Differential transformation is a method that applies the Taylor expansion of the function. For arbitrary function we can Taylor expand it

where $F(n)$ is the coefficient of Taylor expansion. By using the definition, we can explore the properties of this transformation from function $f(x)$ to $F(n)$.

For more references about this topic, read

1. Munganga, J. M. W., Mwambakana, J. N., Maritz, R., Batubenge, T. a., & Moremedi, G. M. (2014). Introduction of the differential transform method to solve differential equations at undergraduate level. International Journal of Mathematical Education in Science and Technology, 45(5), 781–794. http://doi.org/10.1080/0020739X.2013.877609
2. Arikoglu, A., & Ozkol, I. (2008). Solutions of integral and integro-differential equation systems by using differential transform method. Computers and Mathematics with Applications, 56(9), 2411–2417. http://doi.org/10.1016/j.camwa.2008.05.017