In this thesis, we investigate two separated topics, one is the Whittaker-Hill equation, the other one is difference Nevanlinna theory about vanishing and infinite periods.

For the Whittaker-Hill equation, we study the general solution of Whittaker-Hill equation by applying the confluent hypergeometric functions as basis in Chapter 2. According to certain initial conditions and boundary conditions, we obtain four special solutions which are either even or odd and can either be periodic or be semi-periodic. In particular, we extend Ince's work in 1923 ([35]). Moreover, we show that the solutions of Whittaker-Hill equation with finite exponent of convergence of zero-sequence are precisely the terminating solutions. We also study the instability intervals of the Whittaker-Hill op...[ Read more ]

In this thesis, we investigate two separated topics, one is the Whittaker-Hill equation, the other one is difference Nevanlinna theory about vanishing and infinite periods.

For the Whittaker-Hill equation, we study the general solution of Whittaker-Hill equation by applying the confluent hypergeometric functions as basis in Chapter 2. According to certain initial conditions and boundary conditions, we obtain four special solutions which are either even or odd and can either be periodic or be semi-periodic. In particular, we extend Ince's work in 1923 ([35]). Moreover, we show that the solutions of Whittaker-Hill equation with finite exponent of convergence of zero-sequence are precisely the terminating solutions. We also study the instability intervals of the Whittaker-Hill operator in Chapter 3. Our results generalize the results of P. Djakov, B. Mityagin ([18] and [19]).

By extending the idea of a difference operator with a fixed step to a varying-steps difference operator, we have established a difference Nevanlinna theory for meromorphic functions with the steps tending to zero (vanishing period) and a difference Nevanlinna theory for finite order meromorphic functions with the steps tending to infinity (infinite period) in Chapter 4. We can reestablish the classical little Picard theorem by the vanishing period theory, but we require additional finite order growth restriction for meromorphic functions from the infinite period theory. Then we show some applications of our theories to exhibit connections between discrete equations and their continuous analogues.