Painleve Equations and Orthogonal Polynomials

In this thesis we classify all of the special function solutions to Painleve equations and all their associated equations produced using their Hamiltonian structures. We then use these special solutions to highlight the connection between the Painleve equations and the coefficients of some three-term recurrence relations for some specific orthogonal polynomials. The key idea of this newly developed method is the recognition of certain orthogonal polynomial moments as a particular special function. This means we can compare the matrix of moments with the Wronskian solutions, which the Painleve equations are famous for. Once this connection is found we can simply read o the all important recurrence coefficients in a closed form. In certain cases, we can even improve upon this as some of the weights allow a simplification of the recurrence coefficients to polynomials and with it, the new sequences orthogonal polynomials are simplified too.