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International Journal of Pure and Applied MathematicsVolume 84 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)url: http://www.ijpam.eudoi: http://dx.doi.org/10.12732/ijpam.v84i2.6 Abstract: In this paper, we follow the approach of Miller (see [1]) and obtaingenerating relations of Laguerre 2D polynomials (L2DP) by extending the real-ization ↑ω,µ to study multiplier representations of a Lie group G(0,1). Certain(known or new) generating relations for the polynomials related to L2DP areobtained as special cases.
AMS Subject Classification: 33C45, 33C50, 33C80Key Words: Laguerre 2D polynomials, Lie algebra, generating relations Recently, Wunsche (see [3, 4, 5, 6]) introduced Laguerre 2D polynomials (L2DP)Lm,n(z, z∗) and discussed their properties and their explicit representations.Thesepolynomials are very effective mean for the representations of many results inquantum optics (quasi probabilities in Fock-state basis, ordering problems, mo-ments) and more over in other regions of physics.
The L2DP Lm,n(z, z∗), (m, n = 0, 1, 2, .) are defined as polynomials of two independent variables (z, z∗) which in application are generally a pair ofcomplex conjugated variables in the following way Wunsche (see [4], p. 3181, The representation of the Laguerre 2D polynomials by the usual Laguerre poly-nomials Lαn(u) is given by [4], p. 3181, equation (2.5) Generating relations involving two variable Laguerre polynomials by using twoLaguerre polynomials by using Weisner’s (see [7]) group-theoretic method wasdiscussed by Khan and Yasmin, see [8]. Also Khan in [9] derived some gener-ating relations involving L2DP.Here we follow the approach of Miller (see [1])and obtain generating relations of L2DP by extending the realizations of ↑ω,µto multiplier representations of a Lie group G(0,1).
We note that the following isomorphism (see [1], p. 36) where L[G(0,1)] is the Lie algebra of a complex four-dimensional Lie groupG(0,1), multiplicative matrix group with elements (see [1], p. 9) The group G(1,0)is called the complex harmonic oscillator group (see [2], Chapter 10). Abasis for L[G(0,1)] is provided by the matrices (see [1], p. 9) [j3, j±] = ±j±, [j+, j−] = −ε, [ε, j±] = [ε, j3] = 0.
The machinery constructed in (see [1], Chapters 1, 2 and 4) will be applied tofind realization of the irreducible representation ↑ω,µ of ℓ(0, 1) where ω,µ∈Csuch that µ = 0. The spectrum S of ↑ω,µ is the set S = {−ω + k, k-anon negative integer}.
In particular we are looking for the function fm,n(z, z∗, p, s) = Zm,n(z, z∗)pmsnsuch that J + = µfm+1,n, J−fm,n = (m + ω)fm−1,n, C0,1fm,n = (J+J− − EJ3)fm,n = µωfm,n for all m∈S. The commutation relations satisfied by the operators J±, J3, Eare [J3, J±] = ±J±, [J+, J−] = −E, [J±, E] = [J3, E] = 0 The number of possible solutions of equation (2.5) is tremendous. We assumethat these operators take the form and note these operators satisfy the commutation relations (7).
We can assume that ω = 0 and µ = 1 without any loss of generality for the theory of special functions. In terms of the functions Zm,n(z, z∗) relation (6)become Again, if we take the function fm,n(z, z∗, p, s) = Zm,n(z, z∗)pmsn such that J +´ = µfm,n+1, J−´fm,n = (n + ω)fm,n−1, C´0,1fm,n = (J+´J−´− E´J3´)fm,n = µωfm,n for all n∈S,then the differential operators J±´, J3´, E´are given by and satisfy the commutation relations identical to (7).
Just like before taking ω = 0 and µ = 1, relation (10) become We see from (9) and (12) that Zm,n(z, z∗) = Lm,n(z, z∗) where Lm,n(z, z∗) isgiven by (1). The functions form a basis for a realization of the representation ↑0,1 of ℓ(0, 1).This realizationof ℓ(0, 1) can be extended to a local multiplier representation T (g),g∈G(0, 1)defined on F the space of all functions analytic in a neighborhood of the point(z0, z∗0, p0, s0) = (1, 1, 1, 1).
