable may be realized from Lie algebraic representations of an algebra of endomorphisms analogue to the radial algebra representation obtained in continuum by Sommen (cf. [15]).

arXiv:1304.7191v4 [math.CV] 5 Nov 2013
This paper is organized as follows: In Section 2 it will be given the motivation to study, in the framework of Clifford algebras, sequences of polynomials generated from a set of finite difference operators. In Section 3 the construction of irreducible representations for the spaces of Cliffordvector-valued polynomials on the lattice hZ n based on the Howe dual pair (SO(n), su(1, 1)) (cf. [11]) will be considered.
To find the Fourier decomposition for the space of Clifford-vector-valued polynomials on the lattice hZ n , we will start to determine the positive and negative series representations for the Lie group SU (1,1) in interconnection with the forward/backward discretizations E ± h of the classical Euler operator E = n j=1 x j ∂ x j . Afterwards, the action of the SO(n) × su(1, 1)-module on the lattice hZ n will produce a sequence of invariant and irreducible subspaces under the discrete series representations of SU (1,1).
The results obtained in Section 3 will be used in Section 4 to describe the space of Cliffordvector-valued polynomials on the lattice hZ n as hypergeometric series expansions and to characterize the homogeneous solutions of the differential-difference operator ∂ t + E + h − E − h in terms of the semigroup (E h (t)) t≥0 carrying the one-parameter representation E h (t) = exp(tE − h − tE + h ) of SU(1, 1).

