Applications of Hermite-Pade approximants to water waves and the harmonic oscillator on a lattice

Three types of Hermite-Pade approximants are considered, known respectively as quadratic, integral and differential Pade approximants. The singularity structure of each type of approximant is described. It is more complicated than that of the standard Pade approximant, and this property may often be used to estimate the types of singularity of a function from its power series expansion, as well as evaluate it on its branch cuts. Two applications in different fields are described which illustrate these properties of the above types of Hermite-Pade approximants. The first concerns the characteristic values of Mathieu's equation which are related to the energy eigenvalues of the harmonic oscillator on a lattice. The second concerns the investigation of the singularity structure and values of various physical quantities associated with periodic and solitary water waves.