I'm having a hard time finding "modern" mathematics textbooks on discrete calculus. Reference [1] has some good material but its audience are mostly computer scientists.
[1] Knuth, D. E., Graham, R. L., Patashnik, O. (1994). Concrete Mathematics: A Foundation for Computer Science. Germany: Addison-Wesley.
Here is a pdf https://www.dropbox.com/s/k4hdqqixming8ls/EDC.pdf. It is called Elementary Discrete Calculus.