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Lie algebra representations and 2-index 4-variable 1-parameter Hermite polynomials


Abstract


This paper is an attempt to stress the usefulness of multi-variable special functions by expressing them in terms of the corresponding Lie algebra or Lie group. The problem of framing the 2-index 4-variable 1-parameter Hermite polynomials (2I4V1PHP) into the context of the irreducible representations  \uparrow _{\omega,\mu} of \mathcal{G}(0,1) and  \uparrow^{'} _{\omega,\mu} of \mathcal{K}_{5} is considered. Certain relations involving 2I4V1PHP H_{m,n}(x,y,z,u;\rho) are obtained using the approach adopted by Miller. Certain examples involving other forms of Hermite polynomials are derived as special cases. Further, some properties of the 2I4V1PHP H_{m,n}(x,y,z,u;\rho) are obtained by using a quadratic combination of four operators defined on a Lie algebra of endomorphisms of a vector space.

DOI Code: 10.1285/i15900932v39n1p65

Keywords: 2-index 4-variable 1-parameter Hermite polynomials; Lie group; Lie algebra; representation theory; implicit formulae

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