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Lie algebra \(\mathcal{K}_{5}\) and 3-variable Laguerre-Hermite polynomials. (English) Zbl 1426.33028

The authors study special functions from the viewpoint of being matrix elements/basis vectors connected with unitary irreducible representations of Lie groups.
The basic environment uses a 5-dimensional complex Lie algebra \(\mathcal{K}_5\), generated by commutation relations between a basis of 5 elements \(\{\mathcal{I}^{\pm},\mathcal{I}^3,\mathcal{E},\mathcal{Q}\}\) \[ [\mathcal{I}^3,\mathcal{I}^{\pm}]=\pm\mathcal{I}^{\pm},\ [\mathcal{I}^3,\mathcal{Q}]=2\mathcal{Q},\ [\mathcal{I}^{+},\mathcal{I}^{-}]=\mathcal{E},\ [\mathcal{I}^{-},\mathcal{Q}]=2\mathcal{I}^{+}, \] \[ [\mathcal{I}^{+},\mathcal{Q}]=\Theta,\ [\mathcal{I}^{\pm},\mathcal{E}]=[\mathcal{I}^3,\mathcal{E}]=[\mathcal{Q},\mathcal{E}]=\Theta, \] where \(\Theta\) is the \(5\times 5\) zero matrix.
And the 5-dimensional complex Lie group \(K_5\) with elements \(g(q,a,b,c,\tau)\) with complex entries and multiplication law \[ g(q,a,b,c,\tau)g(q',a',b',c',\tau')=g(q+e^{2\tau}q',a+a'+e^{\tau}cb',b+e^{\tau}b'+2e^{2\tau}cq',c+e^{-\tau}c',\tau+\tau'). \]
The 3-variable Laguerre-Hermite polynomials (3VLHP) are explicitly given by \[ {}_LH_n(x,y,z)=n!\sum_{k=0}^{[\frac{n}{2}]}\,\sum_{r=0}^{n-2k}\,\frac{(-1)^rx^ry^{n-2k-r}z^k}{k!(r!)^2(n-2k-r)!}. \]
The authors then show that the functions \({}_LH_n(x,y,z)t^n\) form a basis for a realization of the representation \(\uparrow_{0,1}^{'}\) of \(\mathcal{K}_5\) and for \(g\in \mathcal{K}_5\) they prove
Theorem 2.1. The following implicit summation formula involving the 3VLHP \({}_LH_n(x,y,z)\) holds \begin{multline*} (1-4qzt^2)^{-(k+1)/2}\exp{\left( a+k\tau+\frac{q(y-x)^2t^2+b(y-x)t+b^2zt^2}{1-4qzt^2}\right)} \\ \times {}_LH_n\left(x,(1-4qzt^2)^{-1/2}\left(y+x\left((1-4qzt^2)^{1/2}-1\right)+2bzt+\frac{c}{t}(1-4qzt^2)\right),z\right) \\ =\sum_{\ell=0}^{\infty}\,A_{\ell k}(g){}_LH_{\ell}(x,y,z)t^{\ell-k}\ (k=0,1,2,\ldots;\ q,a,b,c,\tau\in\mathbb{C}), \end{multline*} where the \(A_{\ell k}(g)\) are given by \[ A_{\ell k}(g)=\exp{(a+k\tau)}c^{k-\ell}L_{\ell}^{(k-\ell)}(-bc), \] with \(L_{\ell}^{(k-\ell)}\) the associated Laguerre polynomials.
The layout of the paper is as follows
§1.
Introduction and preliminaries (\(2\frac{1}{2}\) pages)
§2.
Lie algebra \(\mathcal{K}_5\) and implicit formulae (\(3\frac{1}{2}\) pages) – contains the proof of Theorem 2.1
§3.
Examples (\(2\) pages)
(a)
for \(z=-\frac{1}{2}\), the 3VLHP reduce to the 2-variable Laguerre-Hermite polynomials with generating function \[ J_0(2\sqrt{xt})\exp{(yt-\frac{1}{2}t^2)}=\sum_{n=0}^{\infty}\,{}_LH_n^{\ast}(x,y)t^n. \]
(b)
for \(z=0\), the 3VLHP reduce to the 2-variable Laguerre polynomials \(L_n(x,y)\) with generating function \[ J_0(2\sqrt{xt})\exp{(yt)}=\sum_{n=0}^{\infty}\,L_n(x,y)\frac{t^n}{n!}. \]
(c)
for \(x=0\), the 3VLHP reduces to the 2-variable Hermite-Kempé de Féret polynomials \(H_n(y,z)\) defined by \[ \exp{(yt+zt^2)}=\sum_{n=0}^{\infty}\,H_n(y,z)\frac{t^n}{n!}, \] leading to implicit summation formulae \begin{multline*} \exp{(byt+b^2zt^2)}H_k(y+2bzt+\frac{c}{t},z)\\ =\sum_{\ell =0}^{\infty}\,c^{k-\ell}L_{\ell}^{(k-\ell)}(-bc)H_{\ell}(y,z)t^{\ell-k},\ k=0,1,\ldots;\ a,b,c,\tau\in\mathbb{C}, \end{multline*} \begin{multline*} (1-4qzt^2)^{-(k+1)/2}\exp{\left(\frac{qy^2t^2+byt+b^2zt^2}{1-4qzt^2}\right)}H_k\left((1-4qzt^2)^{-1/2}(y+2bzt),z\right)\\ =\sum_{\ell=0}^{\infty}\,\frac{(-q)^{(\ell-k)/2}}{(\ell-k)!}H_{\ell-k}\left(\frac{b}{2(-q)^{1/2}}\right)HJ_{\ell}(y,z)t^{\ell-k},\ k=0,1,\ldots; q,b\in\mathbb{C},\ell\geq k\geq 0, \end{multline*} \begin{multline*} (1-4qzt^2)^{-(k+1)/2}\exp{\left(\frac{qy^2t^2}{1-4qzt^2}\right)}H_k\left((1-4qzt^2)^{-1/2}(y+\frac{c}{t} (1-4qzt^2)), z\right)\\ =\sum_{\ell=0}^{\infty}\sum_{j=-\infty}^{\infty}\,c^{k-\ell}k!\frac{(qc^2)^j}{j!(2j+k-\ell)!(\ell-2j)!}H_{\ell}(y,z)t^{\ell-k},\ k=0,1,\ldots; q,c\in\mathbb{C}. \end{multline*}
§4.
Concluding remarks
In this paragraph, the authors study 3-variable Hermite-Laguerre polynomials (3VHLP; note the change in order of the H and L). These are given explicitly as a sum of 2-variable Hermite and by a generating function using 2-variable Laguerre functions.
§5.
References (\(19\) items)

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33C80 Connections of hypergeometric functions with groups and algebras, and related topics
33E20 Other functions defined by series and integrals
35R03 PDEs on Heisenberg groups, Lie groups, Carnot groups, etc.
22E60 Lie algebras of Lie groups
22E70 Applications of Lie groups to the sciences; explicit representations
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