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In this paper, a result for double index sequences is established by using a quadratic combination of four operators defined on a Lie algebra of endomorphisms of a vector space. This result is used to give new proofs of some properties of the Bessel and Tricomi functions of two indices. Further, this result is extended for multi-index sequences. [ABSTRACT FROM AUTHOR]

The representations of the Lie algebras generate in a natural way all known classical special polynomials. This allows one to greatly simplify the theory of orthogonal polynomials by expressing them in terms of the corresponding Lie algebra or Lie group. In this article, the problem of framing the 1-parameter 2D-Hermite polynomials (1P2DHP)(which are 2D orthogonal polynomials) into the context of the irreducible representationsandof the four-dimensional Lie algebrais considered. This approach stress the mathematical relevance of 2D-orthogonal polynomials and Lie algebras. Certain relations involving the 1P2DHPare obtained using the approach adopted by Miller. The linear differential operators serve as useful tools towards obtaining these relations. The analysis has been carried out by generalizing the formalism relevant to 1P2DHP. Certain examples involving 2D-Hermite polynomialsand Laguerre polynomialsare obtained as special cases. [ABSTRACT FROM AUTHOR]