Using operators (8),the multiplier representation (see [1], p. 17) takes the [T (exp aε)f ](z, z∗, p, s) = exp(a)f ((z, z∗, p, s) [T (exp bj+)f ](z, z∗, p, s) = exp(bzp)f ((z, z∗ − bp, p, s) [T (exp cj−)f ](z, z∗, p, s) = f (z + , z∗, p, s) [T (exp τ j3)f ](z, z∗, p, s) = f (z, z∗, peτ , s) for f ∈F .If g∈G(0, 1) has parameters (a, b, c, τ ), then T (g) = T (exp aε)T (exp bj+)T (exp cj−)T (exp τ j3) [T (g)f ](z, z∗, p, s) = exp(a + bzp)f (z + , z∗ − bp, peτ , s) The matrix element of T (g) with respect to the analytic basis fm,n(z, z∗, p, s) =Lm,n(z, z∗)pmsn are the function Alk(g) uniquely determined by ↑ω,µ of ℓ(0, 1)and we obtain relations Alk(g)fl,n(z, z∗, p, s), k = 0, 1, 2, .
exp(a + τ k + bzp)Lk,n(z + , z∗ − bp) = where k = 0, 1, 2, ., and the matrix element Alk(g) are given by (see [1], p. 87,equation (4.26)) Alk(g) = exp(a + τ k)ck−lLk−l(−bc), k, l ≥ 0.
Substituting (16) into (15), we obtain the generating relation Again taking the operators (11) and proceeding exactly as before, b´, c´, s ∈ C, m, q = 0, 1, 2, .
We consider some special cases of the generating relations obtained in previoussection, which yield many new and known relations for the polynomials relatedto L2DP.
ck−lLk−l(−bc)Ln−l(zz∗)(z∗)k−l(−p)l−kl!, Now in particular, taking z = 1 and replacing z∗ by u, p by −t and n by q in(19) we get a result of Miller (see [1], (4.94), p. 112).
II. Again making use of (2) in (18) we get b´, c´, s ∈ C, m, q = 0, 1, 2, .
Again in particular taking s = 1 and replacing z by b1, z∗ by c1, b´ by −b2, c´by c2, m by l, q by l + n and r by j in (20) we get a result of Miller (see[1], (4.28), p. 88).
We have considered the problem of framing L2DP Lm,m(z, z∗) into the contextof the representation ↑ω,µ of the Lie algebra ℓ(0, 1) of the complex harmonicgroup G(0, 1). Generating relations involving L2DP are obtained by usingMillers technique. Some relations for the products of Laguerre polynomialsand identities of Miller are also obtained as special cases.
Further,we observe that these operators J−, J+, J−´J+´ and I = 1 satisfy [J−, J+] = [J−´, J+´] = I, [J−, J+´] = [J−´, J+] = 0, These relations imply that the five operators J−, J+, J−´J+´, I are closed withregard to the commutation relations. Therefore, they form a realization of anabstract five-dimensional Lie algebra which is the Lie algebra of the Heisenberg-Weyl group W (2, R) or to its complex extension W (2, C) for a two-mode system.
Thus L2DP form a certain basis for this realization of the Heisenberg-Weylalgebra or to its complex extension ω(2, C) (see [3] and the references therein).
By the quadratic combinations of the basic operators J−, J+, J−´J+´, I, we canform ten more operators, which form several Lie Algebras.
The study of the L2DP for applications as well as for its connections with various Lie algebras is an interesting problem for further research.
[1] W. Miller, Lie Theory and Special Functions, Academic press, New York [2] W. Miller, Symmetry Groups and their Applications, Academic Press, New [3] A. Wunsche, Laguerre 2D-functions and their applications in quantum optics, J. Phy. A: Math. Gen., 31 (1998), 8267-8287.
[4] A. Wunsche, Transformations of Laguerre 2D polynomials with applica- tions to quasiprobabilities, J. Phy. A: Math. Gen., 32 (1999), 3179-3199.
[5] A. Wunsche, General Hermite and Laguerre two-dimensional polynomials, J. Phy. A: Math. Gen., 33 (2000), 1603-1629.
[6] A. Wunsche, Hermite and Laguerre 2D polynomials, J. Comput. Appl. [7] L. Weisner, Group-theoretic origin of certain generating functions, Pacific J. Math., 5 (1955), 1033-1039.
[8] Subuhi Khan, Ghazala Yasmin, Lie-theoretic generating relations of two variable Laguerre polynomials, Rep. Math. Phys., 51 (2003), 1-7.
[9] Subuhi Khan, Harmonic oscillator group and Laguerre 2D polynomials, Rep. Math. Phys., 52 (2003), 227-234.

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