2-variable generalized Hermite matrix polynomials and Lie algebra representation
Motivated by the definition of 2-variable Hermite Kampé de Fériet polynomials \(H_n(x,y)\) via their generating function, \[ \exp(xt + y t^2) =\sum_{n=0}^\infty H_n(x,y) \frac{t^n}{n!}, \] and by several extensions of classical orthogonal polynomials to matrix parameters, the authors introduce the set of 2-variable generalized Hermite polynomials \(H^\lambda\) via \[ \exp\left(\lambda\left(xt\sqrt{ A/2} + y t^2 I\right)\right) =\sum_{n=0}^\infty H^\lambda_n(x,y;A) \frac{t^n}{n!}, \] where \(A\) is a matrix in \(\mathbb C^{m\times m}\), and \(\lambda\) a complex parameter. In particular, \[ H^1_n(x,y;A)=H_n\left(x\sqrt{A/2},y\right). \] Several algebraic properties of \(H^\lambda\) (such as their explicit representation or the differential equation) are obtained. Also, the authors use the representation theory of the harmonic oscillator Lie algebra \(\mathcal G(0,1)\) in order to derive further generating identities for the 2-variable generalized Hermite polynomials \(H^\lambda\).
scientific article; zbMATH DE number 5904004
scientific article