A theoretical program for constructing Lax pairs for q-Painlevé and difference Painlevé equations from first principles using semi-classical orthogonal polynomial systems on
non-uniform, quadratic lattices is established.
This theory starts with a natural deformation to any of the systems in the Askey table of hypergeometric orthogonal polynomials and it is conjectured that the
dynamical systems it constructs includes all the difference and q-difference Painlevé type equations of the Sakai scheme.

An infinite log-gas formalism is applied to the computation of conditioned gap probabilities that there are n eigenvalues in the gap of size |t|
at the hard and soft edges of random matrix β-ensembles,
which are conjectured to be correct up to and including terms O(log|t|) as t → ∞.

The generating function for the distribution of the number of positive eigenvalues of a Gaussian unitary ensemble is a τ-function of the fourth Painlevé equation, and from this we deduce an exact hypergeometric function evaluation for the variance.

The diagonal spin-spin correlations of the square lattice Ising model, originally expressed as Toeplitz determinants, are given by two distinct Fredholm determinants
- one with an integral operator having an Appell function kernel and another with a summation operator having a Gauss hypergeometric function kernel.
Either determinant allows for a Neumann expansion possessing a natural λ-parameter generalisation and we prove that both expansions are in fact equal,
implying a continuous and a discrete representation of the form factors.

Identification the q-difference equation satisfied by a deformed version of the little q-Jacobi polynomials as a gauge transformation of a special case of the associated linear problem for q-P_{VI}.

A system of bi-orthogonal polynomials and their associated functions corresponding to a regular semi-classical weight on the unit circle constitute a class of general classical solutions to the Garnier systems, and such systems can also be characterised by recurrence relations which are the multi-variable analogues of the discrete fifth Painlevé equation.

Christoffel-Geronimus-Uvarov transformations of a general system of bi-orthogonal polynomials and associated functions on the unit circle are derived.
In the specialisation of the weight function to the regular semi-classical case and if the zeros and poles of the Christoffel-Geronimus-Uvarov factors coincide with the
singularities of the weight then these are the Schlesinger transformations of an isomonodromic system.

Demonstration that the correlation between a spin at the origin and one on the next-to-diagonal in the square lattice Ising model is governed by a component of the isomonodromic system associated with the 1980 solution of Jimbo and Miwa.

The distribution function for the first eigenvalue spacing in the Laguerre unitary ensemble of finite rank random matrices is found in terms of a Painlevé V system,
and the solution of its associated linear isomonodromic system. In the scaling to the hard edge regime an analogous situation occurs where a certain Painlevé III'
system and its associated linear isomonodromic system characterise the scaled distribution.

The spectral average which generalises the local spacing distribution of the eigenvalues of random N× N hermitian matrices in the bulk of
their spectrum as N→∞ is a τ-function of the fifth Painlevé system, τ(s), which has generic parameters and is transcendental
but is characterised by particular boundary conditions about the singular point s=0, which we determine here.
When the average reduces to the local spacing distribution we find that τ-function is of the separatrix, or partially truncated type.

Solutions to the q-Painlevé equation of type A_{6}^{(1)} (a q-difference analogue of the Painlevé II equation)
are constructed from a determinantal formula with basic hypergeometric function elements.
The continuous limit of this q-Painlevé equation to the Painlevé II equation and the limit of its hypergeometric solutions to the Airy functions are demonstrated.

New results are given for the boundary conditions of a generating function for the distribution of eigenvalues of the generalised Dyson circular unitary ensemble in the
neighbourhood of the fixed singularities at t=0,1,∞. The structure of these expansions is related to Jimbo's 1982 general expansions for the τ-function
of the sixth Painlevé equation in the neighbourhood of its fixed singularities, and this theory is itself put in its context of its linear isomonodromy problem.

The boundary conditions for the solution of the σ-form differential equation for a Painlevé V transcendent, which characterises the generating function for the
probability that an interval (0,s) at the hard edge contains k eigenvalues in the Laguerre unitary ensemble are specified for the full parameter space.
An analogous study is carried out for the the hard edge scaling limit of the random matrix average, which we have previously evaluated in terms of a Painlevé III' transcendent in σ-form.
An application of the latter result is given to the rapid evaluation of a Hankel determinant appearing in a recent work of Conrey, Rubinstein and Snaith relating to the derivative of the Riemann zeta function.

Raising and lowering operators for orthogonal polynomials on the unit circle and second order differential and q-difference equations for these polynomials are derived.

We develop a formalism for calculating arbitrary expectation values for any extensive lattice Hamiltonian system using an analytic Lanczos expansion,
and a recently proved exact theorem for ground-state energies.
The ground-state energy, staggered magnetization, and excited state gap of the two-dimensional anisotropic antiferromagnetic Heisenberg model are then calculated using
this expansion for a range of anisotropy parameters, and compared to other moment-based techniques, such as the t expansion, spin-wave theory and series-expansion methods.

Use of the exact bounds to the rate of convergence of the eigenvalue estimates in the symmetric Lanczos process due to Saad 1980,
and an application to the one-dimensional anti-ferromagnetic Heisenberg model.

The exact solution to the problem of reflection and diffraction of atomic de Broglie waves by a travelling evanescent wave is found
- the wavefunctions, the tunnelling losses and the non-adiabatic losses are given exactly in terms of hyper-Bessel functions, and are valid for all detuning and Rabi frequencies,
thus generalizing previous approximate methods. Furthermore, we give the limiting cases of the amplitudes in the uniform semiclassical limit,
which is valid in all regions including near the classical turning points, and in the large and weak coupling cases.
We find our uniform semiclassical limit to be closer to the exact result over the full range of parameter values than the previously reported calculations.