the main objective of this thesis is to study some new families of complex biorthogonal polynomials which arise in a model of non-commutative quantum mechanics (NCQM) in two dimensions. The NCQM model in which we are interested is a system with an extended Landau Hamiltonian, such as the one which arises when an electron is placed in a constant magnetic field. The new polynomials are deformed versions of the well-known complex Hermite polynomials, generated by the non-commutative raising and lowering operators. We work out in detail the construction of the polynomials and compute some of their useful properties, e.g., the associated generating functions and three-term recurrence relations. It is shown that relative to a non- commutative scaling factor, these polynomials form a class of biorthogonal complex polynomials and in a certain well-defined limit, they reduce to the standard complex Hermite polynomials.
The second objective of this thesis is to study the group theoretical properties of certain bilinear combinations of the non-commutative ladder operators. This is done following the well-known manner in which the angular momentum generators of standard quantum mechanics can be obtained from bilinear combinations of the raising and lowering operators of the harmonic oscillator. We study the Lie group structures that are generated by these operators, which again depend on a parameter characterizing the non-commutativity and which, in an appropriate limit, reduce to the standard quantum mechanical operators.