Scope of problems
Let e 1 , e 2 , . . . , e n be an orthogonal basis of R n . The Clifford algebra of signature (0, n), which we will denote by C 0,n , corresponds to the algebra generated from the set of graded anti-commuting relations e j e k + e k e j = −2δ j,k for any j, k = 1, 2, . . . , n.
Under the linear space isomorphism given by the mapping e j 1 e j 2 · · · e jr → dx j 1 dx j 2 · · · dx jr , with 1 ≤ j 1 < j 2 < · · · < j r ≤ n, the resulting algebra with dimension 2 n is isomorphic to the exterior algebra (R n ).
This allows us to represent any vector x = (x 1 , x 2 , . . . , x n ) of R n as an element x = n j=1 x j e j ∈ C 0,n and the translations (x 1 , x 2 , . . . , x j ± h, . . . , x n ) on the grid hZ n ⊂ R n with mesh width h > 0 by the displacements x ± he j over C 0,n . Moreover, the Clifford-vector-valued functions correspond to linear combinations in terms of the r-multivector basis e j 1 e j 2 · · · e jr labeled by subsets J = {j 1 , j 2 , . . . , j r } of {1, 2, . . . , n}, i.e.
Let us now recall some basic facts about finite difference operators. The forward/backward finite differences ∂ ±j h defined on the grid hZ n by Also, they satisfy the product rules Along the paper we will use the bold letters f , g, . . . , m, . . . , w, . . . and so on, when we refer to functions of the above form. The finite difference Dirac operators D ± h defined viz are Clifford-vector-valued and correspond to finite difference approximations of the classical gradient operator in a coordinate-free way. Now let P = R[x] ⊗ C 0,n be the space of Clifford-vector-valued polynomials. We say that Iterating r-times the operator D + h , resp. D − h , it turns out that the action of the semigroup exp(tD + h ) t≥0 , resp. exp(tD − h ) t≥0 , on each m s (x; −h), resp. m s (x; h), gives rise to a binomial expansion. Indeed, the iterated relations D ± h r m s (x; ∓h) = s! (s−r)! m s−r (x; ∓h), resp. D ± h r m s (x; ∓h) = 0, that hold for s ≥ r, resp. s < r, leads to and analogously, to The action of the semigroups exp(tD + h ) t≥0 and exp(tD − h ) t≥0 on P then correspond to the hypercomplex extension of the Taylor series expansion for polynomials (see, e.g., [1, Subsection 3.3]) that gives rise e.g. to Clifford-vector-valued polynomials of Bernoulli type analogous to the ones obtained in [13,Section 2].
This approach corresponds to the discrete counterpart of the Cauchy-Kovaleskaya extension described in [4,Subsection III.2]. For an alternative application of this approach in interrelationship with discrete versions of Fueter polynomials, one refer to [3,Sections 3,4].
The Fock space formalism carrying Hilbert spaces (cf. [5]) reveals that the problem of constructing polynomial sets {m s (x; τ ) : s ∈ N 0 }, with τ = ±h, possessing the Appell set property is equivalent to the construction of operator endomorphisms M + h , M − h ∈ End(P) in such way that the elements of the form yield a basis for P. Hereby, the constants λ s are chosen under the condition λ 0 = 1 and the constraints (4). In terms of the umbral calculus formalism (see [6,12] and the references therein), from the identity operator I : f (x) → f (x) and the commuting bracket [A, B] defined as one can start to construct the set of Clifford-vector-valued polynomials {m s (x; τ ) : s ∈ N 0 } from Weyl-Heisenberg algebra symmetries. In order to proceed, we will define for a given linear polynomial In the discrete Clifford analysis setting (cf. [9, Subsection 1.3]), it is precisely the Weyl-Heisenberg relations (6), resp. (7), that allows us to determine in a unique way M − h , resp. M + h , as providing in this way a Fourier duality between D + h and M − h , resp. D − h and M + h . For discretizations of the Dirac operator D = n j=1 e j ∂ x j , written as a superposition of forward and backward differences, the Fourier duality may be constructed by means of a set of skew-Weyl relations (cf. [2]).
Notice that the Fourier duality terminology comes from invariant theory (cf. [11]). In the language of Clifford analysis this is nothing else than the so-called Fischer duality (cf. [15]).
Since M ± h maps m s (x; ±h) into m s+1 (x; ±h), it remains clear that each Clifford-vectorvalued polynomial m s (x; h), resp. m s (x; −h), is an eigenfunction for the factorized Hamiltonian The Weyl-Heisenberg character between the operators ∂ +j h and W −j combined with the orthogonality constraint (1) provided by the Clifford basis of C 0,n allows us to rewrite the factorized Hamiltonians h in terms of the following forward/backward discretizations of the Euler operator E = j=1 x j ∂ x j : Indeed, from (2) one can rewrite E + h and E − h as h admit the following decompositions: The second quantization approach through the set of Weyl-Heisenberg generators (6) and (7) (cf. [5]) reveals that E ± h are number-type operators with spectrum N 0 . This leads to From the above construction, it turns out that the Weyl-Heisenberg algebra character provided from the graded commuting relations (6), resp. (7), gives the required ladder structure to construct also discrete analogues for Gegenbauer polynomials as a series expansion written in terms of the Appell set {m s (x; τ ) : s ∈ N 0 } (cf. [7]).
Since the eigenfunctions of E + h and E − h do not coincide, in general, it remains natural to ask which polynomial subsets {m s (x; h) : s ∈ N 0 } of P give rise to solutions of the coupled eigenvalue system From the relations (2), it remains clear that forward and backward differences, ∂ +j h and ∂ −j h respectively, commute. In contrast, the operators W −j h , W +j h do not commute in general. This means that for each j = 1, 2, . . . , n the set of operators ∂ +j h , ∂ −j h , W −j h , W +j h and I do not endow a canonical realization of an Weyl-Heisenberg type algebra.
Although the solutions of the above coupled system of eigenvalue equations may not be represented in terms of Weyl-Heisenberg algebra symmetries, the set of generators W −j h , W +j h , W j itself (see the coordinate expressions (5)) will be the departure point of this paper to describe the hidden Lie algebraic symmetries encoded by the solutions of the above coupled system.

