Difference Nevanlinna theories with vanishing and infinite periods

By extending the idea of a difference operator with a fixed step to varying-steps difference operators, 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 this paper. We can recover the classical little Picard theorem from the vanishing period theory, but we require additional finite order growth restriction for meromorphic functions from the infinite period theory. Then we give some applications of our theories to exhibit connections between discrete equations and and their continuous analogues.