Lie algebra $$\mathcal {K}_{5}$$ K 5 and 3-variable Laguerre–Hermite polynomials

Abstract

This article deals with the problem of framing the 3-variable Laguerre–Hermite polynomials (3VLHP) into the context of the representation of the five dimensional Lie algebra $$\mathcal {K}_{5}$$ . Certain implicit formulae involving these polynomials are derived by constructing linear differential operators. The corresponding results for the polynomials related to the 3VLHP are obtained as special cases. Further, implicit formula involving the 3-variable Hermite–Laguerre polynomials (3VHLP) is obtained by using the operational methods.