I am currently being introduced to difference equations by the appendix in my book and I need help clarifying one bit. In particular, it is mentioned to me that for a given difference equation the solutions are given as infinite vectors. My interpretation is that since every number can be re written in polar form, each solution will have a different argument making it an infinite vector. Furthermore, comparing difference equations to differential equations, I assume there is a constant that needs to be found such that to get an answer from which infinitely many more can be "generated" by having different arguments. Here is the extract from the book 
Any help would be greatly appreciated!
As @Michael mentioned, there are infinitely many solutions, each for each initial condition, and for each of these solutions there are series associated with it (a0,...,ad−1). It follows that since addition is associative, the vectors as basis, and the space has closure then the set of vectors of the series of the solutions form a vector space.
Again many thanks to @Micheal