Some properties of generalized multiple Hermite polynomials (Q555190)

The authors introduce and discuss a new class of polynomials (generalized multiple Hermite polynomials) defined as follows. Given real numbers \(p>0\), \(\alpha,\alpha_1,\alpha_2>-1\), \(\alpha_1\not=\alpha_2\), and a positive integer \(k\), they put \[ G_{n,m}^{(\alpha,\alpha_1,\alpha_2)}(x,k;p):=x^{-\alpha}e^{px^k} e^{-\alpha_1 x}\,\frac{d^n}{dx^n}e^{(\alpha_1-\alpha_2)x}\,\frac{d^m}{dx^m} x^{\alpha}e^{-px^k}\,e^{\alpha_2 x}. \] For some particular values of the parameters, the latter give the standard multiple Hermite polynomials and multiple Laguerre I polynomials. The authors obtain an explicit expression of these polynomials and show that \(G_{n,m}^{(\alpha,\alpha_1,\alpha_2)}(x,k;p)\) are of exact degree \((k-1)(n+m)\). They also obtain an operational formula for these polynomials in terms of multiple Laguerre I polynomials, linear generating functions and recurrence formulae. The authors derive several families of bilinear, bilateral and mixed multilateral finite series relationships for such polynomials.