Discrete series representions of SU(1, 1)
Accordingly to [17,Section 6.4], the Lie group SU(1, 1) has two families of discrete series representations. In order to determine it algebraically, let us take a close look for the graded commuting relations involving the number operators E + h and E − h defined in (8). For this purpose, one starts to show that the set of operators W + h , W − h and W defined from the left endomorphisms (5) acting on the space P: generate a Lie algebra isomorphic to su(1, 1). In order to proceed, we will start with the following lemma which interrelates the set of generators Lemma 1. For every j, k = 1, 2, . . . , n we have the following set of graded commuting rules: (2) The operators W +k h , resp. W −k h , and W j are interrelated by for j = k, one obtain for every j, k = 1, 2, . . . , n, the set of graded commuting relations Thus, for all j, k = 1, 2, . . . , n the above set of graded commuting relations are equivalent to for every j = k, it remains to show for every j = 1, 2, . . . , n the graded commuting relations From a direct computation From Lemma 1, one obtain for the set of multiplication operators (5), the Lie algebra isomorphism Thus, one can infer that 1 h W + h , 1 h W − h and 1 h W are the canonical generators of the threedimensional Lie algebra su(1, 1) ∼ = sl(2, R). The remaining commuting relations are given by Now let's turn our attention to the coordinate expressions of E ± h given by (8). Recall that the conditions ∂ +j h w (x k ) = ∂ −j h w (x k ) = δ jk µ provided from construction allows us to recast E + h and E − h in terms of the multiplication operators defined in (5). Indeed, the basic identities lead to the following coordinate expressions: and consequently, to the following coordinate expressions involving the sum/difference between E + h and E − h : The following set of results are also straightforward and will give the key ingredients to construct the positive/negative series representations encoded by SU(1,1).
I are the canonical generators of the Lie algebra su(1, 1). The remainder commuting relations are given by Proof . First, notice that direct combination of relations (10) with the coordinate expressions (11) and (12) results into the following set of graded commuting relations carrying the In order to prove the graded commuting relations E + h + n 2 I, 1 h and based on the coordinate expressions (11). In concrete The graded commuting relations provided from (10) yield and therefore, the relations (13) are equivalent to I follow straightforwardly from direct combination of the coordinate expression obtained in (12) for In the proof of Proposition 1 and in the subsequent results, we will make use of the following lemma which follows straightforwardly from induction on N. Proposition 1. For any s ∈ N we have the following graded commuting relations: Proof . Recall that when [A, B] = ±B, Lemma 3 reduces to Combination of Lemma 2 with the above identity carrying the substitutions For the proof of Finally, the direct application of Lemma 3 carrying the substitutions Now let m 0 (x; h) be a Clifford-vector-valued polynomial that satisfies the set of equations Then it is straightforward to see that the basis functions of the form w s ( satisfy the following set of ladder operator relations: We are now in conditions to construct the positive, resp. negative, part for the discrete series representation carrying the group SU(1, 1) in the same order of ideas of [ determines all irreducible unitary representations π λ of SU(1, 1) on the enveloping algebra U (su(1, 1)) through its eigenvalues λ. Moreover, the positive series representation π + λ labelled by λ is thus determined by the set of ladder operators h underlies the representation space 2 (N 0 ) carrying the family of subspaces (H s;h ) s∈N 0 of the form while the negative ones is thus determined by the set of ladder operators and underlies the representation space 2 (N 0 ) carrying the family of subspaces (H s;−h ) s∈N 0 of the form A short computation based on the ladder operator properties (14) and (15) shows that the positive/negative discrete series representations π + λ /π − λ of the Lie group SU(1, 1) are labeled by the constants λ = n 2 4 − n 2 − 2s, with s ∈ N 0 .

The action of SO(n) on the lattice
With the aim of understanding the action of SO(n) on the lattice hZ n as a representation theory carrying canonical generators which are invariant with respect to the orthogonal Lie algebra so(n), we will first recall some basic concepts and observations about SO(n) and so(n) in the context of Clifford algebras. Let us denote by B(x, y) = − 1 2 (xy + yx) the bilinear form generated by the Clifford vector representations x = n j=1 x j e j and y = n j=1 y j e j of R n . The set of matrices T : R n → R n with determinant equals 1 for which B(T x, T y) = B(x, y) forms a group under the operation of composition. This group is called the special orthogonal group of rotations and it is denoted by SO(n).
Let us take a close look for the SO(n)-action on the lattice hZ n given by the left regular representation In order to proceed, one can select from SO(n) the 1-parameter subgroups elements T ±h jk (θ) = exp θS ±h jk generated by exponentiation from the Lie algebra elements S ±h jk ∈ so(n) (1 ≤ j < k ≤ n) in such way that each S ±h jk is skew-symmetric, i.e. S ±h jk = −S ±h kj . In particular, the canonical elements of the form that correspond to the discrete counterparts of the classical angular momentum operators L jk = x j ∂ x k − x k ∂ x j , endow the left representations of SO(n) acting on each subspace H s;h , resp. H s;−h of P, that is Λ T +h jk (θ) = exp θS +h jk , resp. Λ T −h jk (θ) = exp θS −h jk . It is straightforwardly to see from (8) that the operator E + h − E − h commutes with the skewsymmetric elements (18) belonging in this way to the center of the enveloping algebra U (so(n)). Indeed, the Clifford-vector-valued extension of the classical Sheffer map Ψ x from R[x] to P defined by linearity from the mapping and hence to the intertwining property below at the level of SO(n): This in turn shows that the 1-parameter representation T +h jk (θ), resp. T −h jk (θ) of SO(n), on hZ n is canonically isomorphic to to the 1-parameter representation T jk (θ) of SO(n) on R n .

The Howe dual pair (SO(n), su(1, 1))
Along this section we will study subspaces of Clifford-vector-valued polynomials which are invariant under the action of the SO(n) × su(1, 1)-module. The ladder properties (14) and (15) reveal that for each s ∈ N 0 the representation π + In order to collect the infinite multiplicities of π + λ /π − λ carrying the eigenvalues λ = n 2 4 − n 2 − 2s of the Casimir operator (16), it remains to investigate the su(1, 1)-action on P regarded as a SO(n) × su(1, 1)-module that yields the Howe dual pair (SO(n), su (1, 1)). For a sake of readability, the Howe dual pair construction will be only sketched. Further details arising this construction can be found in [10,Chapters 4 & 5]. First, notice that the above set of properties produces the following infinite triangle as a chain diagram carrying the families of subspaces (H s;h ) s∈N 0 , resp. (H s;−h ) s∈N 0 : In the above triangle diagram, the representations π + -act as isomorphisms that shift each individual summand from the right to the left. The first line gives the direct sum decomposition of P in terms of the SO(n)-invariant pieces H s;h , resp. H s;−h , through hZ n , i.e.
Also, for each s ∈ N 0 the (s + 1)-row (which is infinite-dimensional) give rise to su(1, 1)modules isomorphic to the SO(n)-module H s;h , resp. H s;−h , while the (s + 1)-column provides the splitting of the subspace H s;h , resp. H s;−h , as a direct sum in terms of the irreducible pieces These chain decompositions lead, in a multiplicity free way, to the following Fourier decompositions of P: Moreover, the constants γ s are given by Hereby (a) r = a(a + 1) · · · (a + r − 1) denotes the Pochhammer symbol.
Proof . To prove this, let us consider the family {m s (x; h) : s ∈ N 0 } of Clifford-vector-valued polynomials, each of them given as a solution of the eigenvalue equations From the Fourier decompositions labeled by (19) each w s (x; ±h) ∈ H s;h may be written in a unique way as From the ladder operator relations (14) and (15), one obtain, for any r = 0, 1, . . . , s, the map- lead to the recursive relations and moreover, to the following linear expansions where the coefficients d r,s are given by d r,s = (r + 1)(2s + n − r − 2)c r+1,s + c r,s . Hence m s (x; h) satisfies the constraint (20) if and only if d r,s = 0 holds for every r = 0, 1, . . . , s, that is, each c r,s is determined from the condition c 0,s = 1 and from the constraint .
Now it remains to show the relations m s (x; h) = γ s w s (x; ±h). Iterating r times the operator E + h − E − h , one obtain from (14) the recursive relations

Application to Cauchy problems
On this section it will be studied families of Clifford-vector-valued polynomials given as solutions of the following homogeneous Cauchy problem in [0, ∞) × hZ n : The solution of the Cauchy problem (22) may be written formally as g(t, is a one-parameter representation of the Lie group SU(1, 1).
Note that the graded commuting property The next sequence of results will makes clear that, for any t ≥ 0, the operator E h (t) leaves the space of Clifford-vector-valued polynomials P invariant.
Lemma 4. We have the following intertwining properties carrying the semigroup operator Proof . In order to start proving the intertwining property one can start to compute, for each s ∈ N, the graded commutator 2E A short computation based on Lemma 2 shows that

This leads to
Therefore, the above graded commuting relation is equivalent to the intertwining property This leads to and hence, to the intertwining property Moreover the semigroup (E h (t)) t≥0 leaves invariant the space of Clifford-vector-valued polynomials P.
Proof . Recall that from the Fourier decomposition (19) any f (x) ∈ P may be written uniquely as From the representation of E + h + E − h + nI given by (12), one can conclude that the righthand side of the above relation is a Clifford-vector-valued polynomial but also an eigenfunction of tE − h + (1 − t)E + h carrying the eigenvalue s ∈ N 0 . Therefore E h (t)f (